ab-angle->ABCF B

Percentage Accurate: 53.7% → 67.3%

Time: 7.5s

Alternatives: 23

Speedup: 5.5×

• 32.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	return ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	return ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_0)) * Math.cos(t_0);
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle / 180.0)
	return ((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin(t_0)) * math.cos(t_0)

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	return Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	tmp = ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]]

Initial Program: 53.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	return ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	return ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_0)) * Math.cos(t_0);
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle / 180.0)
	return ((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin(t_0)) * math.cos(t_0)

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	return Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	tmp = ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 67.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \left(angle \cdot 0.005555555555555556\right) \cdot \pi\\ \mathbf{if}\;a\_m \leq 2.85 \cdot 10^{+150}:\\ \;\;\;\;\left(\left(b + a\_m\right) \cdot \left(\left(b - a\_m\right) \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right) \cdot \cos t\_0\\ \mathbf{elif}\;a\_m \leq 2.75 \cdot 10^{+259}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\left(b - a\_m\right) \cdot \left(\left(a\_m + b\right) \cdot angle\right)\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a\_m \cdot a\_m\right) \cdot \sin t\_0\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (* (* angle 0.005555555555555556) PI)))
   (if (<= a_m 2.85e+150)
     (*
      (*
       (+ b a_m)
       (* (- b a_m) (* 2.0 (sin (* 0.005555555555555556 (* PI angle))))))
      (cos t_0))
     (if (<= a_m 2.75e+259)
       (* 0.011111111111111112 (* (* (- b a_m) (* (+ a_m b) angle)) PI))
       (*
        (* (* a_m a_m) (sin t_0))
        (* (cos (* (* angle PI) -0.005555555555555556)) -2.0))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = (angle * 0.005555555555555556) * ((double) M_PI);
	double tmp;
	if (a_m <= 2.85e+150) {
		tmp = ((b + a_m) * ((b - a_m) * (2.0 * sin((0.005555555555555556 * (((double) M_PI) * angle)))))) * cos(t_0);
	} else if (a_m <= 2.75e+259) {
		tmp = 0.011111111111111112 * (((b - a_m) * ((a_m + b) * angle)) * ((double) M_PI));
	} else {
		tmp = ((a_m * a_m) * sin(t_0)) * (cos(((angle * ((double) M_PI)) * -0.005555555555555556)) * -2.0);
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double t_0 = (angle * 0.005555555555555556) * Math.PI;
	double tmp;
	if (a_m <= 2.85e+150) {
		tmp = ((b + a_m) * ((b - a_m) * (2.0 * Math.sin((0.005555555555555556 * (Math.PI * angle)))))) * Math.cos(t_0);
	} else if (a_m <= 2.75e+259) {
		tmp = 0.011111111111111112 * (((b - a_m) * ((a_m + b) * angle)) * Math.PI);
	} else {
		tmp = ((a_m * a_m) * Math.sin(t_0)) * (Math.cos(((angle * Math.PI) * -0.005555555555555556)) * -2.0);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	t_0 = (angle * 0.005555555555555556) * math.pi
	tmp = 0
	if a_m <= 2.85e+150:
		tmp = ((b + a_m) * ((b - a_m) * (2.0 * math.sin((0.005555555555555556 * (math.pi * angle)))))) * math.cos(t_0)
	elif a_m <= 2.75e+259:
		tmp = 0.011111111111111112 * (((b - a_m) * ((a_m + b) * angle)) * math.pi)
	else:
		tmp = ((a_m * a_m) * math.sin(t_0)) * (math.cos(((angle * math.pi) * -0.005555555555555556)) * -2.0)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = Float64(Float64(angle * 0.005555555555555556) * pi)
	tmp = 0.0
	if (a_m <= 2.85e+150)
		tmp = Float64(Float64(Float64(b + a_m) * Float64(Float64(b - a_m) * Float64(2.0 * sin(Float64(0.005555555555555556 * Float64(pi * angle)))))) * cos(t_0));
	elseif (a_m <= 2.75e+259)
		tmp = Float64(0.011111111111111112 * Float64(Float64(Float64(b - a_m) * Float64(Float64(a_m + b) * angle)) * pi));
	else
		tmp = Float64(Float64(Float64(a_m * a_m) * sin(t_0)) * Float64(cos(Float64(Float64(angle * pi) * -0.005555555555555556)) * -2.0));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	t_0 = (angle * 0.005555555555555556) * pi;
	tmp = 0.0;
	if (a_m <= 2.85e+150)
		tmp = ((b + a_m) * ((b - a_m) * (2.0 * sin((0.005555555555555556 * (pi * angle)))))) * cos(t_0);
	elseif (a_m <= 2.75e+259)
		tmp = 0.011111111111111112 * (((b - a_m) * ((a_m + b) * angle)) * pi);
	else
		tmp = ((a_m * a_m) * sin(t_0)) * (cos(((angle * pi) * -0.005555555555555556)) * -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[(N[(angle * 0.005555555555555556), $MachinePrecision] * Pi), $MachinePrecision]}, If[LessEqual[a$95$m, 2.85e+150], N[(N[(N[(b + a$95$m), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(2.0 * N[Sin[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 2.75e+259], N[(0.011111111111111112 * N[(N[(N[(b - a$95$m), $MachinePrecision] * N[(N[(a$95$m + b), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a$95$m * a$95$m), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[N[(N[(angle * Pi), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < 2.8500000000000001e150

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\left(\left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(\left(b \cdot b - \color{blue}{{a}^{2}}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\left(\left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\left(a + b\right)} \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\left(a + b\right)} \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(a + b\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot 2\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      14. lower--.f64 57.6

          \[\leadsto \left(\left(\left(a + b\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Applied rewrites 57.6%

      \[\leadsto \left(\color{blue}{\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\left(a + b\right) \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(a + b\right) \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(a + b\right)} \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(b + a\right)} \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(b + a\right)} \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\color{blue}{\left(\left(b - a\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. associate-*l* N/A

         \[\leadsto \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lower-*.f64 67.3

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\pi \cdot \frac{angle}{180}\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      14. lift-PI.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      16. lift-PI.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\color{blue}{\pi} \cdot \frac{angle}{180}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      17. mult-flip N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      19. *-commutative N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      20. associate-*l* N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\left(\pi \cdot \frac{1}{180}\right) \cdot angle\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\color{blue}{\left(\frac{1}{180} \cdot \pi\right)} \cdot angle\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right) \cdot \cos \color{blue}{\left(\pi \cdot \frac{angle}{180}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right) \cdot \cos \color{blue}{\left(\frac{angle}{180} \cdot \pi\right)} \]

      3. lower-*.f64 67.3

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right) \cdot \cos \color{blue}{\left(\frac{angle}{180} \cdot \pi\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right) \cdot \cos \left(\color{blue}{\frac{angle}{180}} \cdot \pi\right) \]

      5. mult-flip N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right) \cdot \cos \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \pi\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right) \cdot \cos \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \pi\right) \]

      7. lower-*.f64 67.4

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right) \cdot \cos \left(\color{blue}{\left(angle \cdot 0.005555555555555556\right)} \cdot \pi\right) \]

   7. Applied rewrites 67.4%

      \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right) \cdot \color{blue}{\cos \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)} \]

   if 2.8500000000000001e150 < a < 2.75000000000000015e259

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      17. lower-*.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      20. lower-+.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot angle\right) \cdot \pi\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot angle\right) \cdot \pi\right) \]

      4. associate-*l* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \pi\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \pi\right) \]

      6. lower-*.f64 62.2

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \pi\right) \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \pi\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot angle\right)\right) \cdot \pi\right) \]

      9. lower-+.f64 62.2

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot angle\right)\right) \cdot \pi\right) \]

   8. Applied rewrites 62.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot angle\right)\right) \cdot \pi\right) \]

   if 2.75000000000000015e259 < a

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. cos-fabs-rev N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\left|\pi \cdot \frac{angle}{180}\right|\right)} \]

      3. cos-neg-rev N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left|\pi \cdot \frac{angle}{180}\right|\right)\right)} \]

      4. sin-+PI/2-rev N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left|\pi \cdot \frac{angle}{180}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left|\pi \cdot \frac{angle}{180}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\left|\pi \cdot \frac{angle}{180}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\left|\color{blue}{\pi \cdot \frac{angle}{180}}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\left|\pi \cdot \color{blue}{\frac{angle}{180}}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\left|\color{blue}{\frac{\pi \cdot angle}{180}}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      10. fabs-div N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\frac{\left|\pi \cdot angle\right|}{\left|180\right|}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\frac{\left|\pi \cdot angle\right|}{\color{blue}{180}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      12. distribute-neg-frac2 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left|\pi \cdot angle\right|}{\mathsf{neg}\left(180\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left|\pi \cdot angle\right|}{\mathsf{neg}\left(180\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      14. lower-fabs.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left|\pi \cdot angle\right|}}{\mathsf{neg}\left(180\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|\color{blue}{angle \cdot \pi}\right|}{\mathsf{neg}\left(180\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|\color{blue}{angle \cdot \pi}\right|}{\mathsf{neg}\left(180\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      17. metadata-eval N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{\color{blue}{-180}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      18. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{\color{blue}{\pi}}{2}\right) \]

      19. mult-flip N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \color{blue}{\pi \cdot \frac{1}{2}}\right) \]

      20. *-commutative N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \color{blue}{\frac{1}{2} \cdot \pi}\right) \]

      21. lower-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \color{blue}{\frac{1}{2} \cdot \pi}\right) \]

      22. metadata-eval 53.0

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \color{blue}{0.5} \cdot \pi\right) \]

   3. Applied rewrites 53.0%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + 0.5 \cdot \pi\right)} \]

   4. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{-1}{180} \cdot \left|angle \cdot \mathsf{PI}\left(\right)\right| + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left({a}^{2} \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{-1}{180} \cdot \left|angle \cdot \mathsf{PI}\left(\right)\right| + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left({a}^{2} \cdot \color{blue}{\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{-1}{180} \cdot \left|angle \cdot \mathsf{PI}\left(\right)\right| + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto -2 \cdot \left({a}^{2} \cdot \left(\color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sin \left(\frac{-1}{180} \cdot \left|angle \cdot \mathsf{PI}\left(\right)\right| + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left({a}^{2} \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{-1}{180} \cdot \left|angle \cdot \mathsf{PI}\left(\right)\right| + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      5. lower-sin.f64 N/A

         \[\leadsto -2 \cdot \left({a}^{2} \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{-1}{180} \cdot \left|angle \cdot \mathsf{PI}\left(\right)\right| + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left({a}^{2} \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\color{blue}{\frac{-1}{180} \cdot \left|angle \cdot \mathsf{PI}\left(\right)\right|} + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left({a}^{2} \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{-1}{180} \cdot \color{blue}{\left|angle \cdot \mathsf{PI}\left(\right)\right|} + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. lower-PI.f64 N/A

         \[\leadsto -2 \cdot \left({a}^{2} \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{-1}{180} \cdot \left|angle \cdot \mathsf{PI}\left(\right)\right| + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   6. Applied rewrites 34.4%

      \[\leadsto \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(-0.005555555555555556, \left|angle \cdot \pi\right|, 0.5 \cdot \pi\right)\right)\right)\right)} \]

   8. Applied rewrites 34.9%

      \[\leadsto \left(\left(a \cdot a\right) \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right) \cdot \color{blue}{\left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot -2\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 67.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \mathbf{if}\;angle \leq 4.2 \cdot 10^{+125}:\\ \;\;\;\;\left(\left(b + a\_m\right) \cdot \left(\left(b - a\_m\right) \cdot \left(\sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right) - \sin \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \cos t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b - a\_m\right) \cdot \left(b + a\_m\right)\right)\right) \cdot \sin t\_0\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + 0.5 \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (if (<= angle 4.2e+125)
     (*
      (*
       (+ b a_m)
       (*
        (- b a_m)
        (-
         (sin (* (* angle 0.005555555555555556) PI))
         (sin (* (* angle PI) -0.005555555555555556)))))
      (cos t_0))
     (*
      (* (* 2.0 (* (- b a_m) (+ b a_m))) (sin t_0))
      (sin (+ (/ (fabs (* angle PI)) -180.0) (* 0.5 PI)))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	double tmp;
	if (angle <= 4.2e+125) {
		tmp = ((b + a_m) * ((b - a_m) * (sin(((angle * 0.005555555555555556) * ((double) M_PI))) - sin(((angle * ((double) M_PI)) * -0.005555555555555556))))) * cos(t_0);
	} else {
		tmp = ((2.0 * ((b - a_m) * (b + a_m))) * sin(t_0)) * sin(((fabs((angle * ((double) M_PI))) / -180.0) + (0.5 * ((double) M_PI))));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	double tmp;
	if (angle <= 4.2e+125) {
		tmp = ((b + a_m) * ((b - a_m) * (Math.sin(((angle * 0.005555555555555556) * Math.PI)) - Math.sin(((angle * Math.PI) * -0.005555555555555556))))) * Math.cos(t_0);
	} else {
		tmp = ((2.0 * ((b - a_m) * (b + a_m))) * Math.sin(t_0)) * Math.sin(((Math.abs((angle * Math.PI)) / -180.0) + (0.5 * Math.PI)));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	t_0 = math.pi * (angle / 180.0)
	tmp = 0
	if angle <= 4.2e+125:
		tmp = ((b + a_m) * ((b - a_m) * (math.sin(((angle * 0.005555555555555556) * math.pi)) - math.sin(((angle * math.pi) * -0.005555555555555556))))) * math.cos(t_0)
	else:
		tmp = ((2.0 * ((b - a_m) * (b + a_m))) * math.sin(t_0)) * math.sin(((math.fabs((angle * math.pi)) / -180.0) + (0.5 * math.pi)))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	tmp = 0.0
	if (angle <= 4.2e+125)
		tmp = Float64(Float64(Float64(b + a_m) * Float64(Float64(b - a_m) * Float64(sin(Float64(Float64(angle * 0.005555555555555556) * pi)) - sin(Float64(Float64(angle * pi) * -0.005555555555555556))))) * cos(t_0));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(b - a_m) * Float64(b + a_m))) * sin(t_0)) * sin(Float64(Float64(abs(Float64(angle * pi)) / -180.0) + Float64(0.5 * pi))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	t_0 = pi * (angle / 180.0);
	tmp = 0.0;
	if (angle <= 4.2e+125)
		tmp = ((b + a_m) * ((b - a_m) * (sin(((angle * 0.005555555555555556) * pi)) - sin(((angle * pi) * -0.005555555555555556))))) * cos(t_0);
	else
		tmp = ((2.0 * ((b - a_m) * (b + a_m))) * sin(t_0)) * sin(((abs((angle * pi)) / -180.0) + (0.5 * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[angle, 4.2e+125], N[(N[(N[(b + a$95$m), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(N[Sin[N[(N[(angle * 0.005555555555555556), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] - N[Sin[N[(N[(angle * Pi), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(N[(b - a$95$m), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(N[Abs[N[(angle * Pi), $MachinePrecision]], $MachinePrecision] / -180.0), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < 4.2000000000000001e125

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\left(\left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(\left(b \cdot b - \color{blue}{{a}^{2}}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\left(\left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\left(a + b\right)} \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\left(a + b\right)} \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(a + b\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot 2\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      14. lower--.f64 57.6

          \[\leadsto \left(\left(\left(a + b\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Applied rewrites 57.6%

      \[\leadsto \left(\color{blue}{\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\left(a + b\right) \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(a + b\right) \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(a + b\right)} \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(b + a\right)} \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(b + a\right)} \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\color{blue}{\left(\left(b - a\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. associate-*l* N/A

         \[\leadsto \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lower-*.f64 67.3

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\pi \cdot \frac{angle}{180}\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      14. lift-PI.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      16. lift-PI.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\color{blue}{\pi} \cdot \frac{angle}{180}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      17. mult-flip N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      19. *-commutative N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      20. associate-*l* N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\left(\pi \cdot \frac{1}{180}\right) \cdot angle\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\color{blue}{\left(\frac{1}{180} \cdot \pi\right)} \cdot angle\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(2 \cdot \sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. count-2-rev N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) + \sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. add-flip N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) - \left(\mathsf{neg}\left(\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift-sin.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) - \left(\mathsf{neg}\left(\color{blue}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) - \left(\mathsf{neg}\left(\sin \color{blue}{\left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) - \left(\mathsf{neg}\left(\sin \left(\frac{1}{180} \cdot \color{blue}{\left(\pi \cdot angle\right)}\right)\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) - \left(\mathsf{neg}\left(\sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \pi\right)}\right)\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) - \left(\mathsf{neg}\left(\sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \pi\right)}\right)\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) - \left(\mathsf{neg}\left(\sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)}\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) - \left(\mathsf{neg}\left(\sin \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. mult-flip-rev N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) - \left(\mathsf{neg}\left(\sin \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)}\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) - \left(\mathsf{neg}\left(\sin \left(\frac{\color{blue}{angle \cdot \pi}}{180}\right)\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) - \left(\mathsf{neg}\left(\sin \left(\frac{\color{blue}{\pi \cdot angle}}{180}\right)\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      14. associate-*r/ N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) - \left(\mathsf{neg}\left(\sin \color{blue}{\left(\pi \cdot \frac{angle}{180}\right)}\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      15. lift-PI.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) - \left(\mathsf{neg}\left(\sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      16. lift-PI.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) - \left(\mathsf{neg}\left(\sin \left(\color{blue}{\pi} \cdot \frac{angle}{180}\right)\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) - \left(\mathsf{neg}\left(\sin \left(\pi \cdot \color{blue}{\frac{angle}{180}}\right)\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) - \left(\mathsf{neg}\left(\sin \color{blue}{\left(\pi \cdot \frac{angle}{180}\right)}\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      19. cos-+PI/2-rev N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) - \color{blue}{\cos \left(\pi \cdot \frac{angle}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. Applied rewrites 67.3%

      \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right) - \sin \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   if 4.2000000000000001e125 < angle

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. cos-fabs-rev N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\left|\pi \cdot \frac{angle}{180}\right|\right)} \]

      3. cos-neg-rev N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left|\pi \cdot \frac{angle}{180}\right|\right)\right)} \]

      4. sin-+PI/2-rev N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left|\pi \cdot \frac{angle}{180}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left|\pi \cdot \frac{angle}{180}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\left|\pi \cdot \frac{angle}{180}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\left|\color{blue}{\pi \cdot \frac{angle}{180}}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\left|\pi \cdot \color{blue}{\frac{angle}{180}}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\left|\color{blue}{\frac{\pi \cdot angle}{180}}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      10. fabs-div N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\frac{\left|\pi \cdot angle\right|}{\left|180\right|}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\frac{\left|\pi \cdot angle\right|}{\color{blue}{180}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      12. distribute-neg-frac2 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left|\pi \cdot angle\right|}{\mathsf{neg}\left(180\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left|\pi \cdot angle\right|}{\mathsf{neg}\left(180\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      14. lower-fabs.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left|\pi \cdot angle\right|}}{\mathsf{neg}\left(180\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|\color{blue}{angle \cdot \pi}\right|}{\mathsf{neg}\left(180\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|\color{blue}{angle \cdot \pi}\right|}{\mathsf{neg}\left(180\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      17. metadata-eval N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{\color{blue}{-180}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      18. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{\color{blue}{\pi}}{2}\right) \]

      19. mult-flip N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \color{blue}{\pi \cdot \frac{1}{2}}\right) \]

      20. *-commutative N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \color{blue}{\frac{1}{2} \cdot \pi}\right) \]

      21. lower-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \color{blue}{\frac{1}{2} \cdot \pi}\right) \]

      22. metadata-eval 53.0

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \color{blue}{0.5} \cdot \pi\right) \]

   3. Applied rewrites 53.0%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + 0.5 \cdot \pi\right)} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{1}{2} \cdot \pi\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{1}{2} \cdot \pi\right) \]

      3. unpow2 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{1}{2} \cdot \pi\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{1}{2} \cdot \pi\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{1}{2} \cdot \pi\right) \]

      6. difference-of-squares-rev N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{1}{2} \cdot \pi\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{1}{2} \cdot \pi\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{1}{2} \cdot \pi\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{1}{2} \cdot \pi\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(a + b\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{1}{2} \cdot \pi\right) \]

      11. lower-*.f64 56.8

          \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(a + b\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + 0.5 \cdot \pi\right) \]

      12. lift-+.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{1}{2} \cdot \pi\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(2 \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(b + a\right)}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{1}{2} \cdot \pi\right) \]

      14. lower-+.f64 56.8

          \[\leadsto \left(\left(2 \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(b + a\right)}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + 0.5 \cdot \pi\right) \]

   5. Applied rewrites 56.8%

      \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + 0.5 \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 67.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \mathbf{if}\;angle \leq 4.2 \cdot 10^{+125}:\\ \;\;\;\;\left(\left(b + a\_m\right) \cdot \left(\left(b - a\_m\right) \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right) \cdot \cos t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b - a\_m\right) \cdot \left(b + a\_m\right)\right)\right) \cdot \sin t\_0\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + 0.5 \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (if (<= angle 4.2e+125)
     (*
      (*
       (+ b a_m)
       (* (- b a_m) (* 2.0 (sin (* 0.005555555555555556 (* PI angle))))))
      (cos t_0))
     (*
      (* (* 2.0 (* (- b a_m) (+ b a_m))) (sin t_0))
      (sin (+ (/ (fabs (* angle PI)) -180.0) (* 0.5 PI)))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	double tmp;
	if (angle <= 4.2e+125) {
		tmp = ((b + a_m) * ((b - a_m) * (2.0 * sin((0.005555555555555556 * (((double) M_PI) * angle)))))) * cos(t_0);
	} else {
		tmp = ((2.0 * ((b - a_m) * (b + a_m))) * sin(t_0)) * sin(((fabs((angle * ((double) M_PI))) / -180.0) + (0.5 * ((double) M_PI))));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	double tmp;
	if (angle <= 4.2e+125) {
		tmp = ((b + a_m) * ((b - a_m) * (2.0 * Math.sin((0.005555555555555556 * (Math.PI * angle)))))) * Math.cos(t_0);
	} else {
		tmp = ((2.0 * ((b - a_m) * (b + a_m))) * Math.sin(t_0)) * Math.sin(((Math.abs((angle * Math.PI)) / -180.0) + (0.5 * Math.PI)));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	t_0 = math.pi * (angle / 180.0)
	tmp = 0
	if angle <= 4.2e+125:
		tmp = ((b + a_m) * ((b - a_m) * (2.0 * math.sin((0.005555555555555556 * (math.pi * angle)))))) * math.cos(t_0)
	else:
		tmp = ((2.0 * ((b - a_m) * (b + a_m))) * math.sin(t_0)) * math.sin(((math.fabs((angle * math.pi)) / -180.0) + (0.5 * math.pi)))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	tmp = 0.0
	if (angle <= 4.2e+125)
		tmp = Float64(Float64(Float64(b + a_m) * Float64(Float64(b - a_m) * Float64(2.0 * sin(Float64(0.005555555555555556 * Float64(pi * angle)))))) * cos(t_0));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(b - a_m) * Float64(b + a_m))) * sin(t_0)) * sin(Float64(Float64(abs(Float64(angle * pi)) / -180.0) + Float64(0.5 * pi))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	t_0 = pi * (angle / 180.0);
	tmp = 0.0;
	if (angle <= 4.2e+125)
		tmp = ((b + a_m) * ((b - a_m) * (2.0 * sin((0.005555555555555556 * (pi * angle)))))) * cos(t_0);
	else
		tmp = ((2.0 * ((b - a_m) * (b + a_m))) * sin(t_0)) * sin(((abs((angle * pi)) / -180.0) + (0.5 * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[angle, 4.2e+125], N[(N[(N[(b + a$95$m), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(2.0 * N[Sin[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(N[(b - a$95$m), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(N[Abs[N[(angle * Pi), $MachinePrecision]], $MachinePrecision] / -180.0), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < 4.2000000000000001e125

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\left(\left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(\left(b \cdot b - \color{blue}{{a}^{2}}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\left(\left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\left(a + b\right)} \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\left(a + b\right)} \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(a + b\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot 2\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      14. lower--.f64 57.6

          \[\leadsto \left(\left(\left(a + b\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Applied rewrites 57.6%

      \[\leadsto \left(\color{blue}{\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\left(a + b\right) \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(a + b\right) \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(a + b\right)} \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(b + a\right)} \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(b + a\right)} \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\color{blue}{\left(\left(b - a\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. associate-*l* N/A

         \[\leadsto \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lower-*.f64 67.3

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\pi \cdot \frac{angle}{180}\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      14. lift-PI.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      16. lift-PI.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\color{blue}{\pi} \cdot \frac{angle}{180}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      17. mult-flip N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      19. *-commutative N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      20. associate-*l* N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\left(\pi \cdot \frac{1}{180}\right) \cdot angle\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\color{blue}{\left(\frac{1}{180} \cdot \pi\right)} \cdot angle\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   if 4.2000000000000001e125 < angle

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. cos-fabs-rev N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\left|\pi \cdot \frac{angle}{180}\right|\right)} \]

      3. cos-neg-rev N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left|\pi \cdot \frac{angle}{180}\right|\right)\right)} \]

      4. sin-+PI/2-rev N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left|\pi \cdot \frac{angle}{180}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left|\pi \cdot \frac{angle}{180}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\left|\pi \cdot \frac{angle}{180}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\left|\color{blue}{\pi \cdot \frac{angle}{180}}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\left|\pi \cdot \color{blue}{\frac{angle}{180}}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\left|\color{blue}{\frac{\pi \cdot angle}{180}}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      10. fabs-div N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\frac{\left|\pi \cdot angle\right|}{\left|180\right|}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\frac{\left|\pi \cdot angle\right|}{\color{blue}{180}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      12. distribute-neg-frac2 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left|\pi \cdot angle\right|}{\mathsf{neg}\left(180\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left|\pi \cdot angle\right|}{\mathsf{neg}\left(180\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      14. lower-fabs.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left|\pi \cdot angle\right|}}{\mathsf{neg}\left(180\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|\color{blue}{angle \cdot \pi}\right|}{\mathsf{neg}\left(180\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|\color{blue}{angle \cdot \pi}\right|}{\mathsf{neg}\left(180\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      17. metadata-eval N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{\color{blue}{-180}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      18. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{\color{blue}{\pi}}{2}\right) \]

      19. mult-flip N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \color{blue}{\pi \cdot \frac{1}{2}}\right) \]

      20. *-commutative N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \color{blue}{\frac{1}{2} \cdot \pi}\right) \]

      21. lower-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \color{blue}{\frac{1}{2} \cdot \pi}\right) \]

      22. metadata-eval 53.0

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \color{blue}{0.5} \cdot \pi\right) \]

   3. Applied rewrites 53.0%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + 0.5 \cdot \pi\right)} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{1}{2} \cdot \pi\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{1}{2} \cdot \pi\right) \]

      3. unpow2 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{1}{2} \cdot \pi\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{1}{2} \cdot \pi\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{1}{2} \cdot \pi\right) \]

      6. difference-of-squares-rev N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{1}{2} \cdot \pi\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{1}{2} \cdot \pi\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{1}{2} \cdot \pi\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{1}{2} \cdot \pi\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(a + b\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{1}{2} \cdot \pi\right) \]

      11. lower-*.f64 56.8

          \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(a + b\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + 0.5 \cdot \pi\right) \]

      12. lift-+.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{1}{2} \cdot \pi\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(2 \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(b + a\right)}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{1}{2} \cdot \pi\right) \]

      14. lower-+.f64 56.8

          \[\leadsto \left(\left(2 \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(b + a\right)}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + 0.5 \cdot \pi\right) \]

   5. Applied rewrites 56.8%

      \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + 0.5 \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 67.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 1.2 \cdot 10^{+229}:\\ \;\;\;\;\left(\left(a\_m + b\right) \cdot \left(\left(b - a\_m\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\_m\right) \cdot \left(\left(b - a\_m\right) \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= a_m 1.2e+229)
   (*
    (*
     (+ a_m b)
     (* (- b a_m) (* (sin (* (* 0.005555555555555556 angle) PI)) 2.0)))
    (cos (* PI (/ angle 180.0))))
   (*
    (*
     (+ b a_m)
     (* (- b a_m) (* 2.0 (sin (* 0.005555555555555556 (* PI angle))))))
    1.0)))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (a_m <= 1.2e+229) {
		tmp = ((a_m + b) * ((b - a_m) * (sin(((0.005555555555555556 * angle) * ((double) M_PI))) * 2.0))) * cos((((double) M_PI) * (angle / 180.0)));
	} else {
		tmp = ((b + a_m) * ((b - a_m) * (2.0 * sin((0.005555555555555556 * (((double) M_PI) * angle)))))) * 1.0;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if (a_m <= 1.2e+229) {
		tmp = ((a_m + b) * ((b - a_m) * (Math.sin(((0.005555555555555556 * angle) * Math.PI)) * 2.0))) * Math.cos((Math.PI * (angle / 180.0)));
	} else {
		tmp = ((b + a_m) * ((b - a_m) * (2.0 * Math.sin((0.005555555555555556 * (Math.PI * angle)))))) * 1.0;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if a_m <= 1.2e+229:
		tmp = ((a_m + b) * ((b - a_m) * (math.sin(((0.005555555555555556 * angle) * math.pi)) * 2.0))) * math.cos((math.pi * (angle / 180.0)))
	else:
		tmp = ((b + a_m) * ((b - a_m) * (2.0 * math.sin((0.005555555555555556 * (math.pi * angle)))))) * 1.0
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (a_m <= 1.2e+229)
		tmp = Float64(Float64(Float64(a_m + b) * Float64(Float64(b - a_m) * Float64(sin(Float64(Float64(0.005555555555555556 * angle) * pi)) * 2.0))) * cos(Float64(pi * Float64(angle / 180.0))));
	else
		tmp = Float64(Float64(Float64(b + a_m) * Float64(Float64(b - a_m) * Float64(2.0 * sin(Float64(0.005555555555555556 * Float64(pi * angle)))))) * 1.0);
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if (a_m <= 1.2e+229)
		tmp = ((a_m + b) * ((b - a_m) * (sin(((0.005555555555555556 * angle) * pi)) * 2.0))) * cos((pi * (angle / 180.0)));
	else
		tmp = ((b + a_m) * ((b - a_m) * (2.0 * sin((0.005555555555555556 * (pi * angle)))))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[a$95$m, 1.2e+229], N[(N[(N[(a$95$m + b), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(b + a$95$m), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(2.0 * N[Sin[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.2e229

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\left(b \cdot b - \color{blue}{{a}^{2}}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. difference-of-squares N/A

          \[\leadsto \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(a + b\right)} \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\color{blue}{\left(a + b\right)} \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot 2\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\left(a + b\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot 2\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      17. lower--.f64 N/A

          \[\leadsto \left(\left(a + b\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot 2\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      18. lower-*.f64 67.3

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot 2\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   if 1.2e229 < a

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\left(\left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(\left(b \cdot b - \color{blue}{{a}^{2}}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\left(\left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\left(a + b\right)} \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\left(a + b\right)} \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(a + b\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot 2\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      14. lower--.f64 57.6

          \[\leadsto \left(\left(\left(a + b\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Applied rewrites 57.6%

      \[\leadsto \left(\color{blue}{\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\left(a + b\right) \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(a + b\right) \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(a + b\right)} \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(b + a\right)} \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(b + a\right)} \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\color{blue}{\left(\left(b - a\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. associate-*l* N/A

         \[\leadsto \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lower-*.f64 67.3

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\pi \cdot \frac{angle}{180}\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      14. lift-PI.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      16. lift-PI.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\color{blue}{\pi} \cdot \frac{angle}{180}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      17. mult-flip N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      19. *-commutative N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      20. associate-*l* N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\left(\pi \cdot \frac{1}{180}\right) \cdot angle\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\color{blue}{\left(\frac{1}{180} \cdot \pi\right)} \cdot angle\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Taylor expanded in angle around 0

      \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 65.1%

      \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 67.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 1.16 \cdot 10^{+229}:\\ \;\;\;\;\left(a\_m + b\right) \cdot \left(\left(b - a\_m\right) \cdot \sin \left(\left(\left(angle \cdot {\pi}^{0.6666666666666666}\right) \cdot \sqrt[3]{\pi}\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\_m\right) \cdot \left(\left(b - a\_m\right) \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= a_m 1.16e+229)
   (*
    (+ a_m b)
    (*
     (- b a_m)
     (sin
      (*
       (* (* angle (pow PI 0.6666666666666666)) (cbrt PI))
       0.011111111111111112))))
   (*
    (*
     (+ b a_m)
     (* (- b a_m) (* 2.0 (sin (* 0.005555555555555556 (* PI angle))))))
    1.0)))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (a_m <= 1.16e+229) {
		tmp = (a_m + b) * ((b - a_m) * sin((((angle * pow(((double) M_PI), 0.6666666666666666)) * cbrt(((double) M_PI))) * 0.011111111111111112)));
	} else {
		tmp = ((b + a_m) * ((b - a_m) * (2.0 * sin((0.005555555555555556 * (((double) M_PI) * angle)))))) * 1.0;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if (a_m <= 1.16e+229) {
		tmp = (a_m + b) * ((b - a_m) * Math.sin((((angle * Math.pow(Math.PI, 0.6666666666666666)) * Math.cbrt(Math.PI)) * 0.011111111111111112)));
	} else {
		tmp = ((b + a_m) * ((b - a_m) * (2.0 * Math.sin((0.005555555555555556 * (Math.PI * angle)))))) * 1.0;
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (a_m <= 1.16e+229)
		tmp = Float64(Float64(a_m + b) * Float64(Float64(b - a_m) * sin(Float64(Float64(Float64(angle * (pi ^ 0.6666666666666666)) * cbrt(pi)) * 0.011111111111111112))));
	else
		tmp = Float64(Float64(Float64(b + a_m) * Float64(Float64(b - a_m) * Float64(2.0 * sin(Float64(0.005555555555555556 * Float64(pi * angle)))))) * 1.0);
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[a$95$m, 1.16e+229], N[(N[(a$95$m + b), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[Sin[N[(N[(N[(angle * N[Power[Pi, 0.6666666666666666], $MachinePrecision]), $MachinePrecision] * N[Power[Pi, 1/3], $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b + a$95$m), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(2.0 * N[Sin[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.16000000000000001e229

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{{a}^{2}}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      12. difference-of-squares N/A

          \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      14. lift-cos.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot \frac{1}{90}\right)\right) \]

      2. lift-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      3. add-cube-cbrt N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(\left(angle \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot \frac{1}{90}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(\left(angle \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot \frac{1}{90}\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\color{blue}{\left(angle \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      8. pow1/3 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      9. lift-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      10. pow1/3 N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      11. pow-prod-up N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      12. lower-pow.f64 N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot {\pi}^{\color{blue}{\frac{2}{3}}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      14. lift-PI.f64 N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot {\pi}^{\frac{2}{3}}\right) \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \frac{1}{90}\right)\right) \]

      15. lower-cbrt.f64 67.1

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot {\pi}^{0.6666666666666666}\right) \cdot \color{blue}{\sqrt[3]{\pi}}\right) \cdot 0.011111111111111112\right)\right) \]

   5. Applied rewrites 67.1%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(\left(angle \cdot {\pi}^{0.6666666666666666}\right) \cdot \sqrt[3]{\pi}\right)} \cdot 0.011111111111111112\right)\right) \]

   if 1.16000000000000001e229 < a

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\left(\left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(\left(b \cdot b - \color{blue}{{a}^{2}}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\left(\left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\left(a + b\right)} \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\left(a + b\right)} \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(a + b\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot 2\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      14. lower--.f64 57.6

          \[\leadsto \left(\left(\left(a + b\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Applied rewrites 57.6%

      \[\leadsto \left(\color{blue}{\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\left(a + b\right) \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(a + b\right) \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(a + b\right)} \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(b + a\right)} \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(b + a\right)} \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\color{blue}{\left(\left(b - a\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. associate-*l* N/A

         \[\leadsto \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lower-*.f64 67.3

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\pi \cdot \frac{angle}{180}\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      14. lift-PI.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      16. lift-PI.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\color{blue}{\pi} \cdot \frac{angle}{180}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      17. mult-flip N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      19. *-commutative N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      20. associate-*l* N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\left(\pi \cdot \frac{1}{180}\right) \cdot angle\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\color{blue}{\left(\frac{1}{180} \cdot \pi\right)} \cdot angle\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Taylor expanded in angle around 0

      \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 65.1%

      \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 67.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 5 \cdot 10^{+206}:\\ \;\;\;\;\left(b - a\_m\right) \cdot \left(\sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(b + a\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\_m\right) \cdot \left(\left(b - a\_m\right) \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= a_m 5e+206)
   (* (- b a_m) (* (sin (* (* 0.011111111111111112 angle) PI)) (+ b a_m)))
   (*
    (*
     (+ b a_m)
     (* (- b a_m) (* 2.0 (sin (* 0.005555555555555556 (* PI angle))))))
    1.0)))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (a_m <= 5e+206) {
		tmp = (b - a_m) * (sin(((0.011111111111111112 * angle) * ((double) M_PI))) * (b + a_m));
	} else {
		tmp = ((b + a_m) * ((b - a_m) * (2.0 * sin((0.005555555555555556 * (((double) M_PI) * angle)))))) * 1.0;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if (a_m <= 5e+206) {
		tmp = (b - a_m) * (Math.sin(((0.011111111111111112 * angle) * Math.PI)) * (b + a_m));
	} else {
		tmp = ((b + a_m) * ((b - a_m) * (2.0 * Math.sin((0.005555555555555556 * (Math.PI * angle)))))) * 1.0;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if a_m <= 5e+206:
		tmp = (b - a_m) * (math.sin(((0.011111111111111112 * angle) * math.pi)) * (b + a_m))
	else:
		tmp = ((b + a_m) * ((b - a_m) * (2.0 * math.sin((0.005555555555555556 * (math.pi * angle)))))) * 1.0
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (a_m <= 5e+206)
		tmp = Float64(Float64(b - a_m) * Float64(sin(Float64(Float64(0.011111111111111112 * angle) * pi)) * Float64(b + a_m)));
	else
		tmp = Float64(Float64(Float64(b + a_m) * Float64(Float64(b - a_m) * Float64(2.0 * sin(Float64(0.005555555555555556 * Float64(pi * angle)))))) * 1.0);
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if (a_m <= 5e+206)
		tmp = (b - a_m) * (sin(((0.011111111111111112 * angle) * pi)) * (b + a_m));
	else
		tmp = ((b + a_m) * ((b - a_m) * (2.0 * sin((0.005555555555555556 * (pi * angle)))))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[a$95$m, 5e+206], N[(N[(b - a$95$m), $MachinePrecision] * N[(N[Sin[N[(N[(0.011111111111111112 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b + a$95$m), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(2.0 * N[Sin[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.0000000000000002e206

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{{a}^{2}}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      12. difference-of-squares N/A

          \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      14. lift-cos.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot \left(a + b\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right)} \cdot \left(a + b\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(b - a\right) \cdot \left(\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right) \cdot \left(a + b\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(b - a\right) \cdot \left(\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right) \cdot \left(a + b\right)\right)} \]

      6. lower-*.f64 67.0

         \[\leadsto \left(b - a\right) \cdot \color{blue}{\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \left(a + b\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(b - a\right) \cdot \left(\sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)} \cdot \left(a + b\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(b - a\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)} \cdot \left(a + b\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(b - a\right) \cdot \left(\sin \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \pi\right)}\right) \cdot \left(a + b\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(b - a\right) \cdot \left(\sin \color{blue}{\left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right)} \cdot \left(a + b\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(b - a\right) \cdot \left(\sin \color{blue}{\left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right)} \cdot \left(a + b\right)\right) \]

      12. lower-*.f64 67.6

          \[\leadsto \left(b - a\right) \cdot \left(\sin \left(\color{blue}{\left(0.011111111111111112 \cdot angle\right)} \cdot \pi\right) \cdot \left(a + b\right)\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \left(b - a\right) \cdot \left(\sin \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(a + b\right)}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(b - a\right) \cdot \left(\sin \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(b + a\right)}\right) \]

      15. lower-+.f64 67.6

          \[\leadsto \left(b - a\right) \cdot \left(\sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(b + a\right)}\right) \]

   5. Applied rewrites 67.6%

      \[\leadsto \color{blue}{\left(b - a\right) \cdot \left(\sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(b + a\right)\right)} \]

   if 5.0000000000000002e206 < a

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\left(\left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(\left(b \cdot b - \color{blue}{{a}^{2}}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\left(\left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\left(a + b\right)} \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\left(a + b\right)} \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(a + b\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot 2\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      14. lower--.f64 57.6

          \[\leadsto \left(\left(\left(a + b\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Applied rewrites 57.6%

      \[\leadsto \left(\color{blue}{\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\left(a + b\right) \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(a + b\right) \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(a + b\right)} \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(b + a\right)} \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(b + a\right)} \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\color{blue}{\left(\left(b - a\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. associate-*l* N/A

         \[\leadsto \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lower-*.f64 67.3

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\pi \cdot \frac{angle}{180}\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      14. lift-PI.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      16. lift-PI.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\color{blue}{\pi} \cdot \frac{angle}{180}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      17. mult-flip N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      19. *-commutative N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      20. associate-*l* N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\left(\pi \cdot \frac{1}{180}\right) \cdot angle\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      21. *-commutative N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\color{blue}{\left(\frac{1}{180} \cdot \pi\right)} \cdot angle\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Taylor expanded in angle around 0

      \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 65.1%

      \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 67.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 3.4 \cdot 10^{-43}:\\ \;\;\;\;\left(a\_m + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\_m\right)\right)\right)\right)\\ \mathbf{elif}\;angle \leq 4.4 \cdot 10^{+139}:\\ \;\;\;\;\left(\left(b - a\_m\right) \cdot \left(a\_m + b\right)\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\left(\left(b - a\_m\right) \cdot \left(\left(a\_m + b\right) \cdot \pi\right)\right) \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= angle 3.4e-43)
   (* (+ a_m b) (* 0.011111111111111112 (* angle (* PI (- b a_m)))))
   (if (<= angle 4.4e+139)
     (* (* (- b a_m) (+ a_m b)) (sin (* (* angle PI) 0.011111111111111112)))
     (* angle (* (* (- b a_m) (* (+ a_m b) PI)) 0.011111111111111112)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (angle <= 3.4e-43) {
		tmp = (a_m + b) * (0.011111111111111112 * (angle * (((double) M_PI) * (b - a_m))));
	} else if (angle <= 4.4e+139) {
		tmp = ((b - a_m) * (a_m + b)) * sin(((angle * ((double) M_PI)) * 0.011111111111111112));
	} else {
		tmp = angle * (((b - a_m) * ((a_m + b) * ((double) M_PI))) * 0.011111111111111112);
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if (angle <= 3.4e-43) {
		tmp = (a_m + b) * (0.011111111111111112 * (angle * (Math.PI * (b - a_m))));
	} else if (angle <= 4.4e+139) {
		tmp = ((b - a_m) * (a_m + b)) * Math.sin(((angle * Math.PI) * 0.011111111111111112));
	} else {
		tmp = angle * (((b - a_m) * ((a_m + b) * Math.PI)) * 0.011111111111111112);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if angle <= 3.4e-43:
		tmp = (a_m + b) * (0.011111111111111112 * (angle * (math.pi * (b - a_m))))
	elif angle <= 4.4e+139:
		tmp = ((b - a_m) * (a_m + b)) * math.sin(((angle * math.pi) * 0.011111111111111112))
	else:
		tmp = angle * (((b - a_m) * ((a_m + b) * math.pi)) * 0.011111111111111112)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (angle <= 3.4e-43)
		tmp = Float64(Float64(a_m + b) * Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b - a_m)))));
	elseif (angle <= 4.4e+139)
		tmp = Float64(Float64(Float64(b - a_m) * Float64(a_m + b)) * sin(Float64(Float64(angle * pi) * 0.011111111111111112)));
	else
		tmp = Float64(angle * Float64(Float64(Float64(b - a_m) * Float64(Float64(a_m + b) * pi)) * 0.011111111111111112));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if (angle <= 3.4e-43)
		tmp = (a_m + b) * (0.011111111111111112 * (angle * (pi * (b - a_m))));
	elseif (angle <= 4.4e+139)
		tmp = ((b - a_m) * (a_m + b)) * sin(((angle * pi) * 0.011111111111111112));
	else
		tmp = angle * (((b - a_m) * ((a_m + b) * pi)) * 0.011111111111111112);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[angle, 3.4e-43], N[(N[(a$95$m + b), $MachinePrecision] * N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 4.4e+139], N[(N[(N[(b - a$95$m), $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(angle * N[(N[(N[(b - a$95$m), $MachinePrecision] * N[(N[(a$95$m + b), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < 3.4000000000000001e-43

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{{a}^{2}}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      12. difference-of-squares N/A

          \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      14. lift-cos.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      5. lower--.f64 62.3

         \[\leadsto \left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - \color{blue}{a}\right)\right)\right)\right) \]

   6. Applied rewrites 62.3%

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]

   if 3.4000000000000001e-43 < angle < 4.3999999999999999e139

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

      7. lift-sin.f64 N/A

         \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      8. lift-cos.f64 N/A

         \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right)\right) \]

      9. 2-sin N/A

         \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      10. count-2 N/A

          \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \sin \color{blue}{\left(\pi \cdot \frac{angle}{180} + \pi \cdot \frac{angle}{180}\right)} \]

   3. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)} \]

   if 4.3999999999999999e139 < angle

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      17. lower-*.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      20. lower-+.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \cdot \frac{1}{90} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \cdot \frac{1}{90} \]

      5. associate-*l* N/A

         \[\leadsto \left(angle \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right)\right) \cdot \frac{1}{90} \]

      6. associate-*l* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right) \cdot \frac{1}{90}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right) \cdot \frac{1}{90}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto angle \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      9. *-commutative N/A

         \[\leadsto angle \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      11. lift--.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto angle \cdot \left(\left(\left(b \cdot b - a \cdot a\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      13. unpow2 N/A

          \[\leadsto angle \cdot \left(\left(\left({b}^{2} - a \cdot a\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left({b}^{2} - a \cdot a\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      15. unpow2 N/A

          \[\leadsto angle \cdot \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      16. lift-pow.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      17. lift--.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right) \cdot \color{blue}{\frac{1}{90}}\right) \]

   8. Applied rewrites 54.2%

      \[\leadsto angle \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right) \cdot 0.011111111111111112\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 66.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 4.4 \cdot 10^{+139}:\\ \;\;\;\;\left(a\_m + b\right) \cdot \left(\left(b - a\_m\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\left(\left(b - a\_m\right) \cdot \left(\left(a\_m + b\right) \cdot \pi\right)\right) \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= angle 4.4e+139)
   (* (+ a_m b) (* (- b a_m) (sin (* (* angle PI) 0.011111111111111112))))
   (* angle (* (* (- b a_m) (* (+ a_m b) PI)) 0.011111111111111112))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (angle <= 4.4e+139) {
		tmp = (a_m + b) * ((b - a_m) * sin(((angle * ((double) M_PI)) * 0.011111111111111112)));
	} else {
		tmp = angle * (((b - a_m) * ((a_m + b) * ((double) M_PI))) * 0.011111111111111112);
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if (angle <= 4.4e+139) {
		tmp = (a_m + b) * ((b - a_m) * Math.sin(((angle * Math.PI) * 0.011111111111111112)));
	} else {
		tmp = angle * (((b - a_m) * ((a_m + b) * Math.PI)) * 0.011111111111111112);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if angle <= 4.4e+139:
		tmp = (a_m + b) * ((b - a_m) * math.sin(((angle * math.pi) * 0.011111111111111112)))
	else:
		tmp = angle * (((b - a_m) * ((a_m + b) * math.pi)) * 0.011111111111111112)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (angle <= 4.4e+139)
		tmp = Float64(Float64(a_m + b) * Float64(Float64(b - a_m) * sin(Float64(Float64(angle * pi) * 0.011111111111111112))));
	else
		tmp = Float64(angle * Float64(Float64(Float64(b - a_m) * Float64(Float64(a_m + b) * pi)) * 0.011111111111111112));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if (angle <= 4.4e+139)
		tmp = (a_m + b) * ((b - a_m) * sin(((angle * pi) * 0.011111111111111112)));
	else
		tmp = angle * (((b - a_m) * ((a_m + b) * pi)) * 0.011111111111111112);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[angle, 4.4e+139], N[(N[(a$95$m + b), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[Sin[N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(angle * N[(N[(N[(b - a$95$m), $MachinePrecision] * N[(N[(a$95$m + b), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 4.3999999999999999e139

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{{a}^{2}}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      12. difference-of-squares N/A

          \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      14. lift-cos.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]

   if 4.3999999999999999e139 < angle

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      17. lower-*.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      20. lower-+.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \cdot \frac{1}{90} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \cdot \frac{1}{90} \]

      5. associate-*l* N/A

         \[\leadsto \left(angle \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right)\right) \cdot \frac{1}{90} \]

      6. associate-*l* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right) \cdot \frac{1}{90}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right) \cdot \frac{1}{90}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto angle \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      9. *-commutative N/A

         \[\leadsto angle \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      11. lift--.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto angle \cdot \left(\left(\left(b \cdot b - a \cdot a\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      13. unpow2 N/A

          \[\leadsto angle \cdot \left(\left(\left({b}^{2} - a \cdot a\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left({b}^{2} - a \cdot a\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      15. unpow2 N/A

          \[\leadsto angle \cdot \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      16. lift-pow.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      17. lift--.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right) \cdot \color{blue}{\frac{1}{90}}\right) \]

   8. Applied rewrites 54.2%

      \[\leadsto angle \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right) \cdot 0.011111111111111112\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 66.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.06 \cdot 10^{+110}:\\ \;\;\;\;\left(a\_m + b\right) \cdot \left(\left(b - a\_m\right) \cdot \sin \left(\left(0.011111111111111112 \cdot \pi\right) \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a\_m + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\_m\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= b 1.06e+110)
   (* (+ a_m b) (* (- b a_m) (sin (* (* 0.011111111111111112 PI) angle))))
   (* (+ a_m b) (* 0.011111111111111112 (* angle (* PI (- b a_m)))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (b <= 1.06e+110) {
		tmp = (a_m + b) * ((b - a_m) * sin(((0.011111111111111112 * ((double) M_PI)) * angle)));
	} else {
		tmp = (a_m + b) * (0.011111111111111112 * (angle * (((double) M_PI) * (b - a_m))));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if (b <= 1.06e+110) {
		tmp = (a_m + b) * ((b - a_m) * Math.sin(((0.011111111111111112 * Math.PI) * angle)));
	} else {
		tmp = (a_m + b) * (0.011111111111111112 * (angle * (Math.PI * (b - a_m))));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if b <= 1.06e+110:
		tmp = (a_m + b) * ((b - a_m) * math.sin(((0.011111111111111112 * math.pi) * angle)))
	else:
		tmp = (a_m + b) * (0.011111111111111112 * (angle * (math.pi * (b - a_m))))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (b <= 1.06e+110)
		tmp = Float64(Float64(a_m + b) * Float64(Float64(b - a_m) * sin(Float64(Float64(0.011111111111111112 * pi) * angle))));
	else
		tmp = Float64(Float64(a_m + b) * Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b - a_m)))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if (b <= 1.06e+110)
		tmp = (a_m + b) * ((b - a_m) * sin(((0.011111111111111112 * pi) * angle)));
	else
		tmp = (a_m + b) * (0.011111111111111112 * (angle * (pi * (b - a_m))));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[b, 1.06e+110], N[(N[(a$95$m + b), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[Sin[N[(N[(0.011111111111111112 * Pi), $MachinePrecision] * angle), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m + b), $MachinePrecision] * N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.06000000000000005e110

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{{a}^{2}}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      12. difference-of-squares N/A

          \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      14. lift-cos.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot \frac{1}{90}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(angle \cdot \left(\pi \cdot \frac{1}{90}\right)\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\left(\pi \cdot \frac{1}{90}\right) \cdot angle\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\left(\pi \cdot \frac{1}{90}\right) \cdot angle\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(\frac{1}{90} \cdot \pi\right)} \cdot angle\right)\right) \]

      7. lower-*.f64 67.2

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(0.011111111111111112 \cdot \pi\right)} \cdot angle\right)\right) \]

   5. Applied rewrites 67.2%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\left(0.011111111111111112 \cdot \pi\right) \cdot angle\right)}\right) \]

   if 1.06000000000000005e110 < b

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{{a}^{2}}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      12. difference-of-squares N/A

          \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      14. lift-cos.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      5. lower--.f64 62.3

         \[\leadsto \left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - \color{blue}{a}\right)\right)\right)\right) \]

   6. Applied rewrites 62.3%

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 64.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 5.8 \cdot 10^{+200}:\\ \;\;\;\;\left(b - a\_m\right) \cdot \left(\sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(b + a\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\left(b - a\_m\right) \cdot \left(\left(a\_m + b\right) \cdot angle\right)\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= a_m 5.8e+200)
   (* (- b a_m) (* (sin (* (* 0.011111111111111112 angle) PI)) (+ b a_m)))
   (* 0.011111111111111112 (* (* (- b a_m) (* (+ a_m b) angle)) PI))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (a_m <= 5.8e+200) {
		tmp = (b - a_m) * (sin(((0.011111111111111112 * angle) * ((double) M_PI))) * (b + a_m));
	} else {
		tmp = 0.011111111111111112 * (((b - a_m) * ((a_m + b) * angle)) * ((double) M_PI));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if (a_m <= 5.8e+200) {
		tmp = (b - a_m) * (Math.sin(((0.011111111111111112 * angle) * Math.PI)) * (b + a_m));
	} else {
		tmp = 0.011111111111111112 * (((b - a_m) * ((a_m + b) * angle)) * Math.PI);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if a_m <= 5.8e+200:
		tmp = (b - a_m) * (math.sin(((0.011111111111111112 * angle) * math.pi)) * (b + a_m))
	else:
		tmp = 0.011111111111111112 * (((b - a_m) * ((a_m + b) * angle)) * math.pi)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (a_m <= 5.8e+200)
		tmp = Float64(Float64(b - a_m) * Float64(sin(Float64(Float64(0.011111111111111112 * angle) * pi)) * Float64(b + a_m)));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(Float64(b - a_m) * Float64(Float64(a_m + b) * angle)) * pi));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if (a_m <= 5.8e+200)
		tmp = (b - a_m) * (sin(((0.011111111111111112 * angle) * pi)) * (b + a_m));
	else
		tmp = 0.011111111111111112 * (((b - a_m) * ((a_m + b) * angle)) * pi);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[a$95$m, 5.8e+200], N[(N[(b - a$95$m), $MachinePrecision] * N[(N[Sin[N[(N[(0.011111111111111112 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(N[(b - a$95$m), $MachinePrecision] * N[(N[(a$95$m + b), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.7999999999999998e200

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{{a}^{2}}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      12. difference-of-squares N/A

          \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      14. lift-cos.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot \left(a + b\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right)} \cdot \left(a + b\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(b - a\right) \cdot \left(\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right) \cdot \left(a + b\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(b - a\right) \cdot \left(\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right) \cdot \left(a + b\right)\right)} \]

      6. lower-*.f64 67.0

         \[\leadsto \left(b - a\right) \cdot \color{blue}{\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \left(a + b\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(b - a\right) \cdot \left(\sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)} \cdot \left(a + b\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(b - a\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)} \cdot \left(a + b\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(b - a\right) \cdot \left(\sin \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \pi\right)}\right) \cdot \left(a + b\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(b - a\right) \cdot \left(\sin \color{blue}{\left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right)} \cdot \left(a + b\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(b - a\right) \cdot \left(\sin \color{blue}{\left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right)} \cdot \left(a + b\right)\right) \]

      12. lower-*.f64 67.6

          \[\leadsto \left(b - a\right) \cdot \left(\sin \left(\color{blue}{\left(0.011111111111111112 \cdot angle\right)} \cdot \pi\right) \cdot \left(a + b\right)\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \left(b - a\right) \cdot \left(\sin \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(a + b\right)}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(b - a\right) \cdot \left(\sin \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(b + a\right)}\right) \]

      15. lower-+.f64 67.6

          \[\leadsto \left(b - a\right) \cdot \left(\sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(b + a\right)}\right) \]

   5. Applied rewrites 67.6%

      \[\leadsto \color{blue}{\left(b - a\right) \cdot \left(\sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(b + a\right)\right)} \]

   if 5.7999999999999998e200 < a

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      17. lower-*.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      20. lower-+.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot angle\right) \cdot \pi\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot angle\right) \cdot \pi\right) \]

      4. associate-*l* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \pi\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \pi\right) \]

      6. lower-*.f64 62.2

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \pi\right) \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \pi\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot angle\right)\right) \cdot \pi\right) \]

      9. lower-+.f64 62.2

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot angle\right)\right) \cdot \pi\right) \]

   8. Applied rewrites 62.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot angle\right)\right) \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 64.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 2.4 \cdot 10^{+51}:\\ \;\;\;\;\left(a\_m + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\_m\right)\right)\right)\right)\\ \mathbf{elif}\;angle \leq 9.2 \cdot 10^{+118}:\\ \;\;\;\;b \cdot \left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\left(\left(b - a\_m\right) \cdot \left(\left(a\_m + b\right) \cdot \pi\right)\right) \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= angle 2.4e+51)
   (* (+ a_m b) (* 0.011111111111111112 (* angle (* PI (- b a_m)))))
   (if (<= angle 9.2e+118)
     (* b (* b (sin (* 0.011111111111111112 (* angle PI)))))
     (* angle (* (* (- b a_m) (* (+ a_m b) PI)) 0.011111111111111112)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (angle <= 2.4e+51) {
		tmp = (a_m + b) * (0.011111111111111112 * (angle * (((double) M_PI) * (b - a_m))));
	} else if (angle <= 9.2e+118) {
		tmp = b * (b * sin((0.011111111111111112 * (angle * ((double) M_PI)))));
	} else {
		tmp = angle * (((b - a_m) * ((a_m + b) * ((double) M_PI))) * 0.011111111111111112);
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if (angle <= 2.4e+51) {
		tmp = (a_m + b) * (0.011111111111111112 * (angle * (Math.PI * (b - a_m))));
	} else if (angle <= 9.2e+118) {
		tmp = b * (b * Math.sin((0.011111111111111112 * (angle * Math.PI))));
	} else {
		tmp = angle * (((b - a_m) * ((a_m + b) * Math.PI)) * 0.011111111111111112);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if angle <= 2.4e+51:
		tmp = (a_m + b) * (0.011111111111111112 * (angle * (math.pi * (b - a_m))))
	elif angle <= 9.2e+118:
		tmp = b * (b * math.sin((0.011111111111111112 * (angle * math.pi))))
	else:
		tmp = angle * (((b - a_m) * ((a_m + b) * math.pi)) * 0.011111111111111112)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (angle <= 2.4e+51)
		tmp = Float64(Float64(a_m + b) * Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b - a_m)))));
	elseif (angle <= 9.2e+118)
		tmp = Float64(b * Float64(b * sin(Float64(0.011111111111111112 * Float64(angle * pi)))));
	else
		tmp = Float64(angle * Float64(Float64(Float64(b - a_m) * Float64(Float64(a_m + b) * pi)) * 0.011111111111111112));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if (angle <= 2.4e+51)
		tmp = (a_m + b) * (0.011111111111111112 * (angle * (pi * (b - a_m))));
	elseif (angle <= 9.2e+118)
		tmp = b * (b * sin((0.011111111111111112 * (angle * pi))));
	else
		tmp = angle * (((b - a_m) * ((a_m + b) * pi)) * 0.011111111111111112);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[angle, 2.4e+51], N[(N[(a$95$m + b), $MachinePrecision] * N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 9.2e+118], N[(b * N[(b * N[Sin[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(angle * N[(N[(N[(b - a$95$m), $MachinePrecision] * N[(N[(a$95$m + b), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < 2.3999999999999999e51

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{{a}^{2}}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      12. difference-of-squares N/A

          \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      14. lift-cos.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      5. lower--.f64 62.3

         \[\leadsto \left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - \color{blue}{a}\right)\right)\right)\right) \]

   6. Applied rewrites 62.3%

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]

   if 2.3999999999999999e51 < angle < 9.20000000000000063e118

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{{a}^{2}}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      12. difference-of-squares N/A

          \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      14. lift-cos.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]

   4. Taylor expanded in a around 0

      \[\leadsto \color{blue}{b} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 42.3%

      \[\leadsto \color{blue}{b} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \]

   7. Taylor expanded in a around 0

      \[\leadsto b \cdot \color{blue}{\left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto b \cdot \left(b \cdot \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. lower-sin.f64 N/A

         \[\leadsto b \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto b \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto b \cdot \left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-PI.f64 41.0

         \[\leadsto b \cdot \left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) \]

   9. Applied rewrites 41.0%

      \[\leadsto b \cdot \color{blue}{\left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)} \]

   if 9.20000000000000063e118 < angle

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      17. lower-*.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      20. lower-+.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \cdot \frac{1}{90} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \cdot \frac{1}{90} \]

      5. associate-*l* N/A

         \[\leadsto \left(angle \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right)\right) \cdot \frac{1}{90} \]

      6. associate-*l* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right) \cdot \frac{1}{90}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right) \cdot \frac{1}{90}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto angle \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      9. *-commutative N/A

         \[\leadsto angle \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      11. lift--.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto angle \cdot \left(\left(\left(b \cdot b - a \cdot a\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      13. unpow2 N/A

          \[\leadsto angle \cdot \left(\left(\left({b}^{2} - a \cdot a\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left({b}^{2} - a \cdot a\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      15. unpow2 N/A

          \[\leadsto angle \cdot \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      16. lift-pow.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      17. lift--.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right) \cdot \color{blue}{\frac{1}{90}}\right) \]

   8. Applied rewrites 54.2%

      \[\leadsto angle \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right) \cdot 0.011111111111111112\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 63.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 4.5 \cdot 10^{-84}:\\ \;\;\;\;\left(\sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(b - a\_m\right)\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot \left(angle \cdot \left(b - a\_m\right)\right)\right) \cdot \left(a\_m + b\right)\right) \cdot \pi\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= a_m 4.5e-84)
   (* (* (sin (* (* 0.011111111111111112 angle) PI)) (- b a_m)) b)
   (* (* (* 0.011111111111111112 (* angle (- b a_m))) (+ a_m b)) PI)))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (a_m <= 4.5e-84) {
		tmp = (sin(((0.011111111111111112 * angle) * ((double) M_PI))) * (b - a_m)) * b;
	} else {
		tmp = ((0.011111111111111112 * (angle * (b - a_m))) * (a_m + b)) * ((double) M_PI);
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if (a_m <= 4.5e-84) {
		tmp = (Math.sin(((0.011111111111111112 * angle) * Math.PI)) * (b - a_m)) * b;
	} else {
		tmp = ((0.011111111111111112 * (angle * (b - a_m))) * (a_m + b)) * Math.PI;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if a_m <= 4.5e-84:
		tmp = (math.sin(((0.011111111111111112 * angle) * math.pi)) * (b - a_m)) * b
	else:
		tmp = ((0.011111111111111112 * (angle * (b - a_m))) * (a_m + b)) * math.pi
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (a_m <= 4.5e-84)
		tmp = Float64(Float64(sin(Float64(Float64(0.011111111111111112 * angle) * pi)) * Float64(b - a_m)) * b);
	else
		tmp = Float64(Float64(Float64(0.011111111111111112 * Float64(angle * Float64(b - a_m))) * Float64(a_m + b)) * pi);
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if (a_m <= 4.5e-84)
		tmp = (sin(((0.011111111111111112 * angle) * pi)) * (b - a_m)) * b;
	else
		tmp = ((0.011111111111111112 * (angle * (b - a_m))) * (a_m + b)) * pi;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[a$95$m, 4.5e-84], N[(N[(N[Sin[N[(N[(0.011111111111111112 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[(N[(N[(0.011111111111111112 * N[(angle * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.50000000000000016e-84

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{{a}^{2}}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      12. difference-of-squares N/A

          \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      14. lift-cos.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]

   4. Taylor expanded in a around 0

      \[\leadsto \color{blue}{b} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 42.3%

      \[\leadsto \color{blue}{b} \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto b \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto b \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto b \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \pi\right)}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto b \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto b \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right)}\right) \]

      6. lower-*.f64 42.7

         \[\leadsto b \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(0.011111111111111112 \cdot angle\right)} \cdot \pi\right)\right) \]

   8. Applied rewrites 42.7%

      \[\leadsto b \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right)}\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(b - a\right) \cdot \sin \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right)\right) \cdot b} \]

      3. lower-*.f64 42.7

         \[\leadsto \color{blue}{\left(\left(b - a\right) \cdot \sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right)\right) \cdot b} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(b - a\right) \cdot \sin \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right)\right)} \cdot b \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \left(b - a\right)\right)} \cdot b \]

      6. lower-*.f64 42.7

         \[\leadsto \color{blue}{\left(\sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(b - a\right)\right)} \cdot b \]

      7. +-commutative 42.7

         \[\leadsto \left(\sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot b \]

   10. Applied rewrites 42.7%

       \[\leadsto \color{blue}{\left(\sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot b} \]

   if 4.50000000000000016e-84 < a

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      17. lower-*.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      20. lower-+.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right) \cdot \color{blue}{\pi} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right) \cdot \color{blue}{\pi} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right) \cdot \pi \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right) \cdot \pi \]

      7. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right)\right) \cdot \pi \]

      8. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot \left(b + a\right)\right) \cdot \pi \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot \left(b + a\right)\right) \cdot \pi \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot \left(b + a\right)\right) \cdot \pi \]

      11. lower-*.f64 62.3

          \[\leadsto \left(\left(0.011111111111111112 \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot \left(b + a\right)\right) \cdot \pi \]

      12. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot \left(b + a\right)\right) \cdot \pi \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot \left(a + b\right)\right) \cdot \pi \]

      14. lower-+.f64 62.3

          \[\leadsto \left(\left(0.011111111111111112 \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot \left(a + b\right)\right) \cdot \pi \]

   8. Applied rewrites 62.3%

      \[\leadsto \left(\left(0.011111111111111112 \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot \left(a + b\right)\right) \cdot \color{blue}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 62.9% accurate, 5.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 3.1 \cdot 10^{-47}:\\ \;\;\;\;\left(a\_m + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\_m\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle \cdot \mathsf{fma}\left(-a\_m, a\_m, b \cdot b\right)\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= angle 3.1e-47)
   (* (+ a_m b) (* 0.011111111111111112 (* angle (* PI (- b a_m)))))
   (* 0.011111111111111112 (* (* angle (fma (- a_m) a_m (* b b))) PI))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (angle <= 3.1e-47) {
		tmp = (a_m + b) * (0.011111111111111112 * (angle * (((double) M_PI) * (b - a_m))));
	} else {
		tmp = 0.011111111111111112 * ((angle * fma(-a_m, a_m, (b * b))) * ((double) M_PI));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (angle <= 3.1e-47)
		tmp = Float64(Float64(a_m + b) * Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b - a_m)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle * fma(Float64(-a_m), a_m, Float64(b * b))) * pi));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[angle, 3.1e-47], N[(N[(a$95$m + b), $MachinePrecision] * N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(angle * N[((-a$95$m) * a$95$m + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 3.0999999999999998e-47

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{{a}^{2}}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      12. difference-of-squares N/A

          \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      14. lift-cos.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      5. lower--.f64 62.3

         \[\leadsto \left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - \color{blue}{a}\right)\right)\right)\right) \]

   6. Applied rewrites 62.3%

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]

   if 3.0999999999999998e-47 < angle

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      17. lower-*.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      20. lower-+.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      4. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      5. difference-of-squares-rev N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - a \cdot a\right)\right) \cdot \pi\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - a \cdot a\right)\right) \cdot \pi\right) \]

      8. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      10. sub-flip N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} + \left(\mathsf{neg}\left({a}^{2}\right)\right)\right)\right) \cdot \pi\right) \]

      11. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(\mathsf{neg}\left({a}^{2}\right)\right) + {b}^{2}\right)\right) \cdot \pi\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(\mathsf{neg}\left({a}^{2}\right)\right) + {b}^{2}\right)\right) \cdot \pi\right) \]

      13. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(\mathsf{neg}\left(a \cdot a\right)\right) + {b}^{2}\right)\right) \cdot \pi\right) \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(\mathsf{neg}\left(a\right)\right) \cdot a + {b}^{2}\right)\right) \cdot \pi\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{fma}\left(\mathsf{neg}\left(a\right), a, {b}^{2}\right)\right) \cdot \pi\right) \]

      16. lower-neg.f64 53.1

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \mathsf{fma}\left(-a, a, {b}^{2}\right)\right) \cdot \pi\right) \]

      17. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{fma}\left(-a, a, {b}^{2}\right)\right) \cdot \pi\right) \]

      18. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{fma}\left(-a, a, b \cdot b\right)\right) \cdot \pi\right) \]

      19. lower-*.f64 53.1

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \mathsf{fma}\left(-a, a, b \cdot b\right)\right) \cdot \pi\right) \]

   8. Applied rewrites 53.1%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \mathsf{fma}\left(-a, a, b \cdot b\right)\right) \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 62.9% accurate, 5.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 6.3 \cdot 10^{-52}:\\ \;\;\;\;\left(a\_m + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\_m\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(-a\_m, a\_m, b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= angle 6.3e-52)
   (* (+ a_m b) (* 0.011111111111111112 (* angle (* PI (- b a_m)))))
   (* 0.011111111111111112 (* angle (* PI (fma (- a_m) a_m (* b b)))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (angle <= 6.3e-52) {
		tmp = (a_m + b) * (0.011111111111111112 * (angle * (((double) M_PI) * (b - a_m))));
	} else {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * fma(-a_m, a_m, (b * b))));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (angle <= 6.3e-52)
		tmp = Float64(Float64(a_m + b) * Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b - a_m)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * fma(Float64(-a_m), a_m, Float64(b * b)))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[angle, 6.3e-52], N[(N[(a$95$m + b), $MachinePrecision] * N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(Pi * N[((-a$95$m) * a$95$m + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 6.3000000000000003e-52

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{{a}^{2}}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      12. difference-of-squares N/A

          \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      14. lift-cos.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      5. lower--.f64 62.3

         \[\leadsto \left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - \color{blue}{a}\right)\right)\right)\right) \]

   6. Applied rewrites 62.3%

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]

   if 6.3000000000000003e-52 < angle

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      2. sub-negate-rev N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\mathsf{neg}\left(\left({a}^{2} - {b}^{2}\right)\right)\right)\right)\right) \]

      3. sub-flip N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\mathsf{neg}\left(\left({a}^{2} + \left(\mathsf{neg}\left({b}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\left(\mathsf{neg}\left({a}^{2}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({b}^{2}\right)\right)\right)\right)}\right)\right)\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\left(\mathsf{neg}\left({a}^{2}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{{b}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\left(\mathsf{neg}\left(a \cdot a\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{{b}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\left(\mathsf{neg}\left(a\right)\right) \cdot a + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({b}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\left(\mathsf{neg}\left(a\right)\right) \cdot a + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({b}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\left(\mathsf{neg}\left(a\right)\right) \cdot a + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(b \cdot b\right)\right)\right)\right)\right)\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\left(\mathsf{neg}\left(a\right)\right) \cdot a + \left(\mathsf{neg}\left(b \cdot \left(\mathsf{neg}\left(b\right)\right)\right)\right)\right)\right)\right) \]

      11. distribute-lft-neg-out N/A

          \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\left(\mathsf{neg}\left(a\right)\right) \cdot a + \left(\mathsf{neg}\left(b\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right)\right)\right) \]

      12. sqr-neg-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\left(\mathsf{neg}\left(a\right)\right) \cdot a + b \cdot \color{blue}{b}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\left(\mathsf{neg}\left(a\right)\right) \cdot a + {b}^{\color{blue}{2}}\right)\right)\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\left(\mathsf{neg}\left(a\right)\right) \cdot a + {b}^{\color{blue}{2}}\right)\right)\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(\mathsf{neg}\left(a\right), \color{blue}{a}, {b}^{2}\right)\right)\right) \]

      16. lower-neg.f64 53.1

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(-a, a, {b}^{2}\right)\right)\right) \]

      17. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(-a, a, {b}^{2}\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(-a, a, b \cdot b\right)\right)\right) \]

      19. lower-*.f64 53.1

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(-a, a, b \cdot b\right)\right)\right) \]

   6. Applied rewrites 53.1%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(-a, \color{blue}{a}, b \cdot b\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 62.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 3.4 \cdot 10^{-43}:\\ \;\;\;\;\left(a\_m + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\_m\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b - a\_m\right) \cdot \left(b + a\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= angle 3.4e-43)
   (* (+ a_m b) (* 0.011111111111111112 (* angle (* PI (- b a_m)))))
   (* (* (* 0.011111111111111112 angle) PI) (* (- b a_m) (+ b a_m)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (angle <= 3.4e-43) {
		tmp = (a_m + b) * (0.011111111111111112 * (angle * (((double) M_PI) * (b - a_m))));
	} else {
		tmp = ((0.011111111111111112 * angle) * ((double) M_PI)) * ((b - a_m) * (b + a_m));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if (angle <= 3.4e-43) {
		tmp = (a_m + b) * (0.011111111111111112 * (angle * (Math.PI * (b - a_m))));
	} else {
		tmp = ((0.011111111111111112 * angle) * Math.PI) * ((b - a_m) * (b + a_m));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if angle <= 3.4e-43:
		tmp = (a_m + b) * (0.011111111111111112 * (angle * (math.pi * (b - a_m))))
	else:
		tmp = ((0.011111111111111112 * angle) * math.pi) * ((b - a_m) * (b + a_m))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (angle <= 3.4e-43)
		tmp = Float64(Float64(a_m + b) * Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b - a_m)))));
	else
		tmp = Float64(Float64(Float64(0.011111111111111112 * angle) * pi) * Float64(Float64(b - a_m) * Float64(b + a_m)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if (angle <= 3.4e-43)
		tmp = (a_m + b) * (0.011111111111111112 * (angle * (pi * (b - a_m))));
	else
		tmp = ((0.011111111111111112 * angle) * pi) * ((b - a_m) * (b + a_m));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[angle, 3.4e-43], N[(N[(a$95$m + b), $MachinePrecision] * N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.011111111111111112 * angle), $MachinePrecision] * Pi), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 3.4000000000000001e-43

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{{a}^{2}}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      12. difference-of-squares N/A

          \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      14. lift-cos.f64 N/A

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      5. lower--.f64 62.3

         \[\leadsto \left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - \color{blue}{a}\right)\right)\right)\right) \]

   6. Applied rewrites 62.3%

      \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]

   if 3.4000000000000001e-43 < angle

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   6. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 62.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 2 \cdot 10^{-64}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot \left(angle \cdot \left(b - a\_m\right)\right)\right) \cdot \left(a\_m + b\right)\right) \cdot \pi\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b - a\_m\right) \cdot \left(b + a\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= angle 2e-64)
   (* (* (* 0.011111111111111112 (* angle (- b a_m))) (+ a_m b)) PI)
   (* (* (* 0.011111111111111112 angle) PI) (* (- b a_m) (+ b a_m)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (angle <= 2e-64) {
		tmp = ((0.011111111111111112 * (angle * (b - a_m))) * (a_m + b)) * ((double) M_PI);
	} else {
		tmp = ((0.011111111111111112 * angle) * ((double) M_PI)) * ((b - a_m) * (b + a_m));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if (angle <= 2e-64) {
		tmp = ((0.011111111111111112 * (angle * (b - a_m))) * (a_m + b)) * Math.PI;
	} else {
		tmp = ((0.011111111111111112 * angle) * Math.PI) * ((b - a_m) * (b + a_m));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if angle <= 2e-64:
		tmp = ((0.011111111111111112 * (angle * (b - a_m))) * (a_m + b)) * math.pi
	else:
		tmp = ((0.011111111111111112 * angle) * math.pi) * ((b - a_m) * (b + a_m))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (angle <= 2e-64)
		tmp = Float64(Float64(Float64(0.011111111111111112 * Float64(angle * Float64(b - a_m))) * Float64(a_m + b)) * pi);
	else
		tmp = Float64(Float64(Float64(0.011111111111111112 * angle) * pi) * Float64(Float64(b - a_m) * Float64(b + a_m)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if (angle <= 2e-64)
		tmp = ((0.011111111111111112 * (angle * (b - a_m))) * (a_m + b)) * pi;
	else
		tmp = ((0.011111111111111112 * angle) * pi) * ((b - a_m) * (b + a_m));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[angle, 2e-64], N[(N[(N[(0.011111111111111112 * N[(angle * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision], N[(N[(N[(0.011111111111111112 * angle), $MachinePrecision] * Pi), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 1.99999999999999993e-64

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      17. lower-*.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      20. lower-+.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right) \cdot \color{blue}{\pi} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right) \cdot \color{blue}{\pi} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right) \cdot \pi \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right) \cdot \pi \]

      7. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right)\right) \cdot \pi \]

      8. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot \left(b + a\right)\right) \cdot \pi \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot \left(b + a\right)\right) \cdot \pi \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot \left(b + a\right)\right) \cdot \pi \]

      11. lower-*.f64 62.3

          \[\leadsto \left(\left(0.011111111111111112 \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot \left(b + a\right)\right) \cdot \pi \]

      12. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot \left(b + a\right)\right) \cdot \pi \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot \left(a + b\right)\right) \cdot \pi \]

      14. lower-+.f64 62.3

          \[\leadsto \left(\left(0.011111111111111112 \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot \left(a + b\right)\right) \cdot \pi \]

   8. Applied rewrites 62.3%

      \[\leadsto \left(\left(0.011111111111111112 \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot \left(a + b\right)\right) \cdot \color{blue}{\pi} \]

   if 1.99999999999999993e-64 < angle

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   6. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 62.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 3.4 \cdot 10^{-43}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\left(b - a\_m\right) \cdot \left(\left(a\_m + b\right) \cdot angle\right)\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b - a\_m\right) \cdot \left(b + a\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= angle 3.4e-43)
   (* 0.011111111111111112 (* (* (- b a_m) (* (+ a_m b) angle)) PI))
   (* (* (* 0.011111111111111112 angle) PI) (* (- b a_m) (+ b a_m)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (angle <= 3.4e-43) {
		tmp = 0.011111111111111112 * (((b - a_m) * ((a_m + b) * angle)) * ((double) M_PI));
	} else {
		tmp = ((0.011111111111111112 * angle) * ((double) M_PI)) * ((b - a_m) * (b + a_m));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if (angle <= 3.4e-43) {
		tmp = 0.011111111111111112 * (((b - a_m) * ((a_m + b) * angle)) * Math.PI);
	} else {
		tmp = ((0.011111111111111112 * angle) * Math.PI) * ((b - a_m) * (b + a_m));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if angle <= 3.4e-43:
		tmp = 0.011111111111111112 * (((b - a_m) * ((a_m + b) * angle)) * math.pi)
	else:
		tmp = ((0.011111111111111112 * angle) * math.pi) * ((b - a_m) * (b + a_m))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (angle <= 3.4e-43)
		tmp = Float64(0.011111111111111112 * Float64(Float64(Float64(b - a_m) * Float64(Float64(a_m + b) * angle)) * pi));
	else
		tmp = Float64(Float64(Float64(0.011111111111111112 * angle) * pi) * Float64(Float64(b - a_m) * Float64(b + a_m)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if (angle <= 3.4e-43)
		tmp = 0.011111111111111112 * (((b - a_m) * ((a_m + b) * angle)) * pi);
	else
		tmp = ((0.011111111111111112 * angle) * pi) * ((b - a_m) * (b + a_m));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[angle, 3.4e-43], N[(0.011111111111111112 * N[(N[(N[(b - a$95$m), $MachinePrecision] * N[(N[(a$95$m + b), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.011111111111111112 * angle), $MachinePrecision] * Pi), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 3.4000000000000001e-43

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      17. lower-*.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      20. lower-+.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot angle\right) \cdot \pi\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot angle\right) \cdot \pi\right) \]

      4. associate-*l* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \pi\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \pi\right) \]

      6. lower-*.f64 62.2

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \pi\right) \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \pi\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot angle\right)\right) \cdot \pi\right) \]

      9. lower-+.f64 62.2

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot angle\right)\right) \cdot \pi\right) \]

   8. Applied rewrites 62.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot angle\right)\right) \cdot \pi\right) \]

   if 3.4000000000000001e-43 < angle

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   6. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 62.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 3 \cdot 10^{-47}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\left(b - a\_m\right) \cdot \left(\left(a\_m + b\right) \cdot angle\right)\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\_m\right) \cdot \left(b + a\_m\right)\right)\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= angle 3e-47)
   (* 0.011111111111111112 (* (* (- b a_m) (* (+ a_m b) angle)) PI))
   (* 0.011111111111111112 (* (* angle (* (- b a_m) (+ b a_m))) PI))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (angle <= 3e-47) {
		tmp = 0.011111111111111112 * (((b - a_m) * ((a_m + b) * angle)) * ((double) M_PI));
	} else {
		tmp = 0.011111111111111112 * ((angle * ((b - a_m) * (b + a_m))) * ((double) M_PI));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if (angle <= 3e-47) {
		tmp = 0.011111111111111112 * (((b - a_m) * ((a_m + b) * angle)) * Math.PI);
	} else {
		tmp = 0.011111111111111112 * ((angle * ((b - a_m) * (b + a_m))) * Math.PI);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if angle <= 3e-47:
		tmp = 0.011111111111111112 * (((b - a_m) * ((a_m + b) * angle)) * math.pi)
	else:
		tmp = 0.011111111111111112 * ((angle * ((b - a_m) * (b + a_m))) * math.pi)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (angle <= 3e-47)
		tmp = Float64(0.011111111111111112 * Float64(Float64(Float64(b - a_m) * Float64(Float64(a_m + b) * angle)) * pi));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle * Float64(Float64(b - a_m) * Float64(b + a_m))) * pi));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if (angle <= 3e-47)
		tmp = 0.011111111111111112 * (((b - a_m) * ((a_m + b) * angle)) * pi);
	else
		tmp = 0.011111111111111112 * ((angle * ((b - a_m) * (b + a_m))) * pi);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[angle, 3e-47], N[(0.011111111111111112 * N[(N[(N[(b - a$95$m), $MachinePrecision] * N[(N[(a$95$m + b), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(angle * N[(N[(b - a$95$m), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 3.00000000000000017e-47

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      17. lower-*.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      20. lower-+.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot angle\right) \cdot \pi\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot angle\right) \cdot \pi\right) \]

      4. associate-*l* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \pi\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \pi\right) \]

      6. lower-*.f64 62.2

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \pi\right) \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \pi\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot angle\right)\right) \cdot \pi\right) \]

      9. lower-+.f64 62.2

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot angle\right)\right) \cdot \pi\right) \]

   8. Applied rewrites 62.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot angle\right)\right) \cdot \pi\right) \]

   if 3.00000000000000017e-47 < angle

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      17. lower-*.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      20. lower-+.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 62.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := 2 \cdot \left({b}^{2} - {a\_m}^{2}\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+162}:\\ \;\;\;\;-2 \cdot \left(\left(0.005555555555555556 \cdot \left(a\_m \cdot \left(angle \cdot \pi\right)\right)\right) \cdot a\_m\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+118}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\_m\right) \cdot \left(b + a\_m\right)\right)\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot \left(angle \cdot \left(b - a\_m\right)\right)\right) \cdot b\right) \cdot \pi\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (* 2.0 (- (pow b 2.0) (pow a_m 2.0)))))
   (if (<= t_0 -2e+162)
     (* -2.0 (* (* 0.005555555555555556 (* a_m (* angle PI))) a_m))
     (if (<= t_0 5e+118)
       (* 0.011111111111111112 (* (* angle (* (- b a_m) (+ b a_m))) PI))
       (* (* (* 0.011111111111111112 (* angle (- b a_m))) b) PI)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = 2.0 * (pow(b, 2.0) - pow(a_m, 2.0));
	double tmp;
	if (t_0 <= -2e+162) {
		tmp = -2.0 * ((0.005555555555555556 * (a_m * (angle * ((double) M_PI)))) * a_m);
	} else if (t_0 <= 5e+118) {
		tmp = 0.011111111111111112 * ((angle * ((b - a_m) * (b + a_m))) * ((double) M_PI));
	} else {
		tmp = ((0.011111111111111112 * (angle * (b - a_m))) * b) * ((double) M_PI);
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double t_0 = 2.0 * (Math.pow(b, 2.0) - Math.pow(a_m, 2.0));
	double tmp;
	if (t_0 <= -2e+162) {
		tmp = -2.0 * ((0.005555555555555556 * (a_m * (angle * Math.PI))) * a_m);
	} else if (t_0 <= 5e+118) {
		tmp = 0.011111111111111112 * ((angle * ((b - a_m) * (b + a_m))) * Math.PI);
	} else {
		tmp = ((0.011111111111111112 * (angle * (b - a_m))) * b) * Math.PI;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	t_0 = 2.0 * (math.pow(b, 2.0) - math.pow(a_m, 2.0))
	tmp = 0
	if t_0 <= -2e+162:
		tmp = -2.0 * ((0.005555555555555556 * (a_m * (angle * math.pi))) * a_m)
	elif t_0 <= 5e+118:
		tmp = 0.011111111111111112 * ((angle * ((b - a_m) * (b + a_m))) * math.pi)
	else:
		tmp = ((0.011111111111111112 * (angle * (b - a_m))) * b) * math.pi
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = Float64(2.0 * Float64((b ^ 2.0) - (a_m ^ 2.0)))
	tmp = 0.0
	if (t_0 <= -2e+162)
		tmp = Float64(-2.0 * Float64(Float64(0.005555555555555556 * Float64(a_m * Float64(angle * pi))) * a_m));
	elseif (t_0 <= 5e+118)
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle * Float64(Float64(b - a_m) * Float64(b + a_m))) * pi));
	else
		tmp = Float64(Float64(Float64(0.011111111111111112 * Float64(angle * Float64(b - a_m))) * b) * pi);
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	t_0 = 2.0 * ((b ^ 2.0) - (a_m ^ 2.0));
	tmp = 0.0;
	if (t_0 <= -2e+162)
		tmp = -2.0 * ((0.005555555555555556 * (a_m * (angle * pi))) * a_m);
	elseif (t_0 <= 5e+118)
		tmp = 0.011111111111111112 * ((angle * ((b - a_m) * (b + a_m))) * pi);
	else
		tmp = ((0.011111111111111112 * (angle * (b - a_m))) * b) * pi;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+162], N[(-2.0 * N[(N[(0.005555555555555556 * N[(a$95$m * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+118], N[(0.011111111111111112 * N[(N[(angle * N[(N[(b - a$95$m), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.011111111111111112 * N[(angle * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * Pi), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -1.9999999999999999e162

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. cos-fabs-rev N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\left|\pi \cdot \frac{angle}{180}\right|\right)} \]

      3. cos-neg-rev N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left|\pi \cdot \frac{angle}{180}\right|\right)\right)} \]

      4. sin-+PI/2-rev N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left|\pi \cdot \frac{angle}{180}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left|\pi \cdot \frac{angle}{180}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\left|\pi \cdot \frac{angle}{180}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\left|\color{blue}{\pi \cdot \frac{angle}{180}}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\left|\pi \cdot \color{blue}{\frac{angle}{180}}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\left|\color{blue}{\frac{\pi \cdot angle}{180}}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      10. fabs-div N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\frac{\left|\pi \cdot angle\right|}{\left|180\right|}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\frac{\left|\pi \cdot angle\right|}{\color{blue}{180}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      12. distribute-neg-frac2 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left|\pi \cdot angle\right|}{\mathsf{neg}\left(180\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left|\pi \cdot angle\right|}{\mathsf{neg}\left(180\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      14. lower-fabs.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left|\pi \cdot angle\right|}}{\mathsf{neg}\left(180\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|\color{blue}{angle \cdot \pi}\right|}{\mathsf{neg}\left(180\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|\color{blue}{angle \cdot \pi}\right|}{\mathsf{neg}\left(180\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      17. metadata-eval N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{\color{blue}{-180}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      18. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{\color{blue}{\pi}}{2}\right) \]

      19. mult-flip N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \color{blue}{\pi \cdot \frac{1}{2}}\right) \]

      20. *-commutative N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \color{blue}{\frac{1}{2} \cdot \pi}\right) \]

      21. lower-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \color{blue}{\frac{1}{2} \cdot \pi}\right) \]

      22. metadata-eval 53.0

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \color{blue}{0.5} \cdot \pi\right) \]

   3. Applied rewrites 53.0%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + 0.5 \cdot \pi\right)} \]

   4. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{-1}{180} \cdot \left|angle \cdot \mathsf{PI}\left(\right)\right| + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left({a}^{2} \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{-1}{180} \cdot \left|angle \cdot \mathsf{PI}\left(\right)\right| + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left({a}^{2} \cdot \color{blue}{\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{-1}{180} \cdot \left|angle \cdot \mathsf{PI}\left(\right)\right| + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto -2 \cdot \left({a}^{2} \cdot \left(\color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sin \left(\frac{-1}{180} \cdot \left|angle \cdot \mathsf{PI}\left(\right)\right| + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left({a}^{2} \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{-1}{180} \cdot \left|angle \cdot \mathsf{PI}\left(\right)\right| + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      5. lower-sin.f64 N/A

         \[\leadsto -2 \cdot \left({a}^{2} \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{-1}{180} \cdot \left|angle \cdot \mathsf{PI}\left(\right)\right| + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left({a}^{2} \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\color{blue}{\frac{-1}{180} \cdot \left|angle \cdot \mathsf{PI}\left(\right)\right|} + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left({a}^{2} \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{-1}{180} \cdot \color{blue}{\left|angle \cdot \mathsf{PI}\left(\right)\right|} + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. lower-PI.f64 N/A

         \[\leadsto -2 \cdot \left({a}^{2} \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{-1}{180} \cdot \left|angle \cdot \mathsf{PI}\left(\right)\right| + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   6. Applied rewrites 34.4%

      \[\leadsto \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(-0.005555555555555556, \left|angle \cdot \pi\right|, 0.5 \cdot \pi\right)\right)\right)\right)} \]

   8. Applied rewrites 39.8%

      \[\leadsto -2 \cdot \left(\left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right) \cdot a\right) \cdot \color{blue}{a}\right) \]

   9. Taylor expanded in angle around 0

      \[\leadsto -2 \cdot \left(\left(\frac{1}{180} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot a\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -2 \cdot \left(\left(\frac{1}{180} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot a\right) \]

       2. lower-*.f64 N/A

          \[\leadsto -2 \cdot \left(\left(\frac{1}{180} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot a\right) \]

       3. lower-*.f64 N/A

          \[\leadsto -2 \cdot \left(\left(\frac{1}{180} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot a\right) \]

       4. lower-PI.f64 37.2

          \[\leadsto -2 \cdot \left(\left(0.005555555555555556 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right) \cdot a\right) \]

   11. Applied rewrites 37.2%

       \[\leadsto -2 \cdot \left(\left(0.005555555555555556 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right) \cdot a\right) \]

   if -1.9999999999999999e162 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 4.99999999999999972e118

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      17. lower-*.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      20. lower-+.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

   if 4.99999999999999972e118 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      17. lower-*.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      20. lower-+.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

   7. Taylor expanded in a around 0

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 38.0%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \color{blue}{\pi}\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right)\right) \cdot \color{blue}{\pi} \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right)\right) \cdot \color{blue}{\pi} \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right)\right) \cdot \pi \]

       6. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right)\right) \cdot \pi \]

       7. associate-*r* N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot b\right)\right) \cdot \pi \]

       8. +-commutative N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot b\right)\right) \cdot \pi \]

       9. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b\right) \cdot \pi \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b\right) \cdot \pi \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b\right) \cdot \pi \]

       12. lower-*.f64 41.8

           \[\leadsto \left(\left(0.011111111111111112 \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b\right) \cdot \pi \]

       13. +-commutative 41.8

           \[\leadsto \left(\left(0.011111111111111112 \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b\right) \cdot \pi \]

   11. Applied rewrites 41.8%

       \[\leadsto \left(\left(0.011111111111111112 \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b\right) \cdot \color{blue}{\pi} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 61.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a\_m}^{2}\right) \leq 10^{-288}:\\ \;\;\;\;-2 \cdot \left(\left(0.005555555555555556 \cdot \left(a\_m \cdot \left(angle \cdot \pi\right)\right)\right) \cdot a\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle \cdot b\right) \cdot \left(b - a\_m\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= (* 2.0 (- (pow b 2.0) (pow a_m 2.0))) 1e-288)
   (* -2.0 (* (* 0.005555555555555556 (* a_m (* angle PI))) a_m))
   (* (* (* angle b) (- b a_m)) (* PI 0.011111111111111112))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a_m, 2.0))) <= 1e-288) {
		tmp = -2.0 * ((0.005555555555555556 * (a_m * (angle * ((double) M_PI)))) * a_m);
	} else {
		tmp = ((angle * b) * (b - a_m)) * (((double) M_PI) * 0.011111111111111112);
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if ((2.0 * (Math.pow(b, 2.0) - Math.pow(a_m, 2.0))) <= 1e-288) {
		tmp = -2.0 * ((0.005555555555555556 * (a_m * (angle * Math.PI))) * a_m);
	} else {
		tmp = ((angle * b) * (b - a_m)) * (Math.PI * 0.011111111111111112);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if (2.0 * (math.pow(b, 2.0) - math.pow(a_m, 2.0))) <= 1e-288:
		tmp = -2.0 * ((0.005555555555555556 * (a_m * (angle * math.pi))) * a_m)
	else:
		tmp = ((angle * b) * (b - a_m)) * (math.pi * 0.011111111111111112)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a_m ^ 2.0))) <= 1e-288)
		tmp = Float64(-2.0 * Float64(Float64(0.005555555555555556 * Float64(a_m * Float64(angle * pi))) * a_m));
	else
		tmp = Float64(Float64(Float64(angle * b) * Float64(b - a_m)) * Float64(pi * 0.011111111111111112));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if ((2.0 * ((b ^ 2.0) - (a_m ^ 2.0))) <= 1e-288)
		tmp = -2.0 * ((0.005555555555555556 * (a_m * (angle * pi))) * a_m);
	else
		tmp = ((angle * b) * (b - a_m)) * (pi * 0.011111111111111112);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-288], N[(-2.0 * N[(N[(0.005555555555555556 * N[(a$95$m * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(angle * b), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 1.00000000000000006e-288

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. cos-fabs-rev N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\left|\pi \cdot \frac{angle}{180}\right|\right)} \]

      3. cos-neg-rev N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left|\pi \cdot \frac{angle}{180}\right|\right)\right)} \]

      4. sin-+PI/2-rev N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left|\pi \cdot \frac{angle}{180}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left|\pi \cdot \frac{angle}{180}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\left|\pi \cdot \frac{angle}{180}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\left|\color{blue}{\pi \cdot \frac{angle}{180}}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\left|\pi \cdot \color{blue}{\frac{angle}{180}}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\left|\color{blue}{\frac{\pi \cdot angle}{180}}\right|\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      10. fabs-div N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\frac{\left|\pi \cdot angle\right|}{\left|180\right|}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(\mathsf{neg}\left(\frac{\left|\pi \cdot angle\right|}{\color{blue}{180}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      12. distribute-neg-frac2 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left|\pi \cdot angle\right|}{\mathsf{neg}\left(180\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\color{blue}{\frac{\left|\pi \cdot angle\right|}{\mathsf{neg}\left(180\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      14. lower-fabs.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\color{blue}{\left|\pi \cdot angle\right|}}{\mathsf{neg}\left(180\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|\color{blue}{angle \cdot \pi}\right|}{\mathsf{neg}\left(180\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|\color{blue}{angle \cdot \pi}\right|}{\mathsf{neg}\left(180\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      17. metadata-eval N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{\color{blue}{-180}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      18. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \frac{\color{blue}{\pi}}{2}\right) \]

      19. mult-flip N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \color{blue}{\pi \cdot \frac{1}{2}}\right) \]

      20. *-commutative N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \color{blue}{\frac{1}{2} \cdot \pi}\right) \]

      21. lower-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \color{blue}{\frac{1}{2} \cdot \pi}\right) \]

      22. metadata-eval 53.0

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + \color{blue}{0.5} \cdot \pi\right) \]

   3. Applied rewrites 53.0%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\frac{\left|angle \cdot \pi\right|}{-180} + 0.5 \cdot \pi\right)} \]

   4. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{-1}{180} \cdot \left|angle \cdot \mathsf{PI}\left(\right)\right| + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left({a}^{2} \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{-1}{180} \cdot \left|angle \cdot \mathsf{PI}\left(\right)\right| + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left({a}^{2} \cdot \color{blue}{\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{-1}{180} \cdot \left|angle \cdot \mathsf{PI}\left(\right)\right| + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto -2 \cdot \left({a}^{2} \cdot \left(\color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sin \left(\frac{-1}{180} \cdot \left|angle \cdot \mathsf{PI}\left(\right)\right| + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left({a}^{2} \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{-1}{180} \cdot \left|angle \cdot \mathsf{PI}\left(\right)\right| + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      5. lower-sin.f64 N/A

         \[\leadsto -2 \cdot \left({a}^{2} \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{-1}{180} \cdot \left|angle \cdot \mathsf{PI}\left(\right)\right| + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left({a}^{2} \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\color{blue}{\frac{-1}{180} \cdot \left|angle \cdot \mathsf{PI}\left(\right)\right|} + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left({a}^{2} \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{-1}{180} \cdot \color{blue}{\left|angle \cdot \mathsf{PI}\left(\right)\right|} + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. lower-PI.f64 N/A

         \[\leadsto -2 \cdot \left({a}^{2} \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{-1}{180} \cdot \left|angle \cdot \mathsf{PI}\left(\right)\right| + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   6. Applied rewrites 34.4%

      \[\leadsto \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(-0.005555555555555556, \left|angle \cdot \pi\right|, 0.5 \cdot \pi\right)\right)\right)\right)} \]

   8. Applied rewrites 39.8%

      \[\leadsto -2 \cdot \left(\left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right) \cdot a\right) \cdot \color{blue}{a}\right) \]

   9. Taylor expanded in angle around 0

      \[\leadsto -2 \cdot \left(\left(\frac{1}{180} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot a\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -2 \cdot \left(\left(\frac{1}{180} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot a\right) \]

       2. lower-*.f64 N/A

          \[\leadsto -2 \cdot \left(\left(\frac{1}{180} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot a\right) \]

       3. lower-*.f64 N/A

          \[\leadsto -2 \cdot \left(\left(\frac{1}{180} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot a\right) \]

       4. lower-PI.f64 37.2

          \[\leadsto -2 \cdot \left(\left(0.005555555555555556 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right) \cdot a\right) \]

   11. Applied rewrites 37.2%

       \[\leadsto -2 \cdot \left(\left(0.005555555555555556 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right) \cdot a\right) \]

   if 1.00000000000000006e-288 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      17. lower-*.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      20. lower-+.f64 54.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

   7. Taylor expanded in a around 0

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 38.0%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right) \cdot \color{blue}{\frac{1}{90}} \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right) \cdot \frac{1}{90} \]

       4. associate-*l* N/A

          \[\leadsto \left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{90}\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{90}\right)} \]

       6. lift-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \left(\color{blue}{\pi} \cdot \frac{1}{90}\right) \]

       7. lift-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \left(\pi \cdot \frac{1}{90}\right) \]

       8. *-commutative N/A

          \[\leadsto \left(angle \cdot \left(b \cdot \left(b - a\right)\right)\right) \cdot \left(\pi \cdot \frac{1}{90}\right) \]

       9. +-commutative N/A

          \[\leadsto \left(angle \cdot \left(b \cdot \left(b - a\right)\right)\right) \cdot \left(\pi \cdot \frac{1}{90}\right) \]

       10. associate-*r* N/A

           \[\leadsto \left(\left(angle \cdot b\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{\pi} \cdot \frac{1}{90}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\left(angle \cdot b\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{\pi} \cdot \frac{1}{90}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \left(\left(angle \cdot b\right) \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \frac{1}{90}\right) \]

       13. +-commutative N/A

           \[\leadsto \left(\left(angle \cdot b\right) \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \frac{1}{90}\right) \]

       14. +-commutative N/A

           \[\leadsto \left(\left(angle \cdot b\right) \cdot \left(b - a\right)\right) \cdot \mathsf{Rewrite=>}\left(lower-*.f64, \left(\pi \cdot \frac{1}{90}\right)\right) \]

   11. Applied rewrites 40.6%

       \[\leadsto \left(\left(angle \cdot b\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(\pi \cdot 0.011111111111111112\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 41.8% accurate, 7.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \left(\left(0.011111111111111112 \cdot \left(angle \cdot \left(b - a\_m\right)\right)\right) \cdot b\right) \cdot \pi \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (* (* (* 0.011111111111111112 (* angle (- b a_m))) b) PI))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	return ((0.011111111111111112 * (angle * (b - a_m))) * b) * ((double) M_PI);
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	return ((0.011111111111111112 * (angle * (b - a_m))) * b) * Math.PI;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	return ((0.011111111111111112 * (angle * (b - a_m))) * b) * math.pi

Julia

a_m = abs(a)
function code(a_m, b, angle)
	return Float64(Float64(Float64(0.011111111111111112 * Float64(angle * Float64(b - a_m))) * b) * pi)
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b, angle)
	tmp = ((0.011111111111111112 * (angle * (b - a_m))) * b) * pi;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := N[(N[(N[(0.011111111111111112 * N[(angle * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * Pi), $MachinePrecision]

Derivation

1. Initial program 53.7%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   4. lower-PI.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

   5. lower--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

   7. lower-pow.f64 50.3

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

4. Applied rewrites 50.3%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   6. lower-*.f64 50.3

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   7. lift--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   9. unpow2 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

   12. difference-of-squares-rev N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   13. +-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   14. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   15. lift--.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   16. *-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   17. lower-*.f64 54.2

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   18. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   19. +-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   20. lower-+.f64 54.2

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

6. Applied rewrites 54.2%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

7. Taylor expanded in a around 0

   \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right) \]
8. Step-by-step derivation

9. Applied rewrites 38.0%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right) \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right)} \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \color{blue}{\pi}\right) \]

    3. associate-*r* N/A

       \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right)\right) \cdot \color{blue}{\pi} \]

    4. lower-*.f64 N/A

       \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right)\right) \cdot \color{blue}{\pi} \]

    5. lift-*.f64 N/A

       \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right)\right) \cdot \pi \]

    6. lift-*.f64 N/A

       \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right)\right) \cdot \pi \]

    7. associate-*r* N/A

       \[\leadsto \left(\frac{1}{90} \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot b\right)\right) \cdot \pi \]

    8. +-commutative N/A

       \[\leadsto \left(\frac{1}{90} \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot b\right)\right) \cdot \pi \]

    9. associate-*r* N/A

       \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b\right) \cdot \pi \]

    10. lower-*.f64 N/A

        \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b\right) \cdot \pi \]

    11. lower-*.f64 N/A

        \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b\right) \cdot \pi \]

    12. lower-*.f64 41.8

        \[\leadsto \left(\left(0.011111111111111112 \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b\right) \cdot \pi \]

    13. +-commutative 41.8

        \[\leadsto \left(\left(0.011111111111111112 \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b\right) \cdot \pi \]

11. Applied rewrites 41.8%

    \[\leadsto \left(\left(0.011111111111111112 \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b\right) \cdot \color{blue}{\pi} \]
12. Add Preprocessing

Alternative 22: 41.8% accurate, 7.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b - a\_m\right)\right) \cdot \left(b \cdot \pi\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (* 0.011111111111111112 (* (* angle (- b a_m)) (* b PI))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	return 0.011111111111111112 * ((angle * (b - a_m)) * (b * ((double) M_PI)));
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	return 0.011111111111111112 * ((angle * (b - a_m)) * (b * Math.PI));
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	return 0.011111111111111112 * ((angle * (b - a_m)) * (b * math.pi))

Julia

a_m = abs(a)
function code(a_m, b, angle)
	return Float64(0.011111111111111112 * Float64(Float64(angle * Float64(b - a_m)) * Float64(b * pi)))
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b, angle)
	tmp = 0.011111111111111112 * ((angle * (b - a_m)) * (b * pi));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := N[(0.011111111111111112 * N[(N[(angle * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.7%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   4. lower-PI.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

   5. lower--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

   7. lower-pow.f64 50.3

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

4. Applied rewrites 50.3%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   6. lower-*.f64 50.3

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   7. lift--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   9. unpow2 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

   12. difference-of-squares-rev N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   13. +-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   14. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   15. lift--.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   16. *-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   17. lower-*.f64 54.2

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   18. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   19. +-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   20. lower-+.f64 54.2

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

6. Applied rewrites 54.2%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

7. Taylor expanded in a around 0

   \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right) \]
8. Step-by-step derivation

9. Applied rewrites 38.0%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right) \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \color{blue}{\pi}\right) \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right) \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right) \]

    4. associate-*r* N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(\left(angle \cdot \left(b - a\right)\right) \cdot b\right) \cdot \pi\right) \]

    5. +-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(\left(angle \cdot \left(b - a\right)\right) \cdot b\right) \cdot \pi\right) \]

    6. associate-*l* N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(b \cdot \pi\right)}\right) \]

    7. lower-*.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(b \cdot \pi\right)}\right) \]

    8. lower-*.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{b} \cdot \pi\right)\right) \]

    9. lower-*.f64 41.8

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(b \cdot \color{blue}{\pi}\right)\right) \]

    10. +-commutative 41.8

        \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(b \cdot \pi\right)\right) \]

11. Applied rewrites 41.8%

    \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(b \cdot \pi\right)}\right) \]
12. Add Preprocessing

Alternative 23: 38.0% accurate, 7.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\_m\right) \cdot b\right)\right) \cdot \pi\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (* 0.011111111111111112 (* (* angle (* (- b a_m) b)) PI)))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	return 0.011111111111111112 * ((angle * ((b - a_m) * b)) * ((double) M_PI));
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	return 0.011111111111111112 * ((angle * ((b - a_m) * b)) * Math.PI);
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	return 0.011111111111111112 * ((angle * ((b - a_m) * b)) * math.pi)

Julia

a_m = abs(a)
function code(a_m, b, angle)
	return Float64(0.011111111111111112 * Float64(Float64(angle * Float64(Float64(b - a_m) * b)) * pi))
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b, angle)
	tmp = 0.011111111111111112 * ((angle * ((b - a_m) * b)) * pi);
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := N[(0.011111111111111112 * N[(N[(angle * N[(N[(b - a$95$m), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.7%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   4. lower-PI.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

   5. lower--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

   7. lower-pow.f64 50.3

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

4. Applied rewrites 50.3%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   6. lower-*.f64 50.3

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   7. lift--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   9. unpow2 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

   12. difference-of-squares-rev N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   13. +-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   14. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   15. lift--.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   16. *-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   17. lower-*.f64 54.2

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   18. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   19. +-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   20. lower-+.f64 54.2

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

6. Applied rewrites 54.2%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

7. Taylor expanded in a around 0

   \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right) \]
8. Step-by-step derivation

9. Applied rewrites 38.0%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(FPCore (a b angle)
  :name "ab-angle->ABCF B"
  :precision binary64
  (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* PI (/ angle 180.0)))) (cos (* PI (/ angle 180.0)))))