ab-angle->ABCF B

Percentage Accurate: 54.0% → 66.9%

Time: 15.8s

Alternatives: 12

Speedup: 16.8×

• 32.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 66.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;{a\_m}^{2} \leq 2 \cdot 10^{+92}:\\ \;\;\;\;\cos \left(\frac{angle}{180} \cdot \left(\sqrt{\sqrt{\pi} \cdot \pi} \cdot \sqrt{\sqrt{\pi}}\right)\right) \cdot \left(\left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right) \cdot \left(b - a\_m\right)\right) \cdot \left(b + a\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(\sin \left(\left(\sqrt{\pi} \cdot angle\right) \cdot {\left(\frac{180}{\sqrt{\pi}}\right)}^{-1}\right) \cdot 2\right) \cdot \left(b - a\_m\right)\right) \cdot \left(b + a\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= (pow a_m 2.0) 2e+92)
   (*
    (cos (* (/ angle 180.0) (* (sqrt (* (sqrt PI) PI)) (sqrt (sqrt PI)))))
    (*
     (* (* (sin (* PI (* angle 0.005555555555555556))) 2.0) (- b a_m))
     (+ b a_m)))
   (*
    (cos (* (/ angle 180.0) PI))
    (*
     (*
      (* (sin (* (* (sqrt PI) angle) (pow (/ 180.0 (sqrt PI)) -1.0))) 2.0)
      (- b a_m))
     (+ b a_m)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (pow(a_m, 2.0) <= 2e+92) {
		tmp = cos(((angle / 180.0) * (sqrt((sqrt(((double) M_PI)) * ((double) M_PI))) * sqrt(sqrt(((double) M_PI)))))) * (((sin((((double) M_PI) * (angle * 0.005555555555555556))) * 2.0) * (b - a_m)) * (b + a_m));
	} else {
		tmp = cos(((angle / 180.0) * ((double) M_PI))) * (((sin(((sqrt(((double) M_PI)) * angle) * pow((180.0 / sqrt(((double) M_PI))), -1.0))) * 2.0) * (b - a_m)) * (b + a_m));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if (Math.pow(a_m, 2.0) <= 2e+92) {
		tmp = Math.cos(((angle / 180.0) * (Math.sqrt((Math.sqrt(Math.PI) * Math.PI)) * Math.sqrt(Math.sqrt(Math.PI))))) * (((Math.sin((Math.PI * (angle * 0.005555555555555556))) * 2.0) * (b - a_m)) * (b + a_m));
	} else {
		tmp = Math.cos(((angle / 180.0) * Math.PI)) * (((Math.sin(((Math.sqrt(Math.PI) * angle) * Math.pow((180.0 / Math.sqrt(Math.PI)), -1.0))) * 2.0) * (b - a_m)) * (b + a_m));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if math.pow(a_m, 2.0) <= 2e+92:
		tmp = math.cos(((angle / 180.0) * (math.sqrt((math.sqrt(math.pi) * math.pi)) * math.sqrt(math.sqrt(math.pi))))) * (((math.sin((math.pi * (angle * 0.005555555555555556))) * 2.0) * (b - a_m)) * (b + a_m))
	else:
		tmp = math.cos(((angle / 180.0) * math.pi)) * (((math.sin(((math.sqrt(math.pi) * angle) * math.pow((180.0 / math.sqrt(math.pi)), -1.0))) * 2.0) * (b - a_m)) * (b + a_m))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if ((a_m ^ 2.0) <= 2e+92)
		tmp = Float64(cos(Float64(Float64(angle / 180.0) * Float64(sqrt(Float64(sqrt(pi) * pi)) * sqrt(sqrt(pi))))) * Float64(Float64(Float64(sin(Float64(pi * Float64(angle * 0.005555555555555556))) * 2.0) * Float64(b - a_m)) * Float64(b + a_m)));
	else
		tmp = Float64(cos(Float64(Float64(angle / 180.0) * pi)) * Float64(Float64(Float64(sin(Float64(Float64(sqrt(pi) * angle) * (Float64(180.0 / sqrt(pi)) ^ -1.0))) * 2.0) * Float64(b - a_m)) * Float64(b + a_m)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if ((a_m ^ 2.0) <= 2e+92)
		tmp = cos(((angle / 180.0) * (sqrt((sqrt(pi) * pi)) * sqrt(sqrt(pi))))) * (((sin((pi * (angle * 0.005555555555555556))) * 2.0) * (b - a_m)) * (b + a_m));
	else
		tmp = cos(((angle / 180.0) * pi)) * (((sin(((sqrt(pi) * angle) * ((180.0 / sqrt(pi)) ^ -1.0))) * 2.0) * (b - a_m)) * (b + a_m));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[N[Power[a$95$m, 2.0], $MachinePrecision], 2e+92], N[(N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * N[(N[Sqrt[N[(N[Sqrt[Pi], $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Sqrt[Pi], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[Sin[N[(N[(N[Sqrt[Pi], $MachinePrecision] * angle), $MachinePrecision] * N[Power[N[(180.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a #s(literal 2 binary64)) < 2.0000000000000001e92

   1. Initial program 57.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 63.2

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

   4. Applied rewrites 62.3%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lift-PI.f64 N/A

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

      4. add-sqr-sqrt N/A

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

      5. associate-*l* N/A

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

      6. lift-PI.f64 N/A

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

      7. add-sqr-sqrt N/A

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

      8. cube-unmult N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \left(\sqrt{\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{3}}} \cdot \frac{angle}{180}\right) \]

      9. pow3 N/A

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

      10. add-sqr-sqrt N/A

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

      11. lift-PI.f64 N/A

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

      12. sqrt-prod N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. add-sqr-sqrt N/A

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

   6. Applied rewrites 65.1%

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

   if 2.0000000000000001e92 < (pow.f64 a #s(literal 2 binary64))

   1. Initial program 52.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 72.4

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

   4. Applied rewrites 72.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. associate-/r/ N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 72.8

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

   6. Applied rewrites 72.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. times-frac N/A

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

      10. unpow-prod-down N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 N/A

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

      15. clear-num N/A

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

      16. div-inv N/A

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

      17. clear-num N/A

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

      18. /-rgt-identity N/A

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

      19. lower-/.f64 N/A

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

      20. lower-*.f64 81.1

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

   8. Applied rewrites 81.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. /-rgt-identity 81.2

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

   10. Applied rewrites 81.2%

       \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left({\left(\frac{180}{\sqrt{\pi}}\right)}^{-1} \cdot \color{blue}{\left(\sqrt{\pi} \cdot angle\right)}\right) \cdot 2\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{+92}:\\ \;\;\;\;\cos \left(\frac{angle}{180} \cdot \left(\sqrt{\sqrt{\pi} \cdot \pi} \cdot \sqrt{\sqrt{\pi}}\right)\right) \cdot \left(\left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(\sin \left(\left(\sqrt{\pi} \cdot angle\right) \cdot {\left(\frac{180}{\sqrt{\pi}}\right)}^{-1}\right) \cdot 2\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \cos \left(\frac{angle}{180} \cdot \left(\sqrt{\sqrt{\pi} \cdot \pi} \cdot \sqrt{\sqrt{\pi}}\right)\right) \cdot \left(\left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right) \cdot \left(b - a\_m\right)\right) \cdot \left(b + a\_m\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (*
  (cos (* (/ angle 180.0) (* (sqrt (* (sqrt PI) PI)) (sqrt (sqrt PI)))))
  (*
   (* (* (sin (* PI (* angle 0.005555555555555556))) 2.0) (- b a_m))
   (+ b a_m))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	return cos(((angle / 180.0) * (sqrt((sqrt(((double) M_PI)) * ((double) M_PI))) * sqrt(sqrt(((double) M_PI)))))) * (((sin((((double) M_PI) * (angle * 0.005555555555555556))) * 2.0) * (b - a_m)) * (b + a_m));
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	return Math.cos(((angle / 180.0) * (Math.sqrt((Math.sqrt(Math.PI) * Math.PI)) * Math.sqrt(Math.sqrt(Math.PI))))) * (((Math.sin((Math.PI * (angle * 0.005555555555555556))) * 2.0) * (b - a_m)) * (b + a_m));
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	return math.cos(((angle / 180.0) * (math.sqrt((math.sqrt(math.pi) * math.pi)) * math.sqrt(math.sqrt(math.pi))))) * (((math.sin((math.pi * (angle * 0.005555555555555556))) * 2.0) * (b - a_m)) * (b + a_m))

Julia

a_m = abs(a)
function code(a_m, b, angle)
	return Float64(cos(Float64(Float64(angle / 180.0) * Float64(sqrt(Float64(sqrt(pi) * pi)) * sqrt(sqrt(pi))))) * Float64(Float64(Float64(sin(Float64(pi * Float64(angle * 0.005555555555555556))) * 2.0) * Float64(b - a_m)) * Float64(b + a_m)))
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b, angle)
	tmp = cos(((angle / 180.0) * (sqrt((sqrt(pi) * pi)) * sqrt(sqrt(pi))))) * (((sin((pi * (angle * 0.005555555555555556))) * 2.0) * (b - a_m)) * (b + a_m));
end

Wolfram

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

Derivation

1. Initial program 54.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lift--.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. difference-of-squares N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower--.f64 N/A

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

   18. lower-*.f64 67.4

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

4. Applied rewrites 66.9%

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

   1. rem-square-sqrt N/A

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

   2. sqrt-unprod N/A

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

   3. lift-PI.f64 N/A

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

   4. add-sqr-sqrt N/A

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

   5. associate-*l* N/A

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

   6. lift-PI.f64 N/A

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

   7. add-sqr-sqrt N/A

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

   8. cube-unmult N/A

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \cos \left(\sqrt{\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{3}}} \cdot \frac{angle}{180}\right) \]

   9. pow3 N/A

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

   10. add-sqr-sqrt N/A

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

   11. lift-PI.f64 N/A

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

   12. sqrt-prod N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. lift-PI.f64 N/A

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

   16. lower-sqrt.f64 N/A

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

   17. lower-sqrt.f64 N/A

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

   18. lift-PI.f64 N/A

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

   19. add-sqr-sqrt N/A

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

6. Applied rewrites 69.5%

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

7. Final simplification 69.5%

   \[\leadsto \cos \left(\frac{angle}{180} \cdot \left(\sqrt{\sqrt{\pi} \cdot \pi} \cdot \sqrt{\sqrt{\pi}}\right)\right) \cdot \left(\left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right) \]
8. Add Preprocessing

Alternative 3: 67.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right) \cdot \left(b - a\_m\right)\right) \cdot \left(b + a\_m\right)\\ \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+122}:\\ \;\;\;\;\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \frac{angle}{180}\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0
         (*
          (* (* (sin (* PI (* angle 0.005555555555555556))) 2.0) (- b a_m))
          (+ b a_m))))
   (if (<= (/ angle 180.0) 5e+122)
     (* (cos (* (* PI angle) 0.005555555555555556)) t_0)
     (* (cos (* (* (sqrt PI) (sqrt PI)) (/ angle 180.0))) t_0))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = ((sin((((double) M_PI) * (angle * 0.005555555555555556))) * 2.0) * (b - a_m)) * (b + a_m);
	double tmp;
	if ((angle / 180.0) <= 5e+122) {
		tmp = cos(((((double) M_PI) * angle) * 0.005555555555555556)) * t_0;
	} else {
		tmp = cos(((sqrt(((double) M_PI)) * sqrt(((double) M_PI))) * (angle / 180.0))) * t_0;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double t_0 = ((Math.sin((Math.PI * (angle * 0.005555555555555556))) * 2.0) * (b - a_m)) * (b + a_m);
	double tmp;
	if ((angle / 180.0) <= 5e+122) {
		tmp = Math.cos(((Math.PI * angle) * 0.005555555555555556)) * t_0;
	} else {
		tmp = Math.cos(((Math.sqrt(Math.PI) * Math.sqrt(Math.PI)) * (angle / 180.0))) * t_0;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	t_0 = ((math.sin((math.pi * (angle * 0.005555555555555556))) * 2.0) * (b - a_m)) * (b + a_m)
	tmp = 0
	if (angle / 180.0) <= 5e+122:
		tmp = math.cos(((math.pi * angle) * 0.005555555555555556)) * t_0
	else:
		tmp = math.cos(((math.sqrt(math.pi) * math.sqrt(math.pi)) * (angle / 180.0))) * t_0
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = Float64(Float64(Float64(sin(Float64(pi * Float64(angle * 0.005555555555555556))) * 2.0) * Float64(b - a_m)) * Float64(b + a_m))
	tmp = 0.0
	if (Float64(angle / 180.0) <= 5e+122)
		tmp = Float64(cos(Float64(Float64(pi * angle) * 0.005555555555555556)) * t_0);
	else
		tmp = Float64(cos(Float64(Float64(sqrt(pi) * sqrt(pi)) * Float64(angle / 180.0))) * t_0);
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	t_0 = ((sin((pi * (angle * 0.005555555555555556))) * 2.0) * (b - a_m)) * (b + a_m);
	tmp = 0.0;
	if ((angle / 180.0) <= 5e+122)
		tmp = cos(((pi * angle) * 0.005555555555555556)) * t_0;
	else
		tmp = cos(((sqrt(pi) * sqrt(pi)) * (angle / 180.0))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[(N[(N[(N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e+122], N[(N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999989e122

   1. Initial program 59.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 73.6

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

   4. Applied rewrites 72.8%

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

   5. Taylor expanded in angle around inf

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

      1. lower-cos.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 74.6

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

   7. Applied rewrites 74.6%

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

   if 4.99999999999999989e122 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 21.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 24.1

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

   4. Applied rewrites 26.2%

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

      1. lift-PI.f64 N/A

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

      2. add-sqr-sqrt N/A

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

      3. lower-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lower-sqrt.f64 40.9

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

   6. Applied rewrites 40.9%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \cos \left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)} \cdot \frac{angle}{180}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+122}:\\ \;\;\;\;\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(\left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;{b}^{2} - {a\_m}^{2} \leq -2 \cdot 10^{-198}:\\ \;\;\;\;\left(\left(\left(\pi \cdot angle\right) \cdot a\_m\right) \cdot -0.011111111111111112\right) \cdot a\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b \cdot b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= (- (pow b 2.0) (pow a_m 2.0)) -2e-198)
   (* (* (* (* PI angle) a_m) -0.011111111111111112) a_m)
   (* (* (* (* b b) PI) angle) 0.011111111111111112)))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if ((pow(b, 2.0) - pow(a_m, 2.0)) <= -2e-198) {
		tmp = (((((double) M_PI) * angle) * a_m) * -0.011111111111111112) * a_m;
	} else {
		tmp = (((b * b) * ((double) M_PI)) * angle) * 0.011111111111111112;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a_m, 2.0)) <= -2e-198) {
		tmp = (((Math.PI * angle) * a_m) * -0.011111111111111112) * a_m;
	} else {
		tmp = (((b * b) * Math.PI) * angle) * 0.011111111111111112;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a_m, 2.0)) <= -2e-198:
		tmp = (((math.pi * angle) * a_m) * -0.011111111111111112) * a_m
	else:
		tmp = (((b * b) * math.pi) * angle) * 0.011111111111111112
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a_m ^ 2.0)) <= -2e-198)
		tmp = Float64(Float64(Float64(Float64(pi * angle) * a_m) * -0.011111111111111112) * a_m);
	else
		tmp = Float64(Float64(Float64(Float64(b * b) * pi) * angle) * 0.011111111111111112);
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if (((b ^ 2.0) - (a_m ^ 2.0)) <= -2e-198)
		tmp = (((pi * angle) * a_m) * -0.011111111111111112) * a_m;
	else
		tmp = (((b * b) * pi) * angle) * 0.011111111111111112;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision], -2e-198], N[(N[(N[(N[(Pi * angle), $MachinePrecision] * a$95$m), $MachinePrecision] * -0.011111111111111112), $MachinePrecision] * a$95$m), $MachinePrecision], N[(N[(N[(N[(b * b), $MachinePrecision] * Pi), $MachinePrecision] * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -1.9999999999999998e-198

   1. Initial program 57.8%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 52.0

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 51.8%

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

   10. Applied rewrites 63.2%

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

   12. Applied rewrites 63.3%

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

   if -1.9999999999999998e-198 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 52.5%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-PI.f64 N/A

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

   5. Applied rewrites 55.9%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 50.0%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -2 \cdot 10^{-198}:\\ \;\;\;\;\left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot -0.011111111111111112\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b \cdot b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.011111111111111112\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{+173}:\\ \;\;\;\;\left(\sin \left(\left(0.011111111111111112 \cdot \pi\right) \cdot angle\right) \cdot \left(b - a\_m\right)\right) \cdot \left(b + a\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -1.54320987654321 \cdot 10^{-5}, \pi \cdot \pi, 1\right) \cdot \left(\left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right) \cdot \left(b - a\_m\right)\right) \cdot \left(b + a\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= b 2.5e+173)
   (* (* (sin (* (* 0.011111111111111112 PI) angle)) (- b a_m)) (+ b a_m))
   (*
    (fma (* (* angle angle) -1.54320987654321e-5) (* PI PI) 1.0)
    (*
     (* (* (sin (* PI (* angle 0.005555555555555556))) 2.0) (- b a_m))
     (+ b a_m)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (b <= 2.5e+173) {
		tmp = (sin(((0.011111111111111112 * ((double) M_PI)) * angle)) * (b - a_m)) * (b + a_m);
	} else {
		tmp = fma(((angle * angle) * -1.54320987654321e-5), (((double) M_PI) * ((double) M_PI)), 1.0) * (((sin((((double) M_PI) * (angle * 0.005555555555555556))) * 2.0) * (b - a_m)) * (b + a_m));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (b <= 2.5e+173)
		tmp = Float64(Float64(sin(Float64(Float64(0.011111111111111112 * pi) * angle)) * Float64(b - a_m)) * Float64(b + a_m));
	else
		tmp = Float64(fma(Float64(Float64(angle * angle) * -1.54320987654321e-5), Float64(pi * pi), 1.0) * Float64(Float64(Float64(sin(Float64(pi * Float64(angle * 0.005555555555555556))) * 2.0) * Float64(b - a_m)) * Float64(b + a_m)));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[b, 2.5e+173], N[(N[(N[Sin[N[(N[(0.011111111111111112 * Pi), $MachinePrecision] * angle), $MachinePrecision]], $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(angle * angle), $MachinePrecision] * -1.54320987654321e-5), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.50000000000000017e173

   1. Initial program 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. clear-num N/A

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

      10. associate-*l/ N/A

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

   4. Applied rewrites 56.6%

      \[\leadsto \color{blue}{\frac{1 \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

   6. Applied rewrites 67.9%

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

   if 2.50000000000000017e173 < b

   1. Initial program 32.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 75.0

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

   4. Applied rewrites 75.0%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \color{blue}{\left(\frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \left(\color{blue}{\left(\frac{-1}{64800} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{64800} \cdot {angle}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{64800} \cdot {angle}^{2}}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \]

      9. lower-PI.f64 N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), 1\right) \]

      10. lower-PI.f64 80.0

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \color{blue}{\pi}, 1\right) \]

   7. Applied rewrites 80.0%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{+173}:\\ \;\;\;\;\left(\sin \left(\left(0.011111111111111112 \cdot \pi\right) \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -1.54320987654321 \cdot 10^{-5}, \pi \cdot \pi, 1\right) \cdot \left(\left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{+265}:\\ \;\;\;\;\left(\sin \left(\left(0.011111111111111112 \cdot \pi\right) \cdot angle\right) \cdot \left(b - a\_m\right)\right) \cdot \left(b + a\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(\pi \cdot \pi\right) \cdot \pi, 0.011111111111111112 \cdot \pi\right) \cdot angle}{\frac{1}{\left(b - a\_m\right) \cdot \left(b + a\_m\right)}}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= b 3.8e+265)
   (* (* (sin (* (* 0.011111111111111112 PI) angle)) (- b a_m)) (+ b a_m))
   (/
    (*
     (fma
      (* -2.2862368541380886e-7 (* angle angle))
      (* (* PI PI) PI)
      (* 0.011111111111111112 PI))
     angle)
    (/ 1.0 (* (- b a_m) (+ b a_m))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (b <= 3.8e+265) {
		tmp = (sin(((0.011111111111111112 * ((double) M_PI)) * angle)) * (b - a_m)) * (b + a_m);
	} else {
		tmp = (fma((-2.2862368541380886e-7 * (angle * angle)), ((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)), (0.011111111111111112 * ((double) M_PI))) * angle) / (1.0 / ((b - a_m) * (b + a_m)));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (b <= 3.8e+265)
		tmp = Float64(Float64(sin(Float64(Float64(0.011111111111111112 * pi) * angle)) * Float64(b - a_m)) * Float64(b + a_m));
	else
		tmp = Float64(Float64(fma(Float64(-2.2862368541380886e-7 * Float64(angle * angle)), Float64(Float64(pi * pi) * pi), Float64(0.011111111111111112 * pi)) * angle) / Float64(1.0 / Float64(Float64(b - a_m) * Float64(b + a_m))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[b, 3.8e+265], N[(N[(N[Sin[N[(N[(0.011111111111111112 * Pi), $MachinePrecision] * angle), $MachinePrecision]], $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-2.2862368541380886e-7 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] + N[(0.011111111111111112 * Pi), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] / N[(1.0 / N[(N[(b - a$95$m), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.80000000000000015e265

   1. Initial program 55.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. clear-num N/A

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

      10. associate-*l/ N/A

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

   4. Applied rewrites 56.1%

      \[\leadsto \color{blue}{\frac{1 \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

   6. Applied rewrites 68.5%

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

   if 3.80000000000000015e265 < b

   1. Initial program 33.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. clear-num N/A

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

      10. associate-*l/ N/A

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

   4. Applied rewrites 33.3%

      \[\leadsto \color{blue}{\frac{1 \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}}} \]

   5. Taylor expanded in angle around 0

      \[\leadsto \frac{\color{blue}{angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{-1}{4374000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}} + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{4374000} \cdot {angle}^{2}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)} \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{4374000} \cdot {angle}^{2}}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      8. unpow3 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \mathsf{PI}\left(\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{{\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \mathsf{PI}\left(\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \mathsf{PI}\left(\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      13. lower-PI.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      14. lower-PI.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \mathsf{PI}\left(\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      15. lower-PI.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), \color{blue}{\frac{1}{90} \cdot \mathsf{PI}\left(\right)}\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      17. lower-PI.f64 66.7

          \[\leadsto \frac{\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(\pi \cdot \pi\right) \cdot \pi, 0.011111111111111112 \cdot \color{blue}{\pi}\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

   7. Applied rewrites 66.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(\pi \cdot \pi\right) \cdot \pi, 0.011111111111111112 \cdot \pi\right) \cdot angle}}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{+265}:\\ \;\;\;\;\left(\sin \left(\left(0.011111111111111112 \cdot \pi\right) \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(\pi \cdot \pi\right) \cdot \pi, 0.011111111111111112 \cdot \pi\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(b + a\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.8% accurate, 5.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+14}:\\ \;\;\;\;\left(\left(\left(0.011111111111111112 \cdot \pi\right) \cdot angle\right) \cdot \left(b - a\_m\right)\right) \cdot \left(b + a\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(\pi \cdot \pi\right) \cdot \pi, 0.011111111111111112 \cdot \pi\right) \cdot angle}{\frac{1}{\left(b - a\_m\right) \cdot \left(b + a\_m\right)}}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 1e+14)
   (* (* (* (* 0.011111111111111112 PI) angle) (- b a_m)) (+ b a_m))
   (/
    (*
     (fma
      (* -2.2862368541380886e-7 (* angle angle))
      (* (* PI PI) PI)
      (* 0.011111111111111112 PI))
     angle)
    (/ 1.0 (* (- b a_m) (+ b a_m))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 1e+14) {
		tmp = (((0.011111111111111112 * ((double) M_PI)) * angle) * (b - a_m)) * (b + a_m);
	} else {
		tmp = (fma((-2.2862368541380886e-7 * (angle * angle)), ((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)), (0.011111111111111112 * ((double) M_PI))) * angle) / (1.0 / ((b - a_m) * (b + a_m)));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 1e+14)
		tmp = Float64(Float64(Float64(Float64(0.011111111111111112 * pi) * angle) * Float64(b - a_m)) * Float64(b + a_m));
	else
		tmp = Float64(Float64(fma(Float64(-2.2862368541380886e-7 * Float64(angle * angle)), Float64(Float64(pi * pi) * pi), Float64(0.011111111111111112 * pi)) * angle) / Float64(1.0 / Float64(Float64(b - a_m) * Float64(b + a_m))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e+14], N[(N[(N[(N[(0.011111111111111112 * Pi), $MachinePrecision] * angle), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-2.2862368541380886e-7 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] + N[(0.011111111111111112 * Pi), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] / N[(1.0 / N[(N[(b - a$95$m), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1e14

   1. Initial program 64.2%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 62.2

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

   5. Applied rewrites 62.2%

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

   7. Applied rewrites 75.8%

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

   if 1e14 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 26.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. clear-num N/A

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

      10. associate-*l/ N/A

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

   4. Applied rewrites 26.0%

      \[\leadsto \color{blue}{\frac{1 \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}}} \]

   5. Taylor expanded in angle around 0

      \[\leadsto \frac{\color{blue}{angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{-1}{4374000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}} + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{4374000} \cdot {angle}^{2}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)} \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{4374000} \cdot {angle}^{2}}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      8. unpow3 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \mathsf{PI}\left(\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{{\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \mathsf{PI}\left(\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \mathsf{PI}\left(\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      13. lower-PI.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      14. lower-PI.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \mathsf{PI}\left(\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      15. lower-PI.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), \color{blue}{\frac{1}{90} \cdot \mathsf{PI}\left(\right)}\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

      17. lower-PI.f64 33.4

          \[\leadsto \frac{\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(\pi \cdot \pi\right) \cdot \pi, 0.011111111111111112 \cdot \color{blue}{\pi}\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]

   7. Applied rewrites 33.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(\pi \cdot \pi\right) \cdot \pi, 0.011111111111111112 \cdot \pi\right) \cdot angle}}{\frac{1}{\left(b - a\right) \cdot \left(a + b\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+14}:\\ \;\;\;\;\left(\left(\left(0.011111111111111112 \cdot \pi\right) \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(\pi \cdot \pi\right) \cdot \pi, 0.011111111111111112 \cdot \pi\right) \cdot angle}{\frac{1}{\left(b - a\right) \cdot \left(b + a\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.9% accurate, 13.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 1.55 \cdot 10^{+140}:\\ \;\;\;\;\left(\left(\left(b - a\_m\right) \cdot \left(b + a\_m\right)\right) \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot angle\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\pi \cdot angle\right) \cdot a\_m\right) \cdot -0.011111111111111112\right) \cdot a\_m\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= a_m 1.55e+140)
   (* (* (* (- b a_m) (+ b a_m)) (* 0.011111111111111112 PI)) angle)
   (* (* (* (* PI angle) a_m) -0.011111111111111112) a_m)))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (a_m <= 1.55e+140) {
		tmp = (((b - a_m) * (b + a_m)) * (0.011111111111111112 * ((double) M_PI))) * angle;
	} else {
		tmp = (((((double) M_PI) * angle) * a_m) * -0.011111111111111112) * a_m;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if (a_m <= 1.55e+140) {
		tmp = (((b - a_m) * (b + a_m)) * (0.011111111111111112 * Math.PI)) * angle;
	} else {
		tmp = (((Math.PI * angle) * a_m) * -0.011111111111111112) * a_m;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if a_m <= 1.55e+140:
		tmp = (((b - a_m) * (b + a_m)) * (0.011111111111111112 * math.pi)) * angle
	else:
		tmp = (((math.pi * angle) * a_m) * -0.011111111111111112) * a_m
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (a_m <= 1.55e+140)
		tmp = Float64(Float64(Float64(Float64(b - a_m) * Float64(b + a_m)) * Float64(0.011111111111111112 * pi)) * angle);
	else
		tmp = Float64(Float64(Float64(Float64(pi * angle) * a_m) * -0.011111111111111112) * a_m);
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if (a_m <= 1.55e+140)
		tmp = (((b - a_m) * (b + a_m)) * (0.011111111111111112 * pi)) * angle;
	else
		tmp = (((pi * angle) * a_m) * -0.011111111111111112) * a_m;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[a$95$m, 1.55e+140], N[(N[(N[(N[(b - a$95$m), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision] * N[(0.011111111111111112 * Pi), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision], N[(N[(N[(N[(Pi * angle), $MachinePrecision] * a$95$m), $MachinePrecision] * -0.011111111111111112), $MachinePrecision] * a$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.55e140

   1. Initial program 56.9%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 51.7

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

   5. Applied rewrites 51.7%

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

   7. Applied rewrites 51.8%

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

   if 1.55e140 < a

   1. Initial program 43.2%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 48.7

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

   5. Applied rewrites 48.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 51.3%

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

   10. Applied rewrites 71.0%

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

   12. Applied rewrites 70.9%

       \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot a\right) \cdot -0.011111111111111112\right) \cdot a \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.55 \cdot 10^{+140}:\\ \;\;\;\;\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot angle\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot -0.011111111111111112\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.8% accurate, 13.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 1.55 \cdot 10^{+140}:\\ \;\;\;\;\left(\left(\left(b - a\_m\right) \cdot \left(b + a\_m\right)\right) \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot angle\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\pi \cdot angle\right) \cdot a\_m\right) \cdot -0.011111111111111112\right) \cdot a\_m\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= a_m 1.55e+140)
   (* (* (* (- b a_m) (+ b a_m)) PI) (* 0.011111111111111112 angle))
   (* (* (* (* PI angle) a_m) -0.011111111111111112) a_m)))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (a_m <= 1.55e+140) {
		tmp = (((b - a_m) * (b + a_m)) * ((double) M_PI)) * (0.011111111111111112 * angle);
	} else {
		tmp = (((((double) M_PI) * angle) * a_m) * -0.011111111111111112) * a_m;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if (a_m <= 1.55e+140) {
		tmp = (((b - a_m) * (b + a_m)) * Math.PI) * (0.011111111111111112 * angle);
	} else {
		tmp = (((Math.PI * angle) * a_m) * -0.011111111111111112) * a_m;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if a_m <= 1.55e+140:
		tmp = (((b - a_m) * (b + a_m)) * math.pi) * (0.011111111111111112 * angle)
	else:
		tmp = (((math.pi * angle) * a_m) * -0.011111111111111112) * a_m
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (a_m <= 1.55e+140)
		tmp = Float64(Float64(Float64(Float64(b - a_m) * Float64(b + a_m)) * pi) * Float64(0.011111111111111112 * angle));
	else
		tmp = Float64(Float64(Float64(Float64(pi * angle) * a_m) * -0.011111111111111112) * a_m);
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if (a_m <= 1.55e+140)
		tmp = (((b - a_m) * (b + a_m)) * pi) * (0.011111111111111112 * angle);
	else
		tmp = (((pi * angle) * a_m) * -0.011111111111111112) * a_m;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[a$95$m, 1.55e+140], N[(N[(N[(N[(b - a$95$m), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] * N[(0.011111111111111112 * angle), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(Pi * angle), $MachinePrecision] * a$95$m), $MachinePrecision] * -0.011111111111111112), $MachinePrecision] * a$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.55e140

   1. Initial program 56.9%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 51.7

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

   5. Applied rewrites 51.7%

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

   7. Applied rewrites 51.7%

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

   if 1.55e140 < a

   1. Initial program 43.2%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 48.7

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

   5. Applied rewrites 48.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 51.3%

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

   10. Applied rewrites 71.0%

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

   12. Applied rewrites 70.9%

       \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot a\right) \cdot -0.011111111111111112\right) \cdot a \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.55 \cdot 10^{+140}:\\ \;\;\;\;\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot angle\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot -0.011111111111111112\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.2% accurate, 16.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \left(\left(\left(0.011111111111111112 \cdot \pi\right) \cdot angle\right) \cdot \left(b - a\_m\right)\right) \cdot \left(b + a\_m\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (* (* (* (* 0.011111111111111112 PI) angle) (- b a_m)) (+ b a_m)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

3. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. unpow2 N/A

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

   11. unpow2 N/A

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

   12. difference-of-squares N/A

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

   13. lower-*.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. lower--.f64 51.3

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

5. Applied rewrites 51.3%

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

7. Applied rewrites 61.3%

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

8. Final simplification 61.3%

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

Alternative 11: 39.3% accurate, 21.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \left(\left(-0.011111111111111112 \cdot a\_m\right) \cdot angle\right) \cdot \left(\pi \cdot a\_m\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (* (* (* -0.011111111111111112 a_m) angle) (* PI a_m)))

C

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

Java

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

Python

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

Julia

a_m = abs(a)
function code(a_m, b, angle)
	return Float64(Float64(Float64(-0.011111111111111112 * a_m) * angle) * Float64(pi * a_m))
end

MATLAB

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

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := N[(N[(N[(-0.011111111111111112 * a$95$m), $MachinePrecision] * angle), $MachinePrecision] * N[(Pi * a$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.8%

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

3. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. unpow2 N/A

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

   11. unpow2 N/A

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

   12. difference-of-squares N/A

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

   13. lower-*.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. lower--.f64 51.3

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

5. Applied rewrites 51.3%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 37.3%

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

10. Applied rewrites 42.5%

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

12. Applied rewrites 42.5%

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

13. Final simplification 42.5%

    \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot angle\right) \cdot \left(\pi \cdot a\right) \]
14. Add Preprocessing

Alternative 12: 39.3% accurate, 21.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \left(\left(\pi \cdot angle\right) \cdot a\_m\right) \cdot \left(-0.011111111111111112 \cdot a\_m\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (* (* (* PI angle) a_m) (* -0.011111111111111112 a_m)))

C

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

Java

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

Python

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

Julia

a_m = abs(a)
function code(a_m, b, angle)
	return Float64(Float64(Float64(pi * angle) * a_m) * Float64(-0.011111111111111112 * a_m))
end

MATLAB

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

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := N[(N[(N[(Pi * angle), $MachinePrecision] * a$95$m), $MachinePrecision] * N[(-0.011111111111111112 * a$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.8%

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

3. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. unpow2 N/A

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

   11. unpow2 N/A

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

   12. difference-of-squares N/A

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

   13. lower-*.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. lower--.f64 51.3

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

5. Applied rewrites 51.3%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 37.3%

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

10. Applied rewrites 42.5%

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

11. Final simplification 42.5%

    \[\leadsto \left(\left(\pi \cdot angle\right) \cdot a\right) \cdot \left(-0.011111111111111112 \cdot a\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024240 
(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)))))