ab-angle->ABCF B

Percentage Accurate: 53.8% → 67.3%

Time: 7.0s

Alternatives: 23

Speedup: 6.6×

• 66.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 53.8% accurate, 1.0× speedup

Math

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

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, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \left(0.005555555555555556 \cdot \pi\right) \cdot \left|angle\right|\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 1.1 \cdot 10^{+136}:\\ \;\;\;\;\left(\left(\sin \left(\frac{\pi \cdot \left|angle\right|}{180}\right) \cdot \left(2 \cdot \left(b - a\right)\right)\right) \cdot \left(a + b\right)\right) \cdot \cos t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin t\_0\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 PI) (fabs angle))))
   (*
    (copysign 1.0 angle)
    (if (<= (fabs angle) 1.1e+136)
      (*
       (* (* (sin (/ (* PI (fabs angle)) 180.0)) (* 2.0 (- b a))) (+ a b))
       (cos t_0))
      (* (* (* 2.0 (* (- b a) (+ b a))) (sin t_0)) 1.0)))))

C

double code(double a, double b, double angle) {
	double t_0 = (0.005555555555555556 * ((double) M_PI)) * fabs(angle);
	double tmp;
	if (fabs(angle) <= 1.1e+136) {
		tmp = ((sin(((((double) M_PI) * fabs(angle)) / 180.0)) * (2.0 * (b - a))) * (a + b)) * cos(t_0);
	} else {
		tmp = ((2.0 * ((b - a) * (b + a))) * sin(t_0)) * 1.0;
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (0.005555555555555556 * Math.PI) * Math.abs(angle);
	double tmp;
	if (Math.abs(angle) <= 1.1e+136) {
		tmp = ((Math.sin(((Math.PI * Math.abs(angle)) / 180.0)) * (2.0 * (b - a))) * (a + b)) * Math.cos(t_0);
	} else {
		tmp = ((2.0 * ((b - a) * (b + a))) * Math.sin(t_0)) * 1.0;
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = (0.005555555555555556 * math.pi) * math.fabs(angle)
	tmp = 0
	if math.fabs(angle) <= 1.1e+136:
		tmp = ((math.sin(((math.pi * math.fabs(angle)) / 180.0)) * (2.0 * (b - a))) * (a + b)) * math.cos(t_0)
	else:
		tmp = ((2.0 * ((b - a) * (b + a))) * math.sin(t_0)) * 1.0
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(0.005555555555555556 * pi) * abs(angle))
	tmp = 0.0
	if (abs(angle) <= 1.1e+136)
		tmp = Float64(Float64(Float64(sin(Float64(Float64(pi * abs(angle)) / 180.0)) * Float64(2.0 * Float64(b - a))) * Float64(a + b)) * cos(t_0));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(b - a) * Float64(b + a))) * sin(t_0)) * 1.0);
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = (0.005555555555555556 * pi) * abs(angle);
	tmp = 0.0;
	if (abs(angle) <= 1.1e+136)
		tmp = ((sin(((pi * abs(angle)) / 180.0)) * (2.0 * (b - a))) * (a + b)) * cos(t_0);
	else
		tmp = ((2.0 * ((b - a) * (b + a))) * sin(t_0)) * 1.0;
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(0.005555555555555556 * Pi), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 1.1e+136], N[(N[(N[(N[Sin[N[(N[(Pi * N[Abs[angle], $MachinePrecision]), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 1.1e136

   1. Initial program 53.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 53.9%

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

   3. Applied rewrites 53.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 53.9%

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

   5. Applied rewrites 53.9%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. unpow2 N/A

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

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

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 57.9%

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 57.9%

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

   7. Applied rewrites 57.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 67.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. metadata-eval N/A

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

       5. mult-flip N/A

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

       6. *-commutative N/A

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

       7. associate-*r/ N/A

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

       8. lift-*.f64 N/A

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

       9. lower-/.f64 67.1%

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

   11. Applied rewrites 67.1%

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

   if 1.1e136 < angle

   1. Initial program 53.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 53.9%

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

   3. Applied rewrites 53.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 53.9%

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

   5. Applied rewrites 53.9%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. unpow2 N/A

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

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

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 57.9%

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 57.9%

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

   7. Applied rewrites 57.9%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right)\right) \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 56.5%

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

Alternative 2: 67.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|angle\right|}{180}\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 9 \cdot 10^{+94}:\\ \;\;\;\;\left(\left(\sin \left(t\_0 \cdot \pi\right) \cdot \left(2 \cdot \left(\left|b\right| - a\right)\right)\right) \cdot \left(a + \left|b\right|\right)\right) \cdot \cos \left(\left(0.005555555555555556 \cdot \pi\right) \cdot \left|angle\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \mathsf{fma}\left(-a, a, \left|b\right| \cdot \left|b\right|\right)\right) \cdot \sin \left(\pi \cdot t\_0\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (/ (fabs angle) 180.0)))
   (*
    (copysign 1.0 angle)
    (if (<= (fabs angle) 9e+94)
      (*
       (* (* (sin (* t_0 PI)) (* 2.0 (- (fabs b) a))) (+ a (fabs b)))
       (cos (* (* 0.005555555555555556 PI) (fabs angle))))
      (*
       (* (* 2.0 (fma (- a) a (* (fabs b) (fabs b)))) (sin (* PI t_0)))
       1.0)))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(angle) / 180.0;
	double tmp;
	if (fabs(angle) <= 9e+94) {
		tmp = ((sin((t_0 * ((double) M_PI))) * (2.0 * (fabs(b) - a))) * (a + fabs(b))) * cos(((0.005555555555555556 * ((double) M_PI)) * fabs(angle)));
	} else {
		tmp = ((2.0 * fma(-a, a, (fabs(b) * fabs(b)))) * sin((((double) M_PI) * t_0))) * 1.0;
	}
	return copysign(1.0, angle) * tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(abs(angle) / 180.0)
	tmp = 0.0
	if (abs(angle) <= 9e+94)
		tmp = Float64(Float64(Float64(sin(Float64(t_0 * pi)) * Float64(2.0 * Float64(abs(b) - a))) * Float64(a + abs(b))) * cos(Float64(Float64(0.005555555555555556 * pi) * abs(angle))));
	else
		tmp = Float64(Float64(Float64(2.0 * fma(Float64(-a), a, Float64(abs(b) * abs(b)))) * sin(Float64(pi * t_0))) * 1.0);
	end
	return Float64(copysign(1.0, angle) * tmp)
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[angle], $MachinePrecision] / 180.0), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 9e+94], N[(N[(N[(N[Sin[N[(t$95$0 * Pi), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(N[Abs[b], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a + N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(0.005555555555555556 * Pi), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[((-a) * a + N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 8.99999999999999944e94

   1. Initial program 53.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 53.9%

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

   3. Applied rewrites 53.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 53.9%

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

   5. Applied rewrites 53.9%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. unpow2 N/A

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

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

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 57.9%

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 57.9%

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

   7. Applied rewrites 57.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 67.3%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. metadata-eval N/A

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

       4. mult-flip N/A

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

       5. lower-/.f64 67.2%

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

   11. Applied rewrites 67.2%

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

   if 8.99999999999999944e94 < angle

   1. Initial program 53.8%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 56.4%

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 56.4%

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

   3. Applied rewrites 56.4%

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

   4. Taylor expanded in angle around 0

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

   6. Applied rewrites 55.2%

      \[\leadsto \left(\left(2 \cdot \mathsf{fma}\left(-a, a, b \cdot b\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 67.3% accurate, 1.1× speedup

Math

\[\mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 2.25 \cdot 10^{+136}:\\ \;\;\;\;\left(\cos \left(\left(\pi \cdot \left|angle\right|\right) \cdot -0.005555555555555556\right) \cdot \left(\left(\sin \left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left|angle\right|\right) \cdot \left(b - a\right)\right) \cdot 2\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\left(0.005555555555555556 \cdot \pi\right) \cdot \left|angle\right|\right)\right) \cdot 1\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (*
  (copysign 1.0 angle)
  (if (<= (fabs angle) 2.25e+136)
    (*
     (*
      (cos (* (* PI (fabs angle)) -0.005555555555555556))
      (* (* (sin (* (* PI 0.005555555555555556) (fabs angle))) (- b a)) 2.0))
     (+ a b))
    (*
     (*
      (* 2.0 (* (- b a) (+ b a)))
      (sin (* (* 0.005555555555555556 PI) (fabs angle))))
     1.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(angle) <= 2.25e+136) {
		tmp = (cos(((((double) M_PI) * fabs(angle)) * -0.005555555555555556)) * ((sin(((((double) M_PI) * 0.005555555555555556) * fabs(angle))) * (b - a)) * 2.0)) * (a + b);
	} else {
		tmp = ((2.0 * ((b - a) * (b + a))) * sin(((0.005555555555555556 * ((double) M_PI)) * fabs(angle)))) * 1.0;
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.abs(angle) <= 2.25e+136) {
		tmp = (Math.cos(((Math.PI * Math.abs(angle)) * -0.005555555555555556)) * ((Math.sin(((Math.PI * 0.005555555555555556) * Math.abs(angle))) * (b - a)) * 2.0)) * (a + b);
	} else {
		tmp = ((2.0 * ((b - a) * (b + a))) * Math.sin(((0.005555555555555556 * Math.PI) * Math.abs(angle)))) * 1.0;
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.fabs(angle) <= 2.25e+136:
		tmp = (math.cos(((math.pi * math.fabs(angle)) * -0.005555555555555556)) * ((math.sin(((math.pi * 0.005555555555555556) * math.fabs(angle))) * (b - a)) * 2.0)) * (a + b)
	else:
		tmp = ((2.0 * ((b - a) * (b + a))) * math.sin(((0.005555555555555556 * math.pi) * math.fabs(angle)))) * 1.0
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(angle) <= 2.25e+136)
		tmp = Float64(Float64(cos(Float64(Float64(pi * abs(angle)) * -0.005555555555555556)) * Float64(Float64(sin(Float64(Float64(pi * 0.005555555555555556) * abs(angle))) * Float64(b - a)) * 2.0)) * Float64(a + b));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(b - a) * Float64(b + a))) * sin(Float64(Float64(0.005555555555555556 * pi) * abs(angle)))) * 1.0);
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (abs(angle) <= 2.25e+136)
		tmp = (cos(((pi * abs(angle)) * -0.005555555555555556)) * ((sin(((pi * 0.005555555555555556) * abs(angle))) * (b - a)) * 2.0)) * (a + b);
	else
		tmp = ((2.0 * ((b - a) * (b + a))) * sin(((0.005555555555555556 * pi) * abs(angle)))) * 1.0;
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 2.25e+136], N[(N[(N[Cos[N[(N[(Pi * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sin[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(0.005555555555555556 * Pi), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 2.25e136

   1. Initial program 53.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 53.9%

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

   3. Applied rewrites 53.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 53.9%

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

   5. Applied rewrites 53.9%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. unpow2 N/A

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

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

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 57.9%

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 57.9%

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

   7. Applied rewrites 57.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 67.3%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. +-commutative N/A

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

       6. associate-*r* N/A

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

       7. lower-*.f64 N/A

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

   11. Applied rewrites 67.1%

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

   if 2.25e136 < angle

   1. Initial program 53.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 53.9%

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

   3. Applied rewrites 53.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 53.9%

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

   5. Applied rewrites 53.9%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. unpow2 N/A

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

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

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 57.9%

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 57.9%

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

   7. Applied rewrites 57.9%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right)\right) \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 56.5%

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

Alternative 4: 67.3% accurate, 1.8× speedup

Math

\[\mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 2.25 \cdot 10^{+136}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\sin \left(0.011111111111111112 \cdot \left(\pi \cdot \left|angle\right|\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\left(0.005555555555555556 \cdot \pi\right) \cdot \left|angle\right|\right)\right) \cdot 1\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (*
  (copysign 1.0 angle)
  (if (<= (fabs angle) 2.25e+136)
    (* (+ b a) (* (sin (* 0.011111111111111112 (* PI (fabs angle)))) (- b a)))
    (*
     (*
      (* 2.0 (* (- b a) (+ b a)))
      (sin (* (* 0.005555555555555556 PI) (fabs angle))))
     1.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(angle) <= 2.25e+136) {
		tmp = (b + a) * (sin((0.011111111111111112 * (((double) M_PI) * fabs(angle)))) * (b - a));
	} else {
		tmp = ((2.0 * ((b - a) * (b + a))) * sin(((0.005555555555555556 * ((double) M_PI)) * fabs(angle)))) * 1.0;
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.abs(angle) <= 2.25e+136) {
		tmp = (b + a) * (Math.sin((0.011111111111111112 * (Math.PI * Math.abs(angle)))) * (b - a));
	} else {
		tmp = ((2.0 * ((b - a) * (b + a))) * Math.sin(((0.005555555555555556 * Math.PI) * Math.abs(angle)))) * 1.0;
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.fabs(angle) <= 2.25e+136:
		tmp = (b + a) * (math.sin((0.011111111111111112 * (math.pi * math.fabs(angle)))) * (b - a))
	else:
		tmp = ((2.0 * ((b - a) * (b + a))) * math.sin(((0.005555555555555556 * math.pi) * math.fabs(angle)))) * 1.0
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(angle) <= 2.25e+136)
		tmp = Float64(Float64(b + a) * Float64(sin(Float64(0.011111111111111112 * Float64(pi * abs(angle)))) * Float64(b - a)));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(b - a) * Float64(b + a))) * sin(Float64(Float64(0.005555555555555556 * pi) * abs(angle)))) * 1.0);
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (abs(angle) <= 2.25e+136)
		tmp = (b + a) * (sin((0.011111111111111112 * (pi * abs(angle)))) * (b - a));
	else
		tmp = ((2.0 * ((b - a) * (b + a))) * sin(((0.005555555555555556 * pi) * abs(angle)))) * 1.0;
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 2.25e+136], N[(N[(b + a), $MachinePrecision] * N[(N[Sin[N[(0.011111111111111112 * N[(Pi * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(0.005555555555555556 * Pi), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 2.25e136

   1. Initial program 53.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 67.2%

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

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

      3. lower-+.f64 67.2%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 67.2%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 67.2%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.2%

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

   5. Applied rewrites 67.2%

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

   if 2.25e136 < angle

   1. Initial program 53.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 53.9%

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

   3. Applied rewrites 53.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 53.9%

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

   5. Applied rewrites 53.9%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. unpow2 N/A

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

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

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 57.9%

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 57.9%

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

   7. Applied rewrites 57.9%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right)\right) \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 56.5%

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

Alternative 5: 67.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left|angle\right|\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 6.5 \cdot 10^{+143}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\sin \left(0.011111111111111112 \cdot t\_0\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(0.005555555555555556 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (fabs angle))))
   (*
    (copysign 1.0 angle)
    (if (<= (fabs angle) 6.5e+143)
      (* (+ b a) (* (sin (* 0.011111111111111112 t_0)) (- b a)))
      (* (* 2.0 (* (+ a b) (- b a))) (sin (* 0.005555555555555556 t_0)))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * fabs(angle);
	double tmp;
	if (fabs(angle) <= 6.5e+143) {
		tmp = (b + a) * (sin((0.011111111111111112 * t_0)) * (b - a));
	} else {
		tmp = (2.0 * ((a + b) * (b - a))) * sin((0.005555555555555556 * t_0));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * Math.abs(angle);
	double tmp;
	if (Math.abs(angle) <= 6.5e+143) {
		tmp = (b + a) * (Math.sin((0.011111111111111112 * t_0)) * (b - a));
	} else {
		tmp = (2.0 * ((a + b) * (b - a))) * Math.sin((0.005555555555555556 * t_0));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pi * math.fabs(angle)
	tmp = 0
	if math.fabs(angle) <= 6.5e+143:
		tmp = (b + a) * (math.sin((0.011111111111111112 * t_0)) * (b - a))
	else:
		tmp = (2.0 * ((a + b) * (b - a))) * math.sin((0.005555555555555556 * t_0))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(pi * abs(angle))
	tmp = 0.0
	if (abs(angle) <= 6.5e+143)
		tmp = Float64(Float64(b + a) * Float64(sin(Float64(0.011111111111111112 * t_0)) * Float64(b - a)));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(a + b) * Float64(b - a))) * sin(Float64(0.005555555555555556 * t_0)));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = pi * abs(angle);
	tmp = 0.0;
	if (abs(angle) <= 6.5e+143)
		tmp = (b + a) * (sin((0.011111111111111112 * t_0)) * (b - a));
	else
		tmp = (2.0 * ((a + b) * (b - a))) * sin((0.005555555555555556 * t_0));
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[Abs[angle], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 6.5e+143], N[(N[(b + a), $MachinePrecision] * N[(N[Sin[N[(0.011111111111111112 * t$95$0), $MachinePrecision]], $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.005555555555555556 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 6.4999999999999997e143

   1. Initial program 53.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 67.2%

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

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

      3. lower-+.f64 67.2%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 67.2%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 67.2%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.2%

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

   5. Applied rewrites 67.2%

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

   if 6.4999999999999997e143 < angle

   1. Initial program 53.8%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 56.4%

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 56.4%

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

   3. Applied rewrites 56.4%

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

   5. Applied rewrites 57.8%

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

   6. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower--.f64 56.4%

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

   8. Applied rewrites 56.4%

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

Alternative 6: 67.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \frac{\left|angle\right|}{180}\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\sin \left(0.011111111111111112 \cdot \left(\pi \cdot \left|angle\right|\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\sin \left(\left(\left|angle\right| \cdot 0.011111111111111112\right) \cdot \pi\right) \cdot \left(b - a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ (fabs angle) 180.0))))
   (*
    (copysign 1.0 angle)
    (if (<=
         (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))
         2e-17)
      (*
       (+ b a)
       (* (sin (* 0.011111111111111112 (* PI (fabs angle)))) (- b a)))
      (*
       (+ b a)
       (* (sin (* (* (fabs angle) 0.011111111111111112) PI)) (- b a)))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (fabs(angle) / 180.0);
	double tmp;
	if ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0)) <= 2e-17) {
		tmp = (b + a) * (sin((0.011111111111111112 * (((double) M_PI) * fabs(angle)))) * (b - a));
	} else {
		tmp = (b + a) * (sin(((fabs(angle) * 0.011111111111111112) * ((double) M_PI))) * (b - a));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (Math.abs(angle) / 180.0);
	double tmp;
	if ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_0)) * Math.cos(t_0)) <= 2e-17) {
		tmp = (b + a) * (Math.sin((0.011111111111111112 * (Math.PI * Math.abs(angle)))) * (b - a));
	} else {
		tmp = (b + a) * (Math.sin(((Math.abs(angle) * 0.011111111111111112) * Math.PI)) * (b - a));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pi * (math.fabs(angle) / 180.0)
	tmp = 0
	if (((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin(t_0)) * math.cos(t_0)) <= 2e-17:
		tmp = (b + a) * (math.sin((0.011111111111111112 * (math.pi * math.fabs(angle)))) * (b - a))
	else:
		tmp = (b + a) * (math.sin(((math.fabs(angle) * 0.011111111111111112) * math.pi)) * (b - a))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(abs(angle) / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= 2e-17)
		tmp = Float64(Float64(b + a) * Float64(sin(Float64(0.011111111111111112 * Float64(pi * abs(angle)))) * Float64(b - a)));
	else
		tmp = Float64(Float64(b + a) * Float64(sin(Float64(Float64(abs(angle) * 0.011111111111111112) * pi)) * Float64(b - a)));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = pi * (abs(angle) / 180.0);
	tmp = 0.0;
	if ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= 2e-17)
		tmp = (b + a) * (sin((0.011111111111111112 * (pi * abs(angle)))) * (b - a));
	else
		tmp = (b + a) * (sin(((abs(angle) * 0.011111111111111112) * pi)) * (b - a));
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(N[Abs[angle], $MachinePrecision] / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[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], 2e-17], N[(N[(b + a), $MachinePrecision] * N[(N[Sin[N[(0.011111111111111112 * N[(Pi * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[Sin[N[(N[(N[Abs[angle], $MachinePrecision] * 0.011111111111111112), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < 2.00000000000000014e-17

   1. Initial program 53.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 67.2%

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

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

      3. lower-+.f64 67.2%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 67.2%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 67.2%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.2%

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

   5. Applied rewrites 67.2%

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

   if 2.00000000000000014e-17 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 53.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 67.2%

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

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

      3. lower-+.f64 67.2%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 67.2%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 67.2%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.2%

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

   5. Applied rewrites 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 67.4%

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

   7. Applied rewrites 67.4%

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

Alternative 7: 67.2% accurate, 1.9× speedup

Math

\[\mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 10^{-23}:\\ \;\;\;\;\left(\left(\left|angle\right| \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(\left(\left|angle\right| \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (*
  (copysign 1.0 angle)
  (if (<= (fabs angle) 1e-23)
    (* (* (* (fabs angle) (+ a b)) (- b a)) (* PI 0.011111111111111112))
    (*
     (* (- b a) (+ a b))
     (sin (* (* (fabs angle) PI) 0.011111111111111112))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(angle) <= 1e-23) {
		tmp = ((fabs(angle) * (a + b)) * (b - a)) * (((double) M_PI) * 0.011111111111111112);
	} else {
		tmp = ((b - a) * (a + b)) * sin(((fabs(angle) * ((double) M_PI)) * 0.011111111111111112));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.abs(angle) <= 1e-23) {
		tmp = ((Math.abs(angle) * (a + b)) * (b - a)) * (Math.PI * 0.011111111111111112);
	} else {
		tmp = ((b - a) * (a + b)) * Math.sin(((Math.abs(angle) * Math.PI) * 0.011111111111111112));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.fabs(angle) <= 1e-23:
		tmp = ((math.fabs(angle) * (a + b)) * (b - a)) * (math.pi * 0.011111111111111112)
	else:
		tmp = ((b - a) * (a + b)) * math.sin(((math.fabs(angle) * math.pi) * 0.011111111111111112))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(angle) <= 1e-23)
		tmp = Float64(Float64(Float64(abs(angle) * Float64(a + b)) * Float64(b - a)) * Float64(pi * 0.011111111111111112));
	else
		tmp = Float64(Float64(Float64(b - a) * Float64(a + b)) * sin(Float64(Float64(abs(angle) * pi) * 0.011111111111111112)));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (abs(angle) <= 1e-23)
		tmp = ((abs(angle) * (a + b)) * (b - a)) * (pi * 0.011111111111111112);
	else
		tmp = ((b - a) * (a + b)) * sin(((abs(angle) * pi) * 0.011111111111111112));
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 1e-23], N[(N[(N[(N[Abs[angle], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b - a), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 9.9999999999999996e-24

   1. Initial program 53.8%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

      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. 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. pow2 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 54.5%

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

      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 54.5%

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

   6. Applied rewrites 54.5%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
   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. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 62.0%

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

   8. Applied rewrites 62.0%

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

   if 9.9999999999999996e-24 < angle

   1. Initial program 53.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 57.8%

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

Alternative 8: 67.1% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (* (+ b a) (* (sin (* 0.011111111111111112 (* PI angle))) (- b a))))

C

double code(double a, double b, double angle) {
	return (b + a) * (sin((0.011111111111111112 * (((double) M_PI) * angle))) * (b - a));
}

Java

public static double code(double a, double b, double angle) {
	return (b + a) * (Math.sin((0.011111111111111112 * (Math.PI * angle))) * (b - a));
}

Python

def code(a, b, angle):
	return (b + a) * (math.sin((0.011111111111111112 * (math.pi * angle))) * (b - a))

Julia

function code(a, b, angle)
	return Float64(Float64(b + a) * Float64(sin(Float64(0.011111111111111112 * Float64(pi * angle))) * Float64(b - a)))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (b + a) * (sin((0.011111111111111112 * (pi * angle))) * (b - a));
end

Wolfram

code[a_, b_, angle_] := N[(N[(b + a), $MachinePrecision] * N[(N[Sin[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
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 67.2%

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

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

   3. lower-+.f64 67.2%

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 67.2%

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 67.2%

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 67.2%

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

5. Applied rewrites 67.2%

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

Alternative 9: 64.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_0 := b - \left|a\right|\\ \mathbf{if}\;\left|a\right| \leq 3.5 \cdot 10^{-139}:\\ \;\;\;\;b \cdot \left(\sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle \cdot \left(\left|a\right| + b\right)\right) \cdot t\_0\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (- b (fabs a))))
   (if (<= (fabs a) 3.5e-139)
     (* b (* (sin (* 0.011111111111111112 (* PI angle))) t_0))
     (* (* (* angle (+ (fabs a) b)) t_0) (* PI 0.011111111111111112)))))

C

double code(double a, double b, double angle) {
	double t_0 = b - fabs(a);
	double tmp;
	if (fabs(a) <= 3.5e-139) {
		tmp = b * (sin((0.011111111111111112 * (((double) M_PI) * angle))) * t_0);
	} else {
		tmp = ((angle * (fabs(a) + b)) * t_0) * (((double) M_PI) * 0.011111111111111112);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = b - Math.abs(a);
	double tmp;
	if (Math.abs(a) <= 3.5e-139) {
		tmp = b * (Math.sin((0.011111111111111112 * (Math.PI * angle))) * t_0);
	} else {
		tmp = ((angle * (Math.abs(a) + b)) * t_0) * (Math.PI * 0.011111111111111112);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = b - math.fabs(a)
	tmp = 0
	if math.fabs(a) <= 3.5e-139:
		tmp = b * (math.sin((0.011111111111111112 * (math.pi * angle))) * t_0)
	else:
		tmp = ((angle * (math.fabs(a) + b)) * t_0) * (math.pi * 0.011111111111111112)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(b - abs(a))
	tmp = 0.0
	if (abs(a) <= 3.5e-139)
		tmp = Float64(b * Float64(sin(Float64(0.011111111111111112 * Float64(pi * angle))) * t_0));
	else
		tmp = Float64(Float64(Float64(angle * Float64(abs(a) + b)) * t_0) * Float64(pi * 0.011111111111111112));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = b - abs(a);
	tmp = 0.0;
	if (abs(a) <= 3.5e-139)
		tmp = b * (sin((0.011111111111111112 * (pi * angle))) * t_0);
	else
		tmp = ((angle * (abs(a) + b)) * t_0) * (pi * 0.011111111111111112);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(b - N[Abs[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 3.5e-139], N[(b * N[(N[Sin[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(angle * N[(N[Abs[a], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 3.50000000000000001e-139

   1. Initial program 53.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 67.2%

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

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

      3. lower-+.f64 67.2%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 67.2%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 67.2%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.2%

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

   5. Applied rewrites 67.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 42.2%

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

   if 3.50000000000000001e-139 < a

   1. Initial program 53.8%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

      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. 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. pow2 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 54.5%

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

      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 54.5%

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

   6. Applied rewrites 54.5%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
   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. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 62.0%

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

   8. Applied rewrites 62.0%

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

Alternative 10: 63.8% accurate, 3.9× speedup

Math

\[\mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 0.0001:\\ \;\;\;\;\left(\left(\left|angle\right| \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left|angle\right| \cdot \left(\pi \cdot \mathsf{fma}\left(-a, a, b \cdot b\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (*
  (copysign 1.0 angle)
  (if (<= (fabs angle) 0.0001)
    (* (* (* (fabs angle) (+ a b)) (- b a)) (* PI 0.011111111111111112))
    (* 0.011111111111111112 (* (fabs angle) (* PI (fma (- a) a (* b b))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(angle) <= 0.0001) {
		tmp = ((fabs(angle) * (a + b)) * (b - a)) * (((double) M_PI) * 0.011111111111111112);
	} else {
		tmp = 0.011111111111111112 * (fabs(angle) * (((double) M_PI) * fma(-a, a, (b * b))));
	}
	return copysign(1.0, angle) * tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(angle) <= 0.0001)
		tmp = Float64(Float64(Float64(abs(angle) * Float64(a + b)) * Float64(b - a)) * Float64(pi * 0.011111111111111112));
	else
		tmp = Float64(0.011111111111111112 * Float64(abs(angle) * Float64(pi * fma(Float64(-a), a, Float64(b * b)))));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

Wolfram

code[a_, b_, angle_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 0.0001], N[(N[(N[(N[Abs[angle], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[Abs[angle], $MachinePrecision] * N[(Pi * N[((-a) * a + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 1.00000000000000005e-4

   1. Initial program 53.8%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

      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. 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. pow2 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 54.5%

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

      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 54.5%

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

   6. Applied rewrites 54.5%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
   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. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 62.0%

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

   8. Applied rewrites 62.0%

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

   if 1.00000000000000005e-4 < angle

   1. Initial program 53.8%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. lift-neg.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 53.4%

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 53.4%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(-a, a, b \cdot b\right)\right)\right) \]

   6. Applied rewrites 53.4%

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

Alternative 11: 62.1% accurate, 6.6× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (* (* (* angle (+ a b)) (- b a)) (* PI 0.011111111111111112)))

C

double code(double a, double b, double angle) {
	return ((angle * (a + b)) * (b - a)) * (((double) M_PI) * 0.011111111111111112);
}

Java

public static double code(double a, double b, double angle) {
	return ((angle * (a + b)) * (b - a)) * (Math.PI * 0.011111111111111112);
}

Python

def code(a, b, angle):
	return ((angle * (a + b)) * (b - a)) * (math.pi * 0.011111111111111112)

Julia

function code(a, b, angle)
	return Float64(Float64(Float64(angle * Float64(a + b)) * Float64(b - a)) * Float64(pi * 0.011111111111111112))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((angle * (a + b)) * (b - a)) * (pi * 0.011111111111111112);
end

Wolfram

code[a_, b_, angle_] := N[(N[(N[(angle * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

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

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
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 50.6%

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

4. Applied rewrites 50.6%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
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 50.6%

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

   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. 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. pow2 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 54.5%

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

   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 54.5%

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

6. Applied rewrites 54.5%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
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. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 62.0%

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

8. Applied rewrites 62.0%

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

Alternative 12: 62.0% accurate, 6.6× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (* (* (* 0.011111111111111112 (* (- b a) angle)) (+ a b)) PI))

C

double code(double a, double b, double angle) {
	return ((0.011111111111111112 * ((b - a) * angle)) * (a + b)) * ((double) M_PI);
}

Java

public static double code(double a, double b, double angle) {
	return ((0.011111111111111112 * ((b - a) * angle)) * (a + b)) * Math.PI;
}

Python

def code(a, b, angle):
	return ((0.011111111111111112 * ((b - a) * angle)) * (a + b)) * math.pi

Julia

function code(a, b, angle)
	return Float64(Float64(Float64(0.011111111111111112 * Float64(Float64(b - a) * angle)) * Float64(a + b)) * pi)
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((0.011111111111111112 * ((b - a) * angle)) * (a + b)) * pi;
end

Wolfram

code[a_, b_, angle_] := N[(N[(N[(0.011111111111111112 * N[(N[(b - a), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]

Derivation

1. Initial program 53.8%

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

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
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 50.6%

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

4. Applied rewrites 50.6%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
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 50.6%

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

   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. 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. pow2 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 54.5%

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

   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 54.5%

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

6. Applied rewrites 54.5%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
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. *-commutative N/A

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

   12. lower-*.f64 62.1%

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 62.1%

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

8. Applied rewrites 62.1%

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

Alternative 13: 62.0% accurate, 6.6× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (* 0.011111111111111112 (* (* (- b a) angle) (* (+ a b) PI))))

C

double code(double a, double b, double angle) {
	return 0.011111111111111112 * (((b - a) * angle) * ((a + b) * ((double) M_PI)));
}

Java

public static double code(double a, double b, double angle) {
	return 0.011111111111111112 * (((b - a) * angle) * ((a + b) * Math.PI));
}

Python

def code(a, b, angle):
	return 0.011111111111111112 * (((b - a) * angle) * ((a + b) * math.pi))

Julia

function code(a, b, angle)
	return Float64(0.011111111111111112 * Float64(Float64(Float64(b - a) * angle) * Float64(Float64(a + b) * pi)))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = 0.011111111111111112 * (((b - a) * angle) * ((a + b) * pi));
end

Wolfram

code[a_, b_, angle_] := N[(0.011111111111111112 * N[(N[(N[(b - a), $MachinePrecision] * angle), $MachinePrecision] * N[(N[(a + b), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

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

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
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 50.6%

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

4. Applied rewrites 50.6%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
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 50.6%

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

   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. 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. pow2 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 54.5%

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

   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 54.5%

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

6. Applied rewrites 54.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   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. associate-*r* N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 62.0%

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 62.0%

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

8. Applied rewrites 62.0%

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

Alternative 14: 57.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| - \left|a\right|\\ t_1 := \pi \cdot \frac{\left|angle\right|}{180}\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({\left(\left|b\right|\right)}^{2} - {\left(\left|a\right|\right)}^{2}\right)\right) \cdot \sin t\_1\right) \cdot \cos t\_1 \leq 0:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\left|angle\right| \cdot \left(t\_0 \cdot \left(\left|b\right| + \left|a\right|\right)\right)\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|angle\right| \cdot \left|b\right|\right) \cdot t\_0\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (- (fabs b) (fabs a))) (t_1 (* PI (/ (fabs angle) 180.0))))
   (*
    (copysign 1.0 angle)
    (if (<=
         (*
          (* (* 2.0 (- (pow (fabs b) 2.0) (pow (fabs a) 2.0))) (sin t_1))
          (cos t_1))
         0.0)
      (*
       0.011111111111111112
       (* (* (fabs angle) (* t_0 (+ (fabs b) (fabs a)))) PI))
      (* (* (* (fabs angle) (fabs b)) t_0) (* PI 0.011111111111111112))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) - fabs(a);
	double t_1 = ((double) M_PI) * (fabs(angle) / 180.0);
	double tmp;
	if ((((2.0 * (pow(fabs(b), 2.0) - pow(fabs(a), 2.0))) * sin(t_1)) * cos(t_1)) <= 0.0) {
		tmp = 0.011111111111111112 * ((fabs(angle) * (t_0 * (fabs(b) + fabs(a)))) * ((double) M_PI));
	} else {
		tmp = ((fabs(angle) * fabs(b)) * t_0) * (((double) M_PI) * 0.011111111111111112);
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(b) - Math.abs(a);
	double t_1 = Math.PI * (Math.abs(angle) / 180.0);
	double tmp;
	if ((((2.0 * (Math.pow(Math.abs(b), 2.0) - Math.pow(Math.abs(a), 2.0))) * Math.sin(t_1)) * Math.cos(t_1)) <= 0.0) {
		tmp = 0.011111111111111112 * ((Math.abs(angle) * (t_0 * (Math.abs(b) + Math.abs(a)))) * Math.PI);
	} else {
		tmp = ((Math.abs(angle) * Math.abs(b)) * t_0) * (Math.PI * 0.011111111111111112);
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) - math.fabs(a)
	t_1 = math.pi * (math.fabs(angle) / 180.0)
	tmp = 0
	if (((2.0 * (math.pow(math.fabs(b), 2.0) - math.pow(math.fabs(a), 2.0))) * math.sin(t_1)) * math.cos(t_1)) <= 0.0:
		tmp = 0.011111111111111112 * ((math.fabs(angle) * (t_0 * (math.fabs(b) + math.fabs(a)))) * math.pi)
	else:
		tmp = ((math.fabs(angle) * math.fabs(b)) * t_0) * (math.pi * 0.011111111111111112)
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) - abs(a))
	t_1 = Float64(pi * Float64(abs(angle) / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) * sin(t_1)) * cos(t_1)) <= 0.0)
		tmp = Float64(0.011111111111111112 * Float64(Float64(abs(angle) * Float64(t_0 * Float64(abs(b) + abs(a)))) * pi));
	else
		tmp = Float64(Float64(Float64(abs(angle) * abs(b)) * t_0) * Float64(pi * 0.011111111111111112));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) - abs(a);
	t_1 = pi * (abs(angle) / 180.0);
	tmp = 0.0;
	if ((((2.0 * ((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) * sin(t_1)) * cos(t_1)) <= 0.0)
		tmp = 0.011111111111111112 * ((abs(angle) * (t_0 * (abs(b) + abs(a)))) * pi);
	else
		tmp = ((abs(angle) * abs(b)) * t_0) * (pi * 0.011111111111111112);
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(N[Abs[angle], $MachinePrecision] / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[(2.0 * N[(N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision], 0.0], N[(0.011111111111111112 * N[(N[(N[Abs[angle], $MachinePrecision] * N[(t$95$0 * N[(N[Abs[b], $MachinePrecision] + N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Abs[angle], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < 0.0

   1. Initial program 53.8%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

      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. 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. pow2 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 54.5%

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

      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 54.5%

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

   6. Applied rewrites 54.5%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 53.8%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

      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. 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. pow2 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 54.5%

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

      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 54.5%

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

   6. Applied rewrites 54.5%

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

   7. Taylor expanded in a around 0

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right) \]
   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. associate-*r* N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 40.4%

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

   11. Applied rewrites 40.4%

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

Alternative 15: 44.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| - \left|a\right|\\ t_1 := \pi \cdot \frac{\left|angle\right|}{180}\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({\left(\left|b\right|\right)}^{2} - {\left(\left|a\right|\right)}^{2}\right)\right) \cdot \sin t\_1\right) \cdot \cos t\_1 \leq 0:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\left|angle\right| \cdot \left(t\_0 \cdot \left|b\right|\right)\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|angle\right| \cdot \left|b\right|\right) \cdot t\_0\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (- (fabs b) (fabs a))) (t_1 (* PI (/ (fabs angle) 180.0))))
   (*
    (copysign 1.0 angle)
    (if (<=
         (*
          (* (* 2.0 (- (pow (fabs b) 2.0) (pow (fabs a) 2.0))) (sin t_1))
          (cos t_1))
         0.0)
      (* 0.011111111111111112 (* (* (fabs angle) (* t_0 (fabs b))) PI))
      (* (* (* (fabs angle) (fabs b)) t_0) (* PI 0.011111111111111112))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) - fabs(a);
	double t_1 = ((double) M_PI) * (fabs(angle) / 180.0);
	double tmp;
	if ((((2.0 * (pow(fabs(b), 2.0) - pow(fabs(a), 2.0))) * sin(t_1)) * cos(t_1)) <= 0.0) {
		tmp = 0.011111111111111112 * ((fabs(angle) * (t_0 * fabs(b))) * ((double) M_PI));
	} else {
		tmp = ((fabs(angle) * fabs(b)) * t_0) * (((double) M_PI) * 0.011111111111111112);
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(b) - Math.abs(a);
	double t_1 = Math.PI * (Math.abs(angle) / 180.0);
	double tmp;
	if ((((2.0 * (Math.pow(Math.abs(b), 2.0) - Math.pow(Math.abs(a), 2.0))) * Math.sin(t_1)) * Math.cos(t_1)) <= 0.0) {
		tmp = 0.011111111111111112 * ((Math.abs(angle) * (t_0 * Math.abs(b))) * Math.PI);
	} else {
		tmp = ((Math.abs(angle) * Math.abs(b)) * t_0) * (Math.PI * 0.011111111111111112);
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) - math.fabs(a)
	t_1 = math.pi * (math.fabs(angle) / 180.0)
	tmp = 0
	if (((2.0 * (math.pow(math.fabs(b), 2.0) - math.pow(math.fabs(a), 2.0))) * math.sin(t_1)) * math.cos(t_1)) <= 0.0:
		tmp = 0.011111111111111112 * ((math.fabs(angle) * (t_0 * math.fabs(b))) * math.pi)
	else:
		tmp = ((math.fabs(angle) * math.fabs(b)) * t_0) * (math.pi * 0.011111111111111112)
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) - abs(a))
	t_1 = Float64(pi * Float64(abs(angle) / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) * sin(t_1)) * cos(t_1)) <= 0.0)
		tmp = Float64(0.011111111111111112 * Float64(Float64(abs(angle) * Float64(t_0 * abs(b))) * pi));
	else
		tmp = Float64(Float64(Float64(abs(angle) * abs(b)) * t_0) * Float64(pi * 0.011111111111111112));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) - abs(a);
	t_1 = pi * (abs(angle) / 180.0);
	tmp = 0.0;
	if ((((2.0 * ((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) * sin(t_1)) * cos(t_1)) <= 0.0)
		tmp = 0.011111111111111112 * ((abs(angle) * (t_0 * abs(b))) * pi);
	else
		tmp = ((abs(angle) * abs(b)) * t_0) * (pi * 0.011111111111111112);
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(N[Abs[angle], $MachinePrecision] / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[(2.0 * N[(N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision], 0.0], N[(0.011111111111111112 * N[(N[(N[Abs[angle], $MachinePrecision] * N[(t$95$0 * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Abs[angle], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < 0.0

   1. Initial program 53.8%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

      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. 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. pow2 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 54.5%

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

      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 54.5%

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

   6. Applied rewrites 54.5%

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

   7. Taylor expanded in a around 0

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 38.0%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 53.8%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

      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. 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. pow2 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 54.5%

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

      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 54.5%

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

   6. Applied rewrites 54.5%

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

   7. Taylor expanded in a around 0

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right) \]
   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. associate-*r* N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 40.4%

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

   11. Applied rewrites 40.4%

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

Alternative 16: 43.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| - \left|a\right|\\ \mathbf{if}\;2 \cdot \left({\left(\left|b\right|\right)}^{2} - {\left(\left|a\right|\right)}^{2}\right) \leq 5 \cdot 10^{+135}:\\ \;\;\;\;\left(0.011111111111111112 \cdot angle\right) \cdot \left(\left|b\right| \cdot \left(t\_0 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot \left(t\_0 \cdot angle\right)\right) \cdot \left|b\right|\right) \cdot \pi\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (- (fabs b) (fabs a))))
   (if (<= (* 2.0 (- (pow (fabs b) 2.0) (pow (fabs a) 2.0))) 5e+135)
     (* (* 0.011111111111111112 angle) (* (fabs b) (* t_0 PI)))
     (* (* (* 0.011111111111111112 (* t_0 angle)) (fabs b)) PI))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) - fabs(a);
	double tmp;
	if ((2.0 * (pow(fabs(b), 2.0) - pow(fabs(a), 2.0))) <= 5e+135) {
		tmp = (0.011111111111111112 * angle) * (fabs(b) * (t_0 * ((double) M_PI)));
	} else {
		tmp = ((0.011111111111111112 * (t_0 * angle)) * fabs(b)) * ((double) M_PI);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(b) - Math.abs(a);
	double tmp;
	if ((2.0 * (Math.pow(Math.abs(b), 2.0) - Math.pow(Math.abs(a), 2.0))) <= 5e+135) {
		tmp = (0.011111111111111112 * angle) * (Math.abs(b) * (t_0 * Math.PI));
	} else {
		tmp = ((0.011111111111111112 * (t_0 * angle)) * Math.abs(b)) * Math.PI;
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) - math.fabs(a)
	tmp = 0
	if (2.0 * (math.pow(math.fabs(b), 2.0) - math.pow(math.fabs(a), 2.0))) <= 5e+135:
		tmp = (0.011111111111111112 * angle) * (math.fabs(b) * (t_0 * math.pi))
	else:
		tmp = ((0.011111111111111112 * (t_0 * angle)) * math.fabs(b)) * math.pi
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) - abs(a))
	tmp = 0.0
	if (Float64(2.0 * Float64((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) <= 5e+135)
		tmp = Float64(Float64(0.011111111111111112 * angle) * Float64(abs(b) * Float64(t_0 * pi)));
	else
		tmp = Float64(Float64(Float64(0.011111111111111112 * Float64(t_0 * angle)) * abs(b)) * pi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) - abs(a);
	tmp = 0.0;
	if ((2.0 * ((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) <= 5e+135)
		tmp = (0.011111111111111112 * angle) * (abs(b) * (t_0 * pi));
	else
		tmp = ((0.011111111111111112 * (t_0 * angle)) * abs(b)) * pi;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 * N[(N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+135], N[(N[(0.011111111111111112 * angle), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * N[(t$95$0 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.011111111111111112 * N[(t$95$0 * angle), $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * Pi), $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)))) < 5.00000000000000029e135

   1. Initial program 53.8%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

      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. 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. pow2 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 54.5%

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

      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 54.5%

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

   6. Applied rewrites 54.5%

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

   7. Taylor expanded in a around 0

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right) \]
   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. 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-*l* N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. associate-*l* N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 38.0%

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

   11. Applied rewrites 38.0%

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

   if 5.00000000000000029e135 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 53.8%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

      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. 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. pow2 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 54.5%

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

      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 54.5%

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

   6. Applied rewrites 54.5%

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

   7. Taylor expanded in a around 0

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right) \]
   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. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       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. *-commutative N/A

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

       12. lower-*.f64 41.8%

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

   11. Applied rewrites 41.8%

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

Alternative 17: 43.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| - \left|a\right|\\ t_1 := \pi \cdot \frac{\left|angle\right|}{180}\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({\left(\left|b\right|\right)}^{2} - {\left(\left|a\right|\right)}^{2}\right)\right) \cdot \sin t\_1\right) \cdot \cos t\_1 \leq 2 \cdot 10^{-77}:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left|angle\right|\right) \cdot \left(\left|b\right| \cdot \left(t\_0 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(t\_0 \cdot \left(\left|b\right| \cdot \left|angle\right|\right)\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (- (fabs b) (fabs a))) (t_1 (* PI (/ (fabs angle) 180.0))))
   (*
    (copysign 1.0 angle)
    (if (<=
         (*
          (* (* 2.0 (- (pow (fabs b) 2.0) (pow (fabs a) 2.0))) (sin t_1))
          (cos t_1))
         2e-77)
      (* (* 0.011111111111111112 (fabs angle)) (* (fabs b) (* t_0 PI)))
      (* 0.011111111111111112 (* (* t_0 (* (fabs b) (fabs angle))) PI))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) - fabs(a);
	double t_1 = ((double) M_PI) * (fabs(angle) / 180.0);
	double tmp;
	if ((((2.0 * (pow(fabs(b), 2.0) - pow(fabs(a), 2.0))) * sin(t_1)) * cos(t_1)) <= 2e-77) {
		tmp = (0.011111111111111112 * fabs(angle)) * (fabs(b) * (t_0 * ((double) M_PI)));
	} else {
		tmp = 0.011111111111111112 * ((t_0 * (fabs(b) * fabs(angle))) * ((double) M_PI));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(b) - Math.abs(a);
	double t_1 = Math.PI * (Math.abs(angle) / 180.0);
	double tmp;
	if ((((2.0 * (Math.pow(Math.abs(b), 2.0) - Math.pow(Math.abs(a), 2.0))) * Math.sin(t_1)) * Math.cos(t_1)) <= 2e-77) {
		tmp = (0.011111111111111112 * Math.abs(angle)) * (Math.abs(b) * (t_0 * Math.PI));
	} else {
		tmp = 0.011111111111111112 * ((t_0 * (Math.abs(b) * Math.abs(angle))) * Math.PI);
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) - math.fabs(a)
	t_1 = math.pi * (math.fabs(angle) / 180.0)
	tmp = 0
	if (((2.0 * (math.pow(math.fabs(b), 2.0) - math.pow(math.fabs(a), 2.0))) * math.sin(t_1)) * math.cos(t_1)) <= 2e-77:
		tmp = (0.011111111111111112 * math.fabs(angle)) * (math.fabs(b) * (t_0 * math.pi))
	else:
		tmp = 0.011111111111111112 * ((t_0 * (math.fabs(b) * math.fabs(angle))) * math.pi)
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) - abs(a))
	t_1 = Float64(pi * Float64(abs(angle) / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) * sin(t_1)) * cos(t_1)) <= 2e-77)
		tmp = Float64(Float64(0.011111111111111112 * abs(angle)) * Float64(abs(b) * Float64(t_0 * pi)));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(t_0 * Float64(abs(b) * abs(angle))) * pi));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) - abs(a);
	t_1 = pi * (abs(angle) / 180.0);
	tmp = 0.0;
	if ((((2.0 * ((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) * sin(t_1)) * cos(t_1)) <= 2e-77)
		tmp = (0.011111111111111112 * abs(angle)) * (abs(b) * (t_0 * pi));
	else
		tmp = 0.011111111111111112 * ((t_0 * (abs(b) * abs(angle))) * pi);
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(N[Abs[angle], $MachinePrecision] / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[(2.0 * N[(N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision], 2e-77], N[(N[(0.011111111111111112 * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * N[(t$95$0 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(t$95$0 * N[(N[Abs[b], $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < 1.9999999999999999e-77

   1. Initial program 53.8%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

      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. 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. pow2 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 54.5%

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

      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 54.5%

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

   6. Applied rewrites 54.5%

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

   7. Taylor expanded in a around 0

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right) \]
   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. 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-*l* N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. associate-*l* N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 38.0%

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

   11. Applied rewrites 38.0%

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

   if 1.9999999999999999e-77 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 53.8%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

      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. 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. pow2 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 54.5%

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

      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 54.5%

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

   6. Applied rewrites 54.5%

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

   7. Taylor expanded in a around 0

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

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 40.4%

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

   11. Applied rewrites 40.4%

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

Alternative 18: 43.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| - \left|a\right|\\ t_1 := \pi \cdot \frac{\left|angle\right|}{180}\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({\left(\left|b\right|\right)}^{2} - {\left(\left|a\right|\right)}^{2}\right)\right) \cdot \sin t\_1\right) \cdot \cos t\_1 \leq 10^{-245}:\\ \;\;\;\;\left|angle\right| \cdot \left(\left(\left|b\right| \cdot \left(t\_0 \cdot \pi\right)\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(t\_0 \cdot \left(\left|b\right| \cdot \left|angle\right|\right)\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (- (fabs b) (fabs a))) (t_1 (* PI (/ (fabs angle) 180.0))))
   (*
    (copysign 1.0 angle)
    (if (<=
         (*
          (* (* 2.0 (- (pow (fabs b) 2.0) (pow (fabs a) 2.0))) (sin t_1))
          (cos t_1))
         1e-245)
      (* (fabs angle) (* (* (fabs b) (* t_0 PI)) 0.011111111111111112))
      (* 0.011111111111111112 (* (* t_0 (* (fabs b) (fabs angle))) PI))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) - fabs(a);
	double t_1 = ((double) M_PI) * (fabs(angle) / 180.0);
	double tmp;
	if ((((2.0 * (pow(fabs(b), 2.0) - pow(fabs(a), 2.0))) * sin(t_1)) * cos(t_1)) <= 1e-245) {
		tmp = fabs(angle) * ((fabs(b) * (t_0 * ((double) M_PI))) * 0.011111111111111112);
	} else {
		tmp = 0.011111111111111112 * ((t_0 * (fabs(b) * fabs(angle))) * ((double) M_PI));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(b) - Math.abs(a);
	double t_1 = Math.PI * (Math.abs(angle) / 180.0);
	double tmp;
	if ((((2.0 * (Math.pow(Math.abs(b), 2.0) - Math.pow(Math.abs(a), 2.0))) * Math.sin(t_1)) * Math.cos(t_1)) <= 1e-245) {
		tmp = Math.abs(angle) * ((Math.abs(b) * (t_0 * Math.PI)) * 0.011111111111111112);
	} else {
		tmp = 0.011111111111111112 * ((t_0 * (Math.abs(b) * Math.abs(angle))) * Math.PI);
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) - math.fabs(a)
	t_1 = math.pi * (math.fabs(angle) / 180.0)
	tmp = 0
	if (((2.0 * (math.pow(math.fabs(b), 2.0) - math.pow(math.fabs(a), 2.0))) * math.sin(t_1)) * math.cos(t_1)) <= 1e-245:
		tmp = math.fabs(angle) * ((math.fabs(b) * (t_0 * math.pi)) * 0.011111111111111112)
	else:
		tmp = 0.011111111111111112 * ((t_0 * (math.fabs(b) * math.fabs(angle))) * math.pi)
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) - abs(a))
	t_1 = Float64(pi * Float64(abs(angle) / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) * sin(t_1)) * cos(t_1)) <= 1e-245)
		tmp = Float64(abs(angle) * Float64(Float64(abs(b) * Float64(t_0 * pi)) * 0.011111111111111112));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(t_0 * Float64(abs(b) * abs(angle))) * pi));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) - abs(a);
	t_1 = pi * (abs(angle) / 180.0);
	tmp = 0.0;
	if ((((2.0 * ((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) * sin(t_1)) * cos(t_1)) <= 1e-245)
		tmp = abs(angle) * ((abs(b) * (t_0 * pi)) * 0.011111111111111112);
	else
		tmp = 0.011111111111111112 * ((t_0 * (abs(b) * abs(angle))) * pi);
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(N[Abs[angle], $MachinePrecision] / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[(2.0 * N[(N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision], 1e-245], N[(N[Abs[angle], $MachinePrecision] * N[(N[(N[Abs[b], $MachinePrecision] * N[(t$95$0 * Pi), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(t$95$0 * N[(N[Abs[b], $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < 9.9999999999999993e-246

   1. Initial program 53.8%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

      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. 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. pow2 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 54.5%

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

      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 54.5%

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

   6. Applied rewrites 54.5%

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

   7. Taylor expanded in a around 0

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right) \]
   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. lift-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. associate-*l* N/A

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

       12. lower-*.f64 N/A

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

       13. lower-*.f64 38.1%

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

   11. Applied rewrites 38.1%

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

   if 9.9999999999999993e-246 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 53.8%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

      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. 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. pow2 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 54.5%

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

      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 54.5%

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

   6. Applied rewrites 54.5%

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

   7. Taylor expanded in a around 0

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

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 40.4%

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

   11. Applied rewrites 40.4%

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

Alternative 19: 43.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| - \left|a\right|\\ t_1 := \pi \cdot \frac{\left|angle\right|}{180}\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({\left(\left|b\right|\right)}^{2} - {\left(\left|a\right|\right)}^{2}\right)\right) \cdot \sin t\_1\right) \cdot \cos t\_1 \leq 5 \cdot 10^{-252}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left|angle\right|\right) \cdot \left(\left|b\right| \cdot t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(t\_0 \cdot \left(\left|b\right| \cdot \left|angle\right|\right)\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (- (fabs b) (fabs a))) (t_1 (* PI (/ (fabs angle) 180.0))))
   (*
    (copysign 1.0 angle)
    (if (<=
         (*
          (* (* 2.0 (- (pow (fabs b) 2.0) (pow (fabs a) 2.0))) (sin t_1))
          (cos t_1))
         5e-252)
      (* 0.011111111111111112 (* (* PI (fabs angle)) (* (fabs b) t_0)))
      (* 0.011111111111111112 (* (* t_0 (* (fabs b) (fabs angle))) PI))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) - fabs(a);
	double t_1 = ((double) M_PI) * (fabs(angle) / 180.0);
	double tmp;
	if ((((2.0 * (pow(fabs(b), 2.0) - pow(fabs(a), 2.0))) * sin(t_1)) * cos(t_1)) <= 5e-252) {
		tmp = 0.011111111111111112 * ((((double) M_PI) * fabs(angle)) * (fabs(b) * t_0));
	} else {
		tmp = 0.011111111111111112 * ((t_0 * (fabs(b) * fabs(angle))) * ((double) M_PI));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(b) - Math.abs(a);
	double t_1 = Math.PI * (Math.abs(angle) / 180.0);
	double tmp;
	if ((((2.0 * (Math.pow(Math.abs(b), 2.0) - Math.pow(Math.abs(a), 2.0))) * Math.sin(t_1)) * Math.cos(t_1)) <= 5e-252) {
		tmp = 0.011111111111111112 * ((Math.PI * Math.abs(angle)) * (Math.abs(b) * t_0));
	} else {
		tmp = 0.011111111111111112 * ((t_0 * (Math.abs(b) * Math.abs(angle))) * Math.PI);
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) - math.fabs(a)
	t_1 = math.pi * (math.fabs(angle) / 180.0)
	tmp = 0
	if (((2.0 * (math.pow(math.fabs(b), 2.0) - math.pow(math.fabs(a), 2.0))) * math.sin(t_1)) * math.cos(t_1)) <= 5e-252:
		tmp = 0.011111111111111112 * ((math.pi * math.fabs(angle)) * (math.fabs(b) * t_0))
	else:
		tmp = 0.011111111111111112 * ((t_0 * (math.fabs(b) * math.fabs(angle))) * math.pi)
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) - abs(a))
	t_1 = Float64(pi * Float64(abs(angle) / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) * sin(t_1)) * cos(t_1)) <= 5e-252)
		tmp = Float64(0.011111111111111112 * Float64(Float64(pi * abs(angle)) * Float64(abs(b) * t_0)));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(t_0 * Float64(abs(b) * abs(angle))) * pi));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) - abs(a);
	t_1 = pi * (abs(angle) / 180.0);
	tmp = 0.0;
	if ((((2.0 * ((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) * sin(t_1)) * cos(t_1)) <= 5e-252)
		tmp = 0.011111111111111112 * ((pi * abs(angle)) * (abs(b) * t_0));
	else
		tmp = 0.011111111111111112 * ((t_0 * (abs(b) * abs(angle))) * pi);
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(N[Abs[angle], $MachinePrecision] / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[(2.0 * N[(N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision], 5e-252], N[(0.011111111111111112 * N[(N[(Pi * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(t$95$0 * N[(N[Abs[b], $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < 5.00000000000000008e-252

   1. Initial program 53.8%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

      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. 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. pow2 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 54.5%

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

      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 54.5%

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

   6. Applied rewrites 54.5%

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

   7. Taylor expanded in a around 0

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right) \]
   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. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. lift-*.f64 N/A

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

       6. lower-*.f64 38.0%

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 38.0%

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

   11. Applied rewrites 38.0%

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

   if 5.00000000000000008e-252 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 53.8%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

      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. 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. pow2 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 54.5%

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

      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 54.5%

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

   6. Applied rewrites 54.5%

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

   7. Taylor expanded in a around 0

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

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 40.4%

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

   11. Applied rewrites 40.4%

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

Alternative 20: 43.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| - \left|a\right|\\ t_1 := \pi \cdot \frac{\left|angle\right|}{180}\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({\left(\left|b\right|\right)}^{2} - {\left(\left|a\right|\right)}^{2}\right)\right) \cdot \sin t\_1\right) \cdot \cos t\_1 \leq 0:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\left|angle\right| \cdot \left(t\_0 \cdot \left|b\right|\right)\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(t\_0 \cdot \left|angle\right|\right) \cdot \left(\left|b\right| \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (- (fabs b) (fabs a))) (t_1 (* PI (/ (fabs angle) 180.0))))
   (*
    (copysign 1.0 angle)
    (if (<=
         (*
          (* (* 2.0 (- (pow (fabs b) 2.0) (pow (fabs a) 2.0))) (sin t_1))
          (cos t_1))
         0.0)
      (* 0.011111111111111112 (* (* (fabs angle) (* t_0 (fabs b))) PI))
      (* 0.011111111111111112 (* (* t_0 (fabs angle)) (* (fabs b) PI)))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) - fabs(a);
	double t_1 = ((double) M_PI) * (fabs(angle) / 180.0);
	double tmp;
	if ((((2.0 * (pow(fabs(b), 2.0) - pow(fabs(a), 2.0))) * sin(t_1)) * cos(t_1)) <= 0.0) {
		tmp = 0.011111111111111112 * ((fabs(angle) * (t_0 * fabs(b))) * ((double) M_PI));
	} else {
		tmp = 0.011111111111111112 * ((t_0 * fabs(angle)) * (fabs(b) * ((double) M_PI)));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(b) - Math.abs(a);
	double t_1 = Math.PI * (Math.abs(angle) / 180.0);
	double tmp;
	if ((((2.0 * (Math.pow(Math.abs(b), 2.0) - Math.pow(Math.abs(a), 2.0))) * Math.sin(t_1)) * Math.cos(t_1)) <= 0.0) {
		tmp = 0.011111111111111112 * ((Math.abs(angle) * (t_0 * Math.abs(b))) * Math.PI);
	} else {
		tmp = 0.011111111111111112 * ((t_0 * Math.abs(angle)) * (Math.abs(b) * Math.PI));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) - math.fabs(a)
	t_1 = math.pi * (math.fabs(angle) / 180.0)
	tmp = 0
	if (((2.0 * (math.pow(math.fabs(b), 2.0) - math.pow(math.fabs(a), 2.0))) * math.sin(t_1)) * math.cos(t_1)) <= 0.0:
		tmp = 0.011111111111111112 * ((math.fabs(angle) * (t_0 * math.fabs(b))) * math.pi)
	else:
		tmp = 0.011111111111111112 * ((t_0 * math.fabs(angle)) * (math.fabs(b) * math.pi))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) - abs(a))
	t_1 = Float64(pi * Float64(abs(angle) / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) * sin(t_1)) * cos(t_1)) <= 0.0)
		tmp = Float64(0.011111111111111112 * Float64(Float64(abs(angle) * Float64(t_0 * abs(b))) * pi));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(t_0 * abs(angle)) * Float64(abs(b) * pi)));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) - abs(a);
	t_1 = pi * (abs(angle) / 180.0);
	tmp = 0.0;
	if ((((2.0 * ((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) * sin(t_1)) * cos(t_1)) <= 0.0)
		tmp = 0.011111111111111112 * ((abs(angle) * (t_0 * abs(b))) * pi);
	else
		tmp = 0.011111111111111112 * ((t_0 * abs(angle)) * (abs(b) * pi));
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(N[Abs[angle], $MachinePrecision] / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[(2.0 * N[(N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision], 0.0], N[(0.011111111111111112 * N[(N[(N[Abs[angle], $MachinePrecision] * N[(t$95$0 * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(t$95$0 * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < 0.0

   1. Initial program 53.8%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

      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. 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. pow2 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 54.5%

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

      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 54.5%

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

   6. Applied rewrites 54.5%

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

   7. Taylor expanded in a around 0

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 38.0%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 53.8%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.6%

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

      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. 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. pow2 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 54.5%

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

      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 54.5%

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

   6. Applied rewrites 54.5%

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

   7. Taylor expanded in a around 0

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right) \]
   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. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 41.8%

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

   11. Applied rewrites 41.8%

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

Alternative 21: 39.3% accurate, 6.3× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (* 0.011111111111111112 (* (* PI angle) (* (fabs b) (- (fabs b) (fabs a))))))

C

double code(double a, double b, double angle) {
	return 0.011111111111111112 * ((((double) M_PI) * angle) * (fabs(b) * (fabs(b) - fabs(a))));
}

Java

public static double code(double a, double b, double angle) {
	return 0.011111111111111112 * ((Math.PI * angle) * (Math.abs(b) * (Math.abs(b) - Math.abs(a))));
}

Python

def code(a, b, angle):
	return 0.011111111111111112 * ((math.pi * angle) * (math.fabs(b) * (math.fabs(b) - math.fabs(a))))

Julia

function code(a, b, angle)
	return Float64(0.011111111111111112 * Float64(Float64(pi * angle) * Float64(abs(b) * Float64(abs(b) - abs(a)))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = 0.011111111111111112 * ((pi * angle) * (abs(b) * (abs(b) - abs(a))));
end

Wolfram

code[a_, b_, angle_] := N[(0.011111111111111112 * N[(N[(Pi * angle), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

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

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
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 50.6%

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

4. Applied rewrites 50.6%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
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 50.6%

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

   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. 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. pow2 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 54.5%

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

   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 54.5%

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

6. Applied rewrites 54.5%

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

7. Taylor expanded in a around 0

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right) \]
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. *-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. associate-*r* N/A

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

    5. lift-*.f64 N/A

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

    6. lower-*.f64 38.0%

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

    7. lift-*.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 38.0%

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

11. Applied rewrites 38.0%

    \[\leadsto 0.011111111111111112 \cdot \left(\left(\pi \cdot angle\right) \cdot \color{blue}{\left(b \cdot \left(b - a\right)\right)}\right) \]
12. Add Preprocessing

Alternative 22: 39.3% accurate, 6.3× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (* 0.011111111111111112 (* (* angle (* (- (fabs b) (fabs a)) (fabs b))) PI)))

C

double code(double a, double b, double angle) {
	return 0.011111111111111112 * ((angle * ((fabs(b) - fabs(a)) * fabs(b))) * ((double) M_PI));
}

Java

public static double code(double a, double b, double angle) {
	return 0.011111111111111112 * ((angle * ((Math.abs(b) - Math.abs(a)) * Math.abs(b))) * Math.PI);
}

Python

def code(a, b, angle):
	return 0.011111111111111112 * ((angle * ((math.fabs(b) - math.fabs(a)) * math.fabs(b))) * math.pi)

Julia

function code(a, b, angle)
	return Float64(0.011111111111111112 * Float64(Float64(angle * Float64(Float64(abs(b) - abs(a)) * abs(b))) * pi))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = 0.011111111111111112 * ((angle * ((abs(b) - abs(a)) * abs(b))) * pi);
end

Wolfram

code[a_, b_, angle_] := N[(0.011111111111111112 * N[(N[(angle * N[(N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

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

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
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 50.6%

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

4. Applied rewrites 50.6%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
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 50.6%

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

   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. 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. pow2 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 54.5%

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

   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 54.5%

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

6. Applied rewrites 54.5%

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

7. Taylor expanded in a around 0

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

Alternative 23: 35.3% accurate, 9.4× speedup

Math

\[0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b\right)\right) \cdot \pi\right) \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (* 0.011111111111111112 (* (* angle (* b b)) PI)))

C

double code(double a, double b, double angle) {
	return 0.011111111111111112 * ((angle * (b * b)) * ((double) M_PI));
}

Java

public static double code(double a, double b, double angle) {
	return 0.011111111111111112 * ((angle * (b * b)) * Math.PI);
}

Python

def code(a, b, angle):
	return 0.011111111111111112 * ((angle * (b * b)) * math.pi)

Julia

function code(a, b, angle)
	return Float64(0.011111111111111112 * Float64(Float64(angle * Float64(b * b)) * pi))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = 0.011111111111111112 * ((angle * (b * b)) * pi);
end

Wolfram

code[a_, b_, angle_] := N[(0.011111111111111112 * N[(N[(angle * N[(b * b), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

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

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
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 50.6%

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

4. Applied rewrites 50.6%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
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 50.6%

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

   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. 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. pow2 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 54.5%

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

   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 54.5%

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

6. Applied rewrites 54.5%

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

7. Taylor expanded in a around 0

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot b\right)\right) \cdot \pi\right) \]
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. Taylor expanded in a around 0

    \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b\right)\right) \cdot \pi\right) \]
11. Step-by-step derivation

12. Applied rewrites 35.3%

    \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b\right)\right) \cdot \pi\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025183 
(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)))))