ab-angle->ABCF B

Percentage Accurate: 53.8% → 67.0%

Time: 16.4s

Alternatives: 19

Speedup: 10.3×

• 32.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 1.55 \cdot 10^{+163}:\\ \;\;\;\;\left(a\_m + b\right) \cdot \left(\left(b - a\_m\right) \cdot \sin \left({\pi}^{0.6666666666666666} \cdot \left(\sqrt[3]{\pi} \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a\_m + b\right) \cdot \left(\left(b - a\_m\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= a_m 1.55e+163)
   (*
    (+ a_m b)
    (*
     (- b a_m)
     (sin
      (*
       (pow PI 0.6666666666666666)
       (* (cbrt PI) (* angle 0.011111111111111112))))))
   (*
    (+ a_m b)
    (*
     (- b a_m)
     (*
      angle
      (fma
       -2.2862368541380886e-7
       (* (* angle angle) (* PI (* PI PI)))
       (* PI 0.011111111111111112)))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (a_m <= 1.55e+163) {
		tmp = (a_m + b) * ((b - a_m) * sin((pow(((double) M_PI), 0.6666666666666666) * (cbrt(((double) M_PI)) * (angle * 0.011111111111111112)))));
	} else {
		tmp = (a_m + b) * ((b - a_m) * (angle * fma(-2.2862368541380886e-7, ((angle * angle) * (((double) M_PI) * (((double) M_PI) * ((double) M_PI)))), (((double) M_PI) * 0.011111111111111112))));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (a_m <= 1.55e+163)
		tmp = Float64(Float64(a_m + b) * Float64(Float64(b - a_m) * sin(Float64((pi ^ 0.6666666666666666) * Float64(cbrt(pi) * Float64(angle * 0.011111111111111112))))));
	else
		tmp = Float64(Float64(a_m + b) * Float64(Float64(b - a_m) * Float64(angle * fma(-2.2862368541380886e-7, Float64(Float64(angle * angle) * Float64(pi * Float64(pi * pi))), Float64(pi * 0.011111111111111112)))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[a$95$m, 1.55e+163], N[(N[(a$95$m + b), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[Sin[N[(N[Power[Pi, 0.6666666666666666], $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m + b), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(angle * N[(-2.2862368541380886e-7 * N[(N[(angle * angle), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.55000000000000014e163

   1. Initial program 55.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 65.3%

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

      1. associate-*l* N/A

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

      2. add-cube-cbrt N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. pow1/3 N/A

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

      6. pow1/3 N/A

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

      7. pow-sqr N/A

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

      8. pow-lowering-pow.f64 N/A

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

      9. PI-lowering-PI.f64 N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 N/A

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

      13. cbrt-lowering-cbrt.f64 N/A

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

      14. PI-lowering-PI.f64 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 64.4

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

   6. Applied egg-rr 64.4%

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

   if 1.55000000000000014e163 < a

   1. Initial program 34.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 79.0%

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

   5. Taylor expanded in angle around 0

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

      1. *-lowering-*.f64 N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. PI-lowering-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. PI-lowering-PI.f64 N/A

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

      13. PI-lowering-PI.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. PI-lowering-PI.f64 88.3

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

   7. Simplified 88.3%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.55 \cdot 10^{+163}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left({\pi}^{0.6666666666666666} \cdot \left(\sqrt[3]{\pi} \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := {b}^{2} - {a\_m}^{2}\\ t_1 := \pi \cdot \left(\pi \cdot \pi\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-96}:\\ \;\;\;\;\left(a\_m + b\right) \cdot \left(\left(b - a\_m\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot t\_1, \pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+288}:\\ \;\;\;\;\mathsf{fma}\left(0.011111111111111112 \cdot \left(\pi \cdot angle\right), b \cdot b, 0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a\_m + b\right) \cdot \left(angle \cdot \mathsf{fma}\left(0.011111111111111112, \left(b - a\_m\right) \cdot \pi, \left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(b - a\_m\right) \cdot t\_1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (pow a_m 2.0))) (t_1 (* PI (* PI PI))))
   (if (<= t_0 -2e-96)
     (*
      (+ a_m b)
      (*
       (- b a_m)
       (*
        angle
        (fma
         -2.2862368541380886e-7
         (* (* angle angle) t_1)
         (* PI 0.011111111111111112)))))
     (if (<= t_0 1e+288)
       (fma (* 0.011111111111111112 (* PI angle)) (* b b) 0.0)
       (*
        (+ a_m b)
        (*
         angle
         (fma
          0.011111111111111112
          (* (- b a_m) PI)
          (*
           (* -2.2862368541380886e-7 (* angle angle))
           (* (- b a_m) t_1)))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = pow(b, 2.0) - pow(a_m, 2.0);
	double t_1 = ((double) M_PI) * (((double) M_PI) * ((double) M_PI));
	double tmp;
	if (t_0 <= -2e-96) {
		tmp = (a_m + b) * ((b - a_m) * (angle * fma(-2.2862368541380886e-7, ((angle * angle) * t_1), (((double) M_PI) * 0.011111111111111112))));
	} else if (t_0 <= 1e+288) {
		tmp = fma((0.011111111111111112 * (((double) M_PI) * angle)), (b * b), 0.0);
	} else {
		tmp = (a_m + b) * (angle * fma(0.011111111111111112, ((b - a_m) * ((double) M_PI)), ((-2.2862368541380886e-7 * (angle * angle)) * ((b - a_m) * t_1))));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = Float64((b ^ 2.0) - (a_m ^ 2.0))
	t_1 = Float64(pi * Float64(pi * pi))
	tmp = 0.0
	if (t_0 <= -2e-96)
		tmp = Float64(Float64(a_m + b) * Float64(Float64(b - a_m) * Float64(angle * fma(-2.2862368541380886e-7, Float64(Float64(angle * angle) * t_1), Float64(pi * 0.011111111111111112)))));
	elseif (t_0 <= 1e+288)
		tmp = fma(Float64(0.011111111111111112 * Float64(pi * angle)), Float64(b * b), 0.0);
	else
		tmp = Float64(Float64(a_m + b) * Float64(angle * fma(0.011111111111111112, Float64(Float64(b - a_m) * pi), Float64(Float64(-2.2862368541380886e-7 * Float64(angle * angle)) * Float64(Float64(b - a_m) * t_1)))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-96], N[(N[(a$95$m + b), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(angle * N[(-2.2862368541380886e-7 * N[(N[(angle * angle), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+288], N[(N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + 0.0), $MachinePrecision], N[(N[(a$95$m + b), $MachinePrecision] * N[(angle * N[(0.011111111111111112 * N[(N[(b - a$95$m), $MachinePrecision] * Pi), $MachinePrecision] + N[(N[(-2.2862368541380886e-7 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -1.9999999999999998e-96

   1. Initial program 50.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 62.1%

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

   5. Taylor expanded in angle around 0

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

      1. *-lowering-*.f64 N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. PI-lowering-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. PI-lowering-PI.f64 N/A

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

      13. PI-lowering-PI.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. PI-lowering-PI.f64 66.1

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

   7. Simplified 66.1%

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

   if -1.9999999999999998e-96 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 1e288

   1. Initial program 65.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. PI-lowering-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. --lowering--.f64 61.4

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

   5. Simplified 61.4%

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

   6. Taylor expanded in b around inf

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

   8. Simplified 62.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right), b \cdot b, 0\right)} \]

   if 1e288 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 33.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 78.3%

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

   5. Taylor expanded in angle around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. PI-lowering-PI.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. --lowering--.f64 N/A

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

      15. cube-mult N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 N/A

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

      18. PI-lowering-PI.f64 N/A

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 N/A

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

      21. PI-lowering-PI.f64 N/A

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

      22. PI-lowering-PI.f64 78.4

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

   7. Simplified 78.4%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -2 \cdot 10^{-96}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq 10^{+288}:\\ \;\;\;\;\mathsf{fma}\left(0.011111111111111112 \cdot \left(\pi \cdot angle\right), b \cdot b, 0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(angle \cdot \mathsf{fma}\left(0.011111111111111112, \left(b - a\right) \cdot \pi, \left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := {b}^{2} - {a\_m}^{2}\\ t_1 := \left(a\_m + b\right) \cdot \left(\left(b - a\_m\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-96}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+288}:\\ \;\;\;\;\mathsf{fma}\left(0.011111111111111112 \cdot \left(\pi \cdot angle\right), b \cdot b, 0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (pow a_m 2.0)))
        (t_1
         (*
          (+ a_m b)
          (*
           (- b a_m)
           (*
            angle
            (fma
             -2.2862368541380886e-7
             (* (* angle angle) (* PI (* PI PI)))
             (* PI 0.011111111111111112)))))))
   (if (<= t_0 -2e-96)
     t_1
     (if (<= t_0 1e+288)
       (fma (* 0.011111111111111112 (* PI angle)) (* b b) 0.0)
       t_1))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = pow(b, 2.0) - pow(a_m, 2.0);
	double t_1 = (a_m + b) * ((b - a_m) * (angle * fma(-2.2862368541380886e-7, ((angle * angle) * (((double) M_PI) * (((double) M_PI) * ((double) M_PI)))), (((double) M_PI) * 0.011111111111111112))));
	double tmp;
	if (t_0 <= -2e-96) {
		tmp = t_1;
	} else if (t_0 <= 1e+288) {
		tmp = fma((0.011111111111111112 * (((double) M_PI) * angle)), (b * b), 0.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = Float64((b ^ 2.0) - (a_m ^ 2.0))
	t_1 = Float64(Float64(a_m + b) * Float64(Float64(b - a_m) * Float64(angle * fma(-2.2862368541380886e-7, Float64(Float64(angle * angle) * Float64(pi * Float64(pi * pi))), Float64(pi * 0.011111111111111112)))))
	tmp = 0.0
	if (t_0 <= -2e-96)
		tmp = t_1;
	elseif (t_0 <= 1e+288)
		tmp = fma(Float64(0.011111111111111112 * Float64(pi * angle)), Float64(b * b), 0.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a$95$m + b), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(angle * N[(-2.2862368541380886e-7 * N[(N[(angle * angle), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-96], t$95$1, If[LessEqual[t$95$0, 1e+288], N[(N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + 0.0), $MachinePrecision], t$95$1]]]]

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-96 or 1e288 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 44.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 68.5%

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

   5. Taylor expanded in angle around 0

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

      1. *-lowering-*.f64 N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. PI-lowering-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. PI-lowering-PI.f64 N/A

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

      13. PI-lowering-PI.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. PI-lowering-PI.f64 70.9

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

   7. Simplified 70.9%

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

   if -1.9999999999999998e-96 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 1e288

   1. Initial program 65.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. PI-lowering-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. --lowering--.f64 61.4

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

   5. Simplified 61.4%

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

   6. Taylor expanded in b around inf

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

   8. Simplified 62.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right), b \cdot b, 0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -2 \cdot 10^{-96}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq 10^{+288}:\\ \;\;\;\;\mathsf{fma}\left(0.011111111111111112 \cdot \left(\pi \cdot angle\right), b \cdot b, 0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;{b}^{2} - {a\_m}^{2} \leq -2 \cdot 10^{-172}:\\ \;\;\;\;\left(a\_m \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a\_m \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(b \cdot \left(b - a\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= (- (pow b 2.0) (pow a_m 2.0)) -2e-172)
   (* (* a_m PI) (* -0.011111111111111112 (* a_m angle)))
   (* (* 0.011111111111111112 (* PI angle)) (* b (- b a_m)))))

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-172) {
		tmp = (a_m * ((double) M_PI)) * (-0.011111111111111112 * (a_m * angle));
	} else {
		tmp = (0.011111111111111112 * (((double) M_PI) * angle)) * (b * (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(b, 2.0) - Math.pow(a_m, 2.0)) <= -2e-172) {
		tmp = (a_m * Math.PI) * (-0.011111111111111112 * (a_m * angle));
	} else {
		tmp = (0.011111111111111112 * (Math.PI * angle)) * (b * (b - a_m));
	}
	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-172:
		tmp = (a_m * math.pi) * (-0.011111111111111112 * (a_m * angle))
	else:
		tmp = (0.011111111111111112 * (math.pi * angle)) * (b * (b - a_m))
	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-172)
		tmp = Float64(Float64(a_m * pi) * Float64(-0.011111111111111112 * Float64(a_m * angle)));
	else
		tmp = Float64(Float64(0.011111111111111112 * Float64(pi * angle)) * Float64(b * Float64(b - a_m)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if (((b ^ 2.0) - (a_m ^ 2.0)) <= -2e-172)
		tmp = (a_m * pi) * (-0.011111111111111112 * (a_m * angle));
	else
		tmp = (0.011111111111111112 * (pi * angle)) * (b * (b - a_m));
	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-172], N[(N[(a$95$m * Pi), $MachinePrecision] * N[(-0.011111111111111112 * N[(a$95$m * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision] * N[(b * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $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))) < -2.0000000000000001e-172

   1. Initial program 50.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. PI-lowering-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. --lowering--.f64 39.1

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

   5. Simplified 39.1%

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

   6. Taylor expanded in b around 0

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

      1. *-lowering-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. PI-lowering-PI.f64 39.1

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

   8. Simplified 39.1%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. PI-lowering-PI.f64 N/A

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

      7. *-lowering-*.f64 50.3

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

   10. Applied egg-rr 50.3%

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. PI-lowering-PI.f64 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 51.2

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

   12. Applied egg-rr 51.2%

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

   if -2.0000000000000001e-172 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 53.0%

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

   3. 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. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. PI-lowering-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. --lowering--.f64 59.6

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

   5. Simplified 59.6%

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

   6. Taylor expanded in b around inf

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

   8. Simplified 59.1%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -2 \cdot 10^{-172}:\\ \;\;\;\;\left(a \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(b \cdot \left(b - a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;{b}^{2} - {a\_m}^{2} \leq -2 \cdot 10^{-96}:\\ \;\;\;\;\left(a\_m \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a\_m \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.011111111111111112 \cdot \left(\pi \cdot angle\right), b \cdot b, 0\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= (- (pow b 2.0) (pow a_m 2.0)) -2e-96)
   (* (* a_m PI) (* -0.011111111111111112 (* a_m angle)))
   (fma (* 0.011111111111111112 (* PI angle)) (* b b) 0.0)))

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-96) {
		tmp = (a_m * ((double) M_PI)) * (-0.011111111111111112 * (a_m * angle));
	} else {
		tmp = fma((0.011111111111111112 * (((double) M_PI) * angle)), (b * b), 0.0);
	}
	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-96)
		tmp = Float64(Float64(a_m * pi) * Float64(-0.011111111111111112 * Float64(a_m * angle)));
	else
		tmp = fma(Float64(0.011111111111111112 * Float64(pi * angle)), Float64(b * b), 0.0);
	end
	return 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-96], N[(N[(a$95$m * Pi), $MachinePrecision] * N[(-0.011111111111111112 * N[(a$95$m * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + 0.0), $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-96

   1. Initial program 50.7%

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

   3. 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. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. PI-lowering-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. --lowering--.f64 39.4

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

   5. Simplified 39.4%

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

   6. Taylor expanded in b around 0

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

      1. *-lowering-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. PI-lowering-PI.f64 39.4

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

   8. Simplified 39.4%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. PI-lowering-PI.f64 N/A

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

      7. *-lowering-*.f64 51.6

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

   10. Applied egg-rr 51.6%

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. PI-lowering-PI.f64 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 52.5

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

   12. Applied egg-rr 52.5%

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

   if -1.9999999999999998e-96 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 52.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. PI-lowering-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. --lowering--.f64 58.4

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

   5. Simplified 58.4%

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

   6. Taylor expanded in b around inf

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

   8. Simplified 57.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right), b \cdot b, 0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -2 \cdot 10^{-96}:\\ \;\;\;\;\left(a \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.011111111111111112 \cdot \left(\pi \cdot angle\right), b \cdot b, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;{b}^{2} - {a\_m}^{2} \leq -2 \cdot 10^{-96}:\\ \;\;\;\;\left(a\_m \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a\_m \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= (- (pow b 2.0) (pow a_m 2.0)) -2e-96)
   (* (* a_m PI) (* -0.011111111111111112 (* a_m angle)))
   (* 0.011111111111111112 (* angle (* PI (* b b))))))

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-96) {
		tmp = (a_m * ((double) M_PI)) * (-0.011111111111111112 * (a_m * angle));
	} else {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (b * b)));
	}
	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-96) {
		tmp = (a_m * Math.PI) * (-0.011111111111111112 * (a_m * angle));
	} else {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b * b)));
	}
	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-96:
		tmp = (a_m * math.pi) * (-0.011111111111111112 * (a_m * angle))
	else:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b * b)))
	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-96)
		tmp = Float64(Float64(a_m * pi) * Float64(-0.011111111111111112 * Float64(a_m * angle)));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b * b))));
	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-96)
		tmp = (a_m * pi) * (-0.011111111111111112 * (a_m * angle));
	else
		tmp = 0.011111111111111112 * (angle * (pi * (b * b)));
	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-96], N[(N[(a$95$m * Pi), $MachinePrecision] * N[(-0.011111111111111112 * N[(a$95$m * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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-96

   1. Initial program 50.7%

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

   3. 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. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. PI-lowering-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. --lowering--.f64 39.4

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

   5. Simplified 39.4%

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

   6. Taylor expanded in b around 0

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

      1. *-lowering-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. PI-lowering-PI.f64 39.4

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

   8. Simplified 39.4%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. PI-lowering-PI.f64 N/A

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

      7. *-lowering-*.f64 51.6

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

   10. Applied egg-rr 51.6%

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. PI-lowering-PI.f64 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 52.5

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

   12. Applied egg-rr 52.5%

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

   if -1.9999999999999998e-96 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 52.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. PI-lowering-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. --lowering--.f64 58.4

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

   5. Simplified 58.4%

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

   6. Taylor expanded in b around inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. PI-lowering-PI.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 57.5

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

   8. Simplified 57.5%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -2 \cdot 10^{-96}:\\ \;\;\;\;\left(a \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;{b}^{2} - {a\_m}^{2} \leq -2 \cdot 10^{-96}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a\_m \cdot \left(angle \cdot \left(a\_m \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= (- (pow b 2.0) (pow a_m 2.0)) -2e-96)
   (* -0.011111111111111112 (* a_m (* angle (* a_m PI))))
   (* 0.011111111111111112 (* angle (* PI (* b b))))))

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-96) {
		tmp = -0.011111111111111112 * (a_m * (angle * (a_m * ((double) M_PI))));
	} else {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (b * b)));
	}
	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-96) {
		tmp = -0.011111111111111112 * (a_m * (angle * (a_m * Math.PI)));
	} else {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b * b)));
	}
	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-96:
		tmp = -0.011111111111111112 * (a_m * (angle * (a_m * math.pi)))
	else:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b * b)))
	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-96)
		tmp = Float64(-0.011111111111111112 * Float64(a_m * Float64(angle * Float64(a_m * pi))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b * b))));
	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-96)
		tmp = -0.011111111111111112 * (a_m * (angle * (a_m * pi)));
	else
		tmp = 0.011111111111111112 * (angle * (pi * (b * b)));
	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-96], N[(-0.011111111111111112 * N[(a$95$m * N[(angle * N[(a$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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-96

   1. Initial program 50.7%

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

   3. 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. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. PI-lowering-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. --lowering--.f64 39.4

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

   5. Simplified 39.4%

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

   6. Taylor expanded in b around 0

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

      1. *-lowering-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. PI-lowering-PI.f64 39.4

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

   8. Simplified 39.4%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. PI-lowering-PI.f64 N/A

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

      7. *-lowering-*.f64 51.6

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

   10. Applied egg-rr 51.6%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. PI-lowering-PI.f64 51.7

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

   12. Applied egg-rr 51.7%

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

   if -1.9999999999999998e-96 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 52.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. PI-lowering-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. --lowering--.f64 58.4

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

   5. Simplified 58.4%

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

   6. Taylor expanded in b around inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. PI-lowering-PI.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 57.5

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

   8. Simplified 57.5%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -2 \cdot 10^{-96}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;a\_m \cdot \left(\left(b - a\_m\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a\_m + b\right) \cdot \left(\left(b - a\_m\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= (pow b 2.0) 5e-134)
   (* a_m (* (- b a_m) (sin (* angle (* PI 0.011111111111111112)))))
   (*
    (+ a_m b)
    (*
     (- b a_m)
     (*
      angle
      (fma
       -2.2862368541380886e-7
       (* (* angle angle) (* PI (* PI PI)))
       (* PI 0.011111111111111112)))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (pow(b, 2.0) <= 5e-134) {
		tmp = a_m * ((b - a_m) * sin((angle * (((double) M_PI) * 0.011111111111111112))));
	} else {
		tmp = (a_m + b) * ((b - a_m) * (angle * fma(-2.2862368541380886e-7, ((angle * angle) * (((double) M_PI) * (((double) M_PI) * ((double) M_PI)))), (((double) M_PI) * 0.011111111111111112))));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if ((b ^ 2.0) <= 5e-134)
		tmp = Float64(a_m * Float64(Float64(b - a_m) * sin(Float64(angle * Float64(pi * 0.011111111111111112)))));
	else
		tmp = Float64(Float64(a_m + b) * Float64(Float64(b - a_m) * Float64(angle * fma(-2.2862368541380886e-7, Float64(Float64(angle * angle) * Float64(pi * Float64(pi * pi))), Float64(pi * 0.011111111111111112)))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[N[Power[b, 2.0], $MachinePrecision], 5e-134], N[(a$95$m * N[(N[(b - a$95$m), $MachinePrecision] * N[Sin[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m + b), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(angle * N[(-2.2862368541380886e-7 * N[(N[(angle * angle), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b #s(literal 2 binary64)) < 5.0000000000000003e-134

   1. Initial program 58.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 66.3%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. PI-lowering-PI.f64 67.4

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

   6. Applied egg-rr 67.4%

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

   7. Taylor expanded in b around 0

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

   9. Simplified 66.9%

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

   if 5.0000000000000003e-134 < (pow.f64 b #s(literal 2 binary64))

   1. Initial program 47.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 68.6%

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

   5. Taylor expanded in angle around 0

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

      1. *-lowering-*.f64 N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. PI-lowering-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. PI-lowering-PI.f64 N/A

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

      13. PI-lowering-PI.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. PI-lowering-PI.f64 68.5

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

   7. Simplified 68.5%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;a \cdot \left(\left(b - a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;{a\_m}^{2} \leq 2 \cdot 10^{-104}:\\ \;\;\;\;\sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a\_m + b\right) \cdot \left(\left(b - a\_m\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= (pow a_m 2.0) 2e-104)
   (* (sin (* 0.011111111111111112 (* PI angle))) (* b b))
   (*
    (+ a_m b)
    (*
     (- b a_m)
     (*
      angle
      (fma
       -2.2862368541380886e-7
       (* (* angle angle) (* PI (* PI PI)))
       (* PI 0.011111111111111112)))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (pow(a_m, 2.0) <= 2e-104) {
		tmp = sin((0.011111111111111112 * (((double) M_PI) * angle))) * (b * b);
	} else {
		tmp = (a_m + b) * ((b - a_m) * (angle * fma(-2.2862368541380886e-7, ((angle * angle) * (((double) M_PI) * (((double) M_PI) * ((double) M_PI)))), (((double) M_PI) * 0.011111111111111112))));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if ((a_m ^ 2.0) <= 2e-104)
		tmp = Float64(sin(Float64(0.011111111111111112 * Float64(pi * angle))) * Float64(b * b));
	else
		tmp = Float64(Float64(a_m + b) * Float64(Float64(b - a_m) * Float64(angle * fma(-2.2862368541380886e-7, Float64(Float64(angle * angle) * Float64(pi * Float64(pi * pi))), Float64(pi * 0.011111111111111112)))));
	end
	return 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-104], N[(N[Sin[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m + b), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(angle * N[(-2.2862368541380886e-7 * N[(N[(angle * angle), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a #s(literal 2 binary64)) < 1.99999999999999985e-104

   1. Initial program 62.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 67.7%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. PI-lowering-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 64.3

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

   7. Simplified 64.3%

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

   if 1.99999999999999985e-104 < (pow.f64 a #s(literal 2 binary64))

   1. Initial program 44.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 67.5%

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

   5. Taylor expanded in angle around 0

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

      1. *-lowering-*.f64 N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. PI-lowering-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. PI-lowering-PI.f64 N/A

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

      13. PI-lowering-PI.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. PI-lowering-PI.f64 69.9

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

   7. Simplified 69.9%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{-104}:\\ \;\;\;\;\sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+78}:\\ \;\;\;\;\left(a\_m + b\right) \cdot \left(\left(b - a\_m\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(\left(a\_m + b\right) \cdot \left(b - a\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 2e+78)
   (*
    (+ a_m b)
    (*
     (- b a_m)
     (*
      angle
      (fma
       -2.2862368541380886e-7
       (* (* angle angle) (* PI (* PI PI)))
       (* PI 0.011111111111111112)))))
   (* (sin (* 0.011111111111111112 (* PI angle))) (* (+ a_m b) (- b a_m)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 2e+78) {
		tmp = (a_m + b) * ((b - a_m) * (angle * fma(-2.2862368541380886e-7, ((angle * angle) * (((double) M_PI) * (((double) M_PI) * ((double) M_PI)))), (((double) M_PI) * 0.011111111111111112))));
	} else {
		tmp = sin((0.011111111111111112 * (((double) M_PI) * angle))) * ((a_m + b) * (b - a_m));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 2e+78)
		tmp = Float64(Float64(a_m + b) * Float64(Float64(b - a_m) * Float64(angle * fma(-2.2862368541380886e-7, Float64(Float64(angle * angle) * Float64(pi * Float64(pi * pi))), Float64(pi * 0.011111111111111112)))));
	else
		tmp = Float64(sin(Float64(0.011111111111111112 * Float64(pi * angle))) * Float64(Float64(a_m + b) * 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], 2e+78], N[(N[(a$95$m + b), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(angle * N[(-2.2862368541380886e-7 * N[(N[(angle * angle), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(a$95$m + b), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000002e78

   1. Initial program 57.5%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 76.5%

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

   5. Taylor expanded in angle around 0

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

      1. *-lowering-*.f64 N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. PI-lowering-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. PI-lowering-PI.f64 N/A

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

      13. PI-lowering-PI.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. PI-lowering-PI.f64 75.3

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

   7. Simplified 75.3%

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

   if 2.00000000000000002e78 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 30.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. difference-of-squares N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. --lowering--.f64 N/A

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

      11. 2-sin N/A

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

      12. count-2 N/A

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

      13. sin-lowering-sin.f64 N/A

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

      14. associate-*r/ N/A

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

      15. div-inv N/A

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

      16. associate-*r/ N/A

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

      17. div-inv N/A

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

   4. Applied egg-rr 33.6%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+78}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 1.35 \cdot 10^{+165}:\\ \;\;\;\;\left(a\_m + b\right) \cdot \left(\left(b - a\_m\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a\_m + b\right) \cdot \left(\left(b - a\_m\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= a_m 1.35e+165)
   (* (+ a_m b) (* (- b a_m) (sin (* 0.011111111111111112 (* PI angle)))))
   (*
    (+ a_m b)
    (*
     (- b a_m)
     (*
      angle
      (fma
       -2.2862368541380886e-7
       (* (* angle angle) (* PI (* PI PI)))
       (* PI 0.011111111111111112)))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (a_m <= 1.35e+165) {
		tmp = (a_m + b) * ((b - a_m) * sin((0.011111111111111112 * (((double) M_PI) * angle))));
	} else {
		tmp = (a_m + b) * ((b - a_m) * (angle * fma(-2.2862368541380886e-7, ((angle * angle) * (((double) M_PI) * (((double) M_PI) * ((double) M_PI)))), (((double) M_PI) * 0.011111111111111112))));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (a_m <= 1.35e+165)
		tmp = Float64(Float64(a_m + b) * Float64(Float64(b - a_m) * sin(Float64(0.011111111111111112 * Float64(pi * angle)))));
	else
		tmp = Float64(Float64(a_m + b) * Float64(Float64(b - a_m) * Float64(angle * fma(-2.2862368541380886e-7, Float64(Float64(angle * angle) * Float64(pi * Float64(pi * pi))), Float64(pi * 0.011111111111111112)))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[a$95$m, 1.35e+165], N[(N[(a$95$m + b), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m + b), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(angle * N[(-2.2862368541380886e-7 * N[(N[(angle * angle), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.35e165

   1. Initial program 55.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 65.3%

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

   if 1.35e165 < a

   1. Initial program 34.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 79.0%

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

   5. Taylor expanded in angle around 0

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

      1. *-lowering-*.f64 N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. PI-lowering-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. PI-lowering-PI.f64 N/A

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

      13. PI-lowering-PI.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. PI-lowering-PI.f64 88.3

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

   7. Simplified 88.3%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.35 \cdot 10^{+165}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 5 \cdot 10^{+164}:\\ \;\;\;\;\left(a\_m + b\right) \cdot \left(\left(b - a\_m\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a\_m + b\right) \cdot \left(\left(b - a\_m\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= a_m 5e+164)
   (* (+ a_m b) (* (- b a_m) (sin (* angle (* PI 0.011111111111111112)))))
   (*
    (+ a_m b)
    (*
     (- b a_m)
     (*
      angle
      (fma
       -2.2862368541380886e-7
       (* (* angle angle) (* PI (* PI PI)))
       (* PI 0.011111111111111112)))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (a_m <= 5e+164) {
		tmp = (a_m + b) * ((b - a_m) * sin((angle * (((double) M_PI) * 0.011111111111111112))));
	} else {
		tmp = (a_m + b) * ((b - a_m) * (angle * fma(-2.2862368541380886e-7, ((angle * angle) * (((double) M_PI) * (((double) M_PI) * ((double) M_PI)))), (((double) M_PI) * 0.011111111111111112))));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (a_m <= 5e+164)
		tmp = Float64(Float64(a_m + b) * Float64(Float64(b - a_m) * sin(Float64(angle * Float64(pi * 0.011111111111111112)))));
	else
		tmp = Float64(Float64(a_m + b) * Float64(Float64(b - a_m) * Float64(angle * fma(-2.2862368541380886e-7, Float64(Float64(angle * angle) * Float64(pi * Float64(pi * pi))), Float64(pi * 0.011111111111111112)))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[a$95$m, 5e+164], N[(N[(a$95$m + b), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[Sin[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m + b), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(angle * N[(-2.2862368541380886e-7 * N[(N[(angle * angle), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.9999999999999995e164

   1. Initial program 55.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 65.3%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. PI-lowering-PI.f64 65.8

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

   6. Applied egg-rr 65.8%

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

   if 4.9999999999999995e164 < a

   1. Initial program 34.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 79.0%

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

   5. Taylor expanded in angle around 0

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

      1. *-lowering-*.f64 N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. PI-lowering-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. PI-lowering-PI.f64 N/A

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

      13. PI-lowering-PI.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. PI-lowering-PI.f64 88.3

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

   7. Simplified 88.3%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{+164}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 1.05 \cdot 10^{-86}:\\ \;\;\;\;\left(b \cdot 2\right) \cdot \left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{elif}\;a\_m \leq 3.1 \cdot 10^{+163}:\\ \;\;\;\;\left(\left(b - a\_m\right) \cdot \left(\pi \cdot angle\right)\right) \cdot \left(\left(a\_m + b\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a\_m + b\right) \cdot \left(\left(b - a\_m\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= a_m 1.05e-86)
   (* (* b 2.0) (* b (sin (* PI (* angle 0.005555555555555556)))))
   (if (<= a_m 3.1e+163)
     (* (* (- b a_m) (* PI angle)) (* (+ a_m b) 0.011111111111111112))
     (*
      (+ a_m b)
      (*
       (- b a_m)
       (*
        angle
        (fma
         -2.2862368541380886e-7
         (* (* angle angle) (* PI (* PI PI)))
         (* PI 0.011111111111111112))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (a_m <= 1.05e-86) {
		tmp = (b * 2.0) * (b * sin((((double) M_PI) * (angle * 0.005555555555555556))));
	} else if (a_m <= 3.1e+163) {
		tmp = ((b - a_m) * (((double) M_PI) * angle)) * ((a_m + b) * 0.011111111111111112);
	} else {
		tmp = (a_m + b) * ((b - a_m) * (angle * fma(-2.2862368541380886e-7, ((angle * angle) * (((double) M_PI) * (((double) M_PI) * ((double) M_PI)))), (((double) M_PI) * 0.011111111111111112))));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (a_m <= 1.05e-86)
		tmp = Float64(Float64(b * 2.0) * Float64(b * sin(Float64(pi * Float64(angle * 0.005555555555555556)))));
	elseif (a_m <= 3.1e+163)
		tmp = Float64(Float64(Float64(b - a_m) * Float64(pi * angle)) * Float64(Float64(a_m + b) * 0.011111111111111112));
	else
		tmp = Float64(Float64(a_m + b) * Float64(Float64(b - a_m) * Float64(angle * fma(-2.2862368541380886e-7, Float64(Float64(angle * angle) * Float64(pi * Float64(pi * pi))), Float64(pi * 0.011111111111111112)))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[a$95$m, 1.05e-86], N[(N[(b * 2.0), $MachinePrecision] * N[(b * N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 3.1e+163], N[(N[(N[(b - a$95$m), $MachinePrecision] * N[(Pi * angle), $MachinePrecision]), $MachinePrecision] * N[(N[(a$95$m + b), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m + b), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(angle * N[(-2.2862368541380886e-7 * N[(N[(angle * angle), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.05e-86

   1. Initial program 55.2%

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

   3. Taylor expanded in 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. *-lowering-*.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. cos-lowering-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. PI-lowering-PI.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 N/A

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

   5. Simplified 46.1%

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

   6. Taylor expanded in angle around 0

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

   8. Simplified 49.1%

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

      1. *-lft-identity N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. sin-lowering-sin.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. PI-lowering-PI.f64 N/A

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

      15. div-inv N/A

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

      16. metadata-eval N/A

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

      17. *-lowering-*.f64 53.3

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

   10. Applied egg-rr 53.3%

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

   if 1.05e-86 < a < 3.10000000000000029e163

   1. Initial program 55.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. PI-lowering-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. --lowering--.f64 51.3

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

   5. Simplified 51.3%

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

      1. remove-double-div N/A

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

      2. un-div-inv N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. un-div-inv N/A

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

      9. clear-num N/A

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

      10. /-rgt-identity N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 N/A

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

      15. PI-lowering-PI.f64 N/A

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

      16. --lowering--.f64 N/A

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

      17. un-div-inv N/A

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

      18. remove-double-div N/A

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

      19. *-lowering-*.f64 N/A

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

      20. +-lowering-+.f64 58.0

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

   7. Applied egg-rr 58.0%

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

   if 3.10000000000000029e163 < a

   1. Initial program 34.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 79.0%

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

   5. Taylor expanded in angle around 0

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

      1. *-lowering-*.f64 N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \color{blue}{\left(angle \cdot angle\right)} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \color{blue}{\left(angle \cdot angle\right)} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      13. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\frac{1}{90} \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

      15. PI-lowering-PI.f64 88.3

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \color{blue}{\pi}\right)\right)\right) \]

   7. Simplified 88.3%

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \pi\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.05 \cdot 10^{-86}:\\ \;\;\;\;\left(b \cdot 2\right) \cdot \left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{+163}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\pi \cdot angle\right)\right) \cdot \left(\left(a + b\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 1.06 \cdot 10^{-52}:\\ \;\;\;\;\left(a\_m + b\right) \cdot \left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a\_m + b\right) \cdot \left(\left(b - a\_m\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= a_m 1.06e-52)
   (* (+ a_m b) (* b (sin (* angle (* PI 0.011111111111111112)))))
   (*
    (+ a_m b)
    (*
     (- b a_m)
     (*
      angle
      (fma
       -2.2862368541380886e-7
       (* (* angle angle) (* PI (* PI PI)))
       (* PI 0.011111111111111112)))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (a_m <= 1.06e-52) {
		tmp = (a_m + b) * (b * sin((angle * (((double) M_PI) * 0.011111111111111112))));
	} else {
		tmp = (a_m + b) * ((b - a_m) * (angle * fma(-2.2862368541380886e-7, ((angle * angle) * (((double) M_PI) * (((double) M_PI) * ((double) M_PI)))), (((double) M_PI) * 0.011111111111111112))));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (a_m <= 1.06e-52)
		tmp = Float64(Float64(a_m + b) * Float64(b * sin(Float64(angle * Float64(pi * 0.011111111111111112)))));
	else
		tmp = Float64(Float64(a_m + b) * Float64(Float64(b - a_m) * Float64(angle * fma(-2.2862368541380886e-7, Float64(Float64(angle * angle) * Float64(pi * Float64(pi * pi))), Float64(pi * 0.011111111111111112)))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[a$95$m, 1.06e-52], N[(N[(a$95$m + b), $MachinePrecision] * N[(b * N[Sin[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m + b), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(angle * N[(-2.2862368541380886e-7 * N[(N[(angle * angle), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.06e-52

   1. Initial program 56.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      4. unpow2 N/A

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \]

      6. difference-of-squares N/A

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right)} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(b + a\right)} \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right)} \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      12. 2-sin N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\sin \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right) \]

      13. count-2 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180} + \mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \]

   4. Applied egg-rr 65.6%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{90}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)} \cdot angle\right)\right) \]

      7. PI-lowering-PI.f64 66.8

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(0.011111111111111112 \cdot \color{blue}{\pi}\right) \cdot angle\right)\right) \]

   6. Applied egg-rr 66.8%

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\left(0.011111111111111112 \cdot \pi\right) \cdot angle\right)}\right) \]

   7. Taylor expanded in b around inf

      \[\leadsto \left(b + a\right) \cdot \left(\color{blue}{b} \cdot \sin \left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 51.9%

      \[\leadsto \left(b + a\right) \cdot \left(\color{blue}{b} \cdot \sin \left(\left(0.011111111111111112 \cdot \pi\right) \cdot angle\right)\right) \]

   if 1.06e-52 < a

   1. Initial program 43.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      4. unpow2 N/A

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \]

      6. difference-of-squares N/A

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right)} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(b + a\right)} \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right)} \]

      11. --lowering--.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      12. 2-sin N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\sin \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right) \]

      13. count-2 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180} + \mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \]

   4. Applied egg-rr 71.5%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)} \]

   5. Taylor expanded in angle around 0

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4374000}, {angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \color{blue}{{angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \color{blue}{\left(angle \cdot angle\right)} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \color{blue}{\left(angle \cdot angle\right)} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      13. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\frac{1}{90} \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

      15. PI-lowering-PI.f64 74.8

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \color{blue}{\pi}\right)\right)\right) \]

   7. Simplified 74.8%

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \pi\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.06 \cdot 10^{-52}:\\ \;\;\;\;\left(a + b\right) \cdot \left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 60.9% accurate, 7.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 4 \cdot 10^{-184}:\\ \;\;\;\;\mathsf{fma}\left(0.011111111111111112 \cdot \left(\pi \cdot angle\right), b \cdot b, 0\right)\\ \mathbf{elif}\;a\_m \leq 5.05 \cdot 10^{+222}:\\ \;\;\;\;\left(\left(b - a\_m\right) \cdot \left(\pi \cdot angle\right)\right) \cdot \left(\left(a\_m + b\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(a\_m \cdot a\_m\right) \cdot \left(angle \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \pi \cdot \left(\pi \cdot \pi\right), \pi \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= a_m 4e-184)
   (fma (* 0.011111111111111112 (* PI angle)) (* b b) 0.0)
   (if (<= a_m 5.05e+222)
     (* (* (- b a_m) (* PI angle)) (* (+ a_m b) 0.011111111111111112))
     (*
      -2.0
      (*
       (* a_m a_m)
       (*
        angle
        (fma
         (* (* angle angle) -2.8577960676726107e-8)
         (* PI (* PI PI))
         (* PI 0.005555555555555556))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (a_m <= 4e-184) {
		tmp = fma((0.011111111111111112 * (((double) M_PI) * angle)), (b * b), 0.0);
	} else if (a_m <= 5.05e+222) {
		tmp = ((b - a_m) * (((double) M_PI) * angle)) * ((a_m + b) * 0.011111111111111112);
	} else {
		tmp = -2.0 * ((a_m * a_m) * (angle * fma(((angle * angle) * -2.8577960676726107e-8), (((double) M_PI) * (((double) M_PI) * ((double) M_PI))), (((double) M_PI) * 0.005555555555555556))));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (a_m <= 4e-184)
		tmp = fma(Float64(0.011111111111111112 * Float64(pi * angle)), Float64(b * b), 0.0);
	elseif (a_m <= 5.05e+222)
		tmp = Float64(Float64(Float64(b - a_m) * Float64(pi * angle)) * Float64(Float64(a_m + b) * 0.011111111111111112));
	else
		tmp = Float64(-2.0 * Float64(Float64(a_m * a_m) * Float64(angle * fma(Float64(Float64(angle * angle) * -2.8577960676726107e-8), Float64(pi * Float64(pi * pi)), Float64(pi * 0.005555555555555556)))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[a$95$m, 4e-184], N[(N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + 0.0), $MachinePrecision], If[LessEqual[a$95$m, 5.05e+222], N[(N[(N[(b - a$95$m), $MachinePrecision] * N[(Pi * angle), $MachinePrecision]), $MachinePrecision] * N[(N[(a$95$m + b), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(a$95$m * a$95$m), $MachinePrecision] * N[(angle * N[(N[(N[(angle * angle), $MachinePrecision] * -2.8577960676726107e-8), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 4.0000000000000002e-184

   1. Initial program 56.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      12. --lowering--.f64 54.7

          \[\leadsto \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Simplified 54.7%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{90} \cdot \frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)}{b}\right)} \]

   8. Simplified 49.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right), b \cdot b, 0\right)} \]

   if 4.0000000000000002e-184 < a < 5.0499999999999998e222

   1. Initial program 47.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      12. --lowering--.f64 47.2

          \[\leadsto \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Simplified 47.2%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]
   6. Step-by-step derivation

      1. remove-double-div N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\frac{1}{\frac{1}{\left(b + a\right) \cdot \left(b - a\right)}}} \]

      2. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{\frac{1}{\left(b + a\right) \cdot \left(b - a\right)}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}}}{\frac{1}{\left(b + a\right) \cdot \left(b - a\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}}{\frac{1}{\color{blue}{\left(b - a\right) \cdot \left(b + a\right)}}} \]

      5. associate-/r* N/A

         \[\leadsto \frac{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}}{\color{blue}{\frac{\frac{1}{b - a}}{b + a}}} \]

      6. div-inv N/A

         \[\leadsto \frac{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}}{\color{blue}{\frac{1}{b - a} \cdot \frac{1}{b + a}}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{angle \cdot \mathsf{PI}\left(\right)}{\frac{1}{b - a}} \cdot \frac{\frac{1}{90}}{\frac{1}{b + a}}} \]

      8. un-div-inv N/A

         \[\leadsto \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{\frac{1}{b - a}}\right)} \cdot \frac{\frac{1}{90}}{\frac{1}{b + a}} \]

      9. clear-num N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\frac{b - a}{1}}\right) \cdot \frac{\frac{1}{90}}{\frac{1}{b + a}} \]

      10. /-rgt-identity N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(b - a\right)}\right) \cdot \frac{\frac{1}{90}}{\frac{1}{b + a}} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{\frac{1}{90}}{\frac{1}{b + a}}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b - a\right)\right)} \cdot \frac{\frac{1}{90}}{\frac{1}{b + a}} \]

      13. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot \frac{\frac{1}{90}}{\frac{1}{b + a}} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot \frac{\frac{1}{90}}{\frac{1}{b + a}} \]

      15. PI-lowering-PI.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \frac{\frac{1}{90}}{\frac{1}{b + a}} \]

      16. --lowering--.f64 N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \color{blue}{\left(b - a\right)}\right) \cdot \frac{\frac{1}{90}}{\frac{1}{b + a}} \]

      17. un-div-inv N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot \frac{1}{\frac{1}{b + a}}\right)} \]

      18. remove-double-div N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(b + a\right)}\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(b + a\right)\right)} \]

      20. +-lowering-+.f64 59.4

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \left(0.011111111111111112 \cdot \color{blue}{\left(b + a\right)}\right) \]

   7. Applied egg-rr 59.4%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \left(0.011111111111111112 \cdot \left(b + a\right)\right)} \]

   if 5.0499999999999998e222 < a

   1. Initial program 44.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\frac{{b}^{2} \cdot {b}^{2} - {a}^{2} \cdot {a}^{2}}{{b}^{2} + {a}^{2}}}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      2. clear-num N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\frac{1}{\frac{{b}^{2} + {a}^{2}}{{b}^{2} \cdot {b}^{2} - {a}^{2} \cdot {a}^{2}}}}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\frac{1}{\frac{{b}^{2} + {a}^{2}}{{b}^{2} \cdot {b}^{2} - {a}^{2} \cdot {a}^{2}}}}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. clear-num N/A

         \[\leadsto \left(\left(2 \cdot \frac{1}{\color{blue}{\frac{1}{\frac{{b}^{2} \cdot {b}^{2} - {a}^{2} \cdot {a}^{2}}{{b}^{2} + {a}^{2}}}}}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      5. flip-- N/A

         \[\leadsto \left(\left(2 \cdot \frac{1}{\frac{1}{\color{blue}{{b}^{2} - {a}^{2}}}}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \frac{1}{\color{blue}{\frac{1}{{b}^{2} - {a}^{2}}}}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\left(2 \cdot \frac{1}{\frac{1}{\color{blue}{b \cdot b} - {a}^{2}}}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\left(2 \cdot \frac{1}{\frac{1}{b \cdot b - \color{blue}{a \cdot a}}}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \left(\left(2 \cdot \frac{1}{\frac{1}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}}}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \frac{1}{\frac{1}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}}}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\left(2 \cdot \frac{1}{\frac{1}{\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)}}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      12. --lowering--.f64 69.3

          \[\leadsto \left(\left(2 \cdot \frac{1}{\frac{1}{\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}}}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Applied egg-rr 69.3%

      \[\leadsto \left(\left(2 \cdot \color{blue}{\frac{1}{\frac{1}{\left(b + a\right) \cdot \left(b - a\right)}}}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto -2 \cdot \color{blue}{\left(\left({a}^{2} \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(\left({a}^{2} \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto -2 \cdot \left(\color{blue}{\left({a}^{2} \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto -2 \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto -2 \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto -2 \cdot \left(\left(\left(a \cdot a\right) \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto -2 \cdot \left(\left(\left(a \cdot a\right) \cdot \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto -2 \cdot \left(\left(\left(a \cdot a\right) \cdot \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto -2 \cdot \left(\left(\left(a \cdot a\right) \cdot \cos \left(\color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{180}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      11. PI-lowering-PI.f64 N/A

          \[\leadsto -2 \cdot \left(\left(\left(a \cdot a\right) \cdot \cos \left(\left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{180}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto -2 \cdot \left(\left(\left(a \cdot a\right) \cdot \cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto -2 \cdot \left(\left(\left(a \cdot a\right) \cdot \cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right) \cdot \sin \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto -2 \cdot \left(\left(\left(a \cdot a\right) \cdot \cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right) \cdot \sin \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto -2 \cdot \left(\left(\left(a \cdot a\right) \cdot \cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right) \cdot \sin \left(\color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{180}\right)\right) \]

      16. PI-lowering-PI.f64 69.3

          \[\leadsto -2 \cdot \left(\left(\left(a \cdot a\right) \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\pi}\right) \cdot 0.005555555555555556\right)\right) \]

   7. Simplified 69.3%

      \[\leadsto \color{blue}{-2 \cdot \left(\left(\left(a \cdot a\right) \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)} \]

   8. Taylor expanded in angle around 0

      \[\leadsto -2 \cdot \left(\color{blue}{{a}^{2}} \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto -2 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right) \]

      2. *-lowering-*.f64 58.6

         \[\leadsto -2 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) \]

   10. Simplified 58.6%

       \[\leadsto -2 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) \]

   11. Taylor expanded in angle around 0

       \[\leadsto -2 \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto -2 \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto -2 \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot \left(\color{blue}{\left(\frac{-1}{34992000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}} + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto -2 \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{34992000} \cdot {angle}^{2}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto -2 \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{34992000} \cdot {angle}^{2}}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto -2 \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto -2 \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       7. cube-mult N/A

          \[\leadsto -2 \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto -2 \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto -2 \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       10. PI-lowering-PI.f64 N/A

           \[\leadsto -2 \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto -2 \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto -2 \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       13. PI-lowering-PI.f64 N/A

           \[\leadsto -2 \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       14. PI-lowering-PI.f64 N/A

           \[\leadsto -2 \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto -2 \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto -2 \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}}\right)\right)\right) \]

       17. PI-lowering-PI.f64 72.9

           \[\leadsto -2 \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), \pi \cdot \left(\pi \cdot \pi\right), \color{blue}{\pi} \cdot 0.005555555555555556\right)\right)\right) \]

   13. Simplified 72.9%

       \[\leadsto -2 \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), \pi \cdot \left(\pi \cdot \pi\right), \pi \cdot 0.005555555555555556\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4 \cdot 10^{-184}:\\ \;\;\;\;\mathsf{fma}\left(0.011111111111111112 \cdot \left(\pi \cdot angle\right), b \cdot b, 0\right)\\ \mathbf{elif}\;a \leq 5.05 \cdot 10^{+222}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\pi \cdot angle\right)\right) \cdot \left(\left(a + b\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \pi \cdot \left(\pi \cdot \pi\right), \pi \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 62.5% accurate, 10.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 100:\\ \;\;\;\;\left(\left(b - a\_m\right) \cdot \left(\pi \cdot angle\right)\right) \cdot \left(\left(a\_m + b\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(b \cdot \left(a\_m + b\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 100.0)
   (* (* (- b a_m) (* PI angle)) (* (+ a_m b) 0.011111111111111112))
   (* (* 0.011111111111111112 (* PI angle)) (* b (+ a_m b)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 100.0) {
		tmp = ((b - a_m) * (((double) M_PI) * angle)) * ((a_m + b) * 0.011111111111111112);
	} else {
		tmp = (0.011111111111111112 * (((double) M_PI) * angle)) * (b * (a_m + b));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 100.0) {
		tmp = ((b - a_m) * (Math.PI * angle)) * ((a_m + b) * 0.011111111111111112);
	} else {
		tmp = (0.011111111111111112 * (Math.PI * angle)) * (b * (a_m + b));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if (angle / 180.0) <= 100.0:
		tmp = ((b - a_m) * (math.pi * angle)) * ((a_m + b) * 0.011111111111111112)
	else:
		tmp = (0.011111111111111112 * (math.pi * angle)) * (b * (a_m + b))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 100.0)
		tmp = Float64(Float64(Float64(b - a_m) * Float64(pi * angle)) * Float64(Float64(a_m + b) * 0.011111111111111112));
	else
		tmp = Float64(Float64(0.011111111111111112 * Float64(pi * angle)) * Float64(b * Float64(a_m + b)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if ((angle / 180.0) <= 100.0)
		tmp = ((b - a_m) * (pi * angle)) * ((a_m + b) * 0.011111111111111112);
	else
		tmp = (0.011111111111111112 * (pi * angle)) * (b * (a_m + b));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 100.0], N[(N[(N[(b - a$95$m), $MachinePrecision] * N[(Pi * angle), $MachinePrecision]), $MachinePrecision] * N[(N[(a$95$m + b), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision] * N[(b * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 100

   1. Initial program 58.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      12. --lowering--.f64 61.5

          \[\leadsto \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Simplified 61.5%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]
   6. Step-by-step derivation

      1. remove-double-div N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\frac{1}{\frac{1}{\left(b + a\right) \cdot \left(b - a\right)}}} \]

      2. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)}{\frac{1}{\left(b + a\right) \cdot \left(b - a\right)}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}}}{\frac{1}{\left(b + a\right) \cdot \left(b - a\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}}{\frac{1}{\color{blue}{\left(b - a\right) \cdot \left(b + a\right)}}} \]

      5. associate-/r* N/A

         \[\leadsto \frac{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}}{\color{blue}{\frac{\frac{1}{b - a}}{b + a}}} \]

      6. div-inv N/A

         \[\leadsto \frac{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}}{\color{blue}{\frac{1}{b - a} \cdot \frac{1}{b + a}}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{angle \cdot \mathsf{PI}\left(\right)}{\frac{1}{b - a}} \cdot \frac{\frac{1}{90}}{\frac{1}{b + a}}} \]

      8. un-div-inv N/A

         \[\leadsto \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{\frac{1}{b - a}}\right)} \cdot \frac{\frac{1}{90}}{\frac{1}{b + a}} \]

      9. clear-num N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\frac{b - a}{1}}\right) \cdot \frac{\frac{1}{90}}{\frac{1}{b + a}} \]

      10. /-rgt-identity N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(b - a\right)}\right) \cdot \frac{\frac{1}{90}}{\frac{1}{b + a}} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b - a\right)\right) \cdot \frac{\frac{1}{90}}{\frac{1}{b + a}}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b - a\right)\right)} \cdot \frac{\frac{1}{90}}{\frac{1}{b + a}} \]

      13. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot \frac{\frac{1}{90}}{\frac{1}{b + a}} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot \frac{\frac{1}{90}}{\frac{1}{b + a}} \]

      15. PI-lowering-PI.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \frac{\frac{1}{90}}{\frac{1}{b + a}} \]

      16. --lowering--.f64 N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \color{blue}{\left(b - a\right)}\right) \cdot \frac{\frac{1}{90}}{\frac{1}{b + a}} \]

      17. un-div-inv N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot \frac{1}{\frac{1}{b + a}}\right)} \]

      18. remove-double-div N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(b + a\right)}\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(b + a\right)\right)} \]

      20. +-lowering-+.f64 70.3

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \left(0.011111111111111112 \cdot \color{blue}{\left(b + a\right)}\right) \]

   7. Applied egg-rr 70.3%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(b - a\right)\right) \cdot \left(0.011111111111111112 \cdot \left(b + a\right)\right)} \]

   if 100 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 32.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. 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. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      12. --lowering--.f64 21.4

          \[\leadsto \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Simplified 21.4%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{b}\right) \]
   7. Step-by-step derivation

   8. Simplified 20.4%

      \[\leadsto \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 100:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\pi \cdot angle\right)\right) \cdot \left(\left(a + b\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(b \cdot \left(a + b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 40.2% accurate, 16.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 9 \cdot 10^{-181}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a\_m \cdot a\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a\_m \cdot \left(angle \cdot \left(a\_m \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= a_m 9e-181)
   (* -0.011111111111111112 (* PI (* angle (* a_m a_m))))
   (* -0.011111111111111112 (* a_m (* angle (* a_m PI))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (a_m <= 9e-181) {
		tmp = -0.011111111111111112 * (((double) M_PI) * (angle * (a_m * a_m)));
	} else {
		tmp = -0.011111111111111112 * (a_m * (angle * (a_m * ((double) M_PI))));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if (a_m <= 9e-181) {
		tmp = -0.011111111111111112 * (Math.PI * (angle * (a_m * a_m)));
	} else {
		tmp = -0.011111111111111112 * (a_m * (angle * (a_m * Math.PI)));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if a_m <= 9e-181:
		tmp = -0.011111111111111112 * (math.pi * (angle * (a_m * a_m)))
	else:
		tmp = -0.011111111111111112 * (a_m * (angle * (a_m * math.pi)))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (a_m <= 9e-181)
		tmp = Float64(-0.011111111111111112 * Float64(pi * Float64(angle * Float64(a_m * a_m))));
	else
		tmp = Float64(-0.011111111111111112 * Float64(a_m * Float64(angle * Float64(a_m * pi))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if (a_m <= 9e-181)
		tmp = -0.011111111111111112 * (pi * (angle * (a_m * a_m)));
	else
		tmp = -0.011111111111111112 * (a_m * (angle * (a_m * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[a$95$m, 9e-181], N[(-0.011111111111111112 * N[(Pi * N[(angle * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(a$95$m * N[(angle * N[(a$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 8.9999999999999998e-181

   1. Initial program 56.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      12. --lowering--.f64 54.7

          \[\leadsto \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Simplified 54.7%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left({a}^{2} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left({a}^{2} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\color{blue}{\left({a}^{2} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \]

      7. PI-lowering-PI.f64 35.0

         \[\leadsto -0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \color{blue}{\pi}\right) \]

   8. Simplified 35.0%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]

   if 8.9999999999999998e-181 < a

   1. Initial program 46.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      12. --lowering--.f64 47.3

          \[\leadsto \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Simplified 47.3%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left({a}^{2} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left({a}^{2} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\color{blue}{\left({a}^{2} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \]

      7. PI-lowering-PI.f64 30.3

         \[\leadsto -0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \color{blue}{\pi}\right) \]

   8. Simplified 30.3%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\left(a \cdot a\right) \cdot angle\right)\right)} \]

      2. associate-*l* N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(a \cdot \left(a \cdot angle\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(a \cdot angle\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(a \cdot angle\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot a\right)} \cdot \left(a \cdot angle\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot a\right) \cdot \left(a \cdot angle\right)\right) \]

      7. *-lowering-*.f64 36.8

         \[\leadsto -0.011111111111111112 \cdot \left(\left(\pi \cdot a\right) \cdot \color{blue}{\left(a \cdot angle\right)}\right) \]

   10. Applied egg-rr 36.8%

       \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(\pi \cdot a\right) \cdot \left(a \cdot angle\right)\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{-1}{90} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \color{blue}{\left(angle \cdot a\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left(\left(\mathsf{PI}\left(\right) \cdot a\right) \cdot angle\right) \cdot a\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left(\left(\mathsf{PI}\left(\right) \cdot a\right) \cdot angle\right) \cdot a\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{-1}{90} \cdot \left(\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot a\right) \cdot angle\right)} \cdot a\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot a\right)} \cdot angle\right) \cdot a\right) \]

       6. PI-lowering-PI.f64 36.8

          \[\leadsto -0.011111111111111112 \cdot \left(\left(\left(\color{blue}{\pi} \cdot a\right) \cdot angle\right) \cdot a\right) \]

   12. Applied egg-rr 36.8%

       \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(\left(\pi \cdot a\right) \cdot angle\right) \cdot a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 9 \cdot 10^{-181}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 40.4% accurate, 16.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 2 \cdot 10^{+95}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a\_m \cdot a\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a\_m \cdot \pi\right) \cdot \left(a\_m \cdot angle\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= a_m 2e+95)
   (* -0.011111111111111112 (* PI (* angle (* a_m a_m))))
   (* -0.011111111111111112 (* (* a_m PI) (* a_m angle)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (a_m <= 2e+95) {
		tmp = -0.011111111111111112 * (((double) M_PI) * (angle * (a_m * a_m)));
	} else {
		tmp = -0.011111111111111112 * ((a_m * ((double) M_PI)) * (a_m * angle));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if (a_m <= 2e+95) {
		tmp = -0.011111111111111112 * (Math.PI * (angle * (a_m * a_m)));
	} else {
		tmp = -0.011111111111111112 * ((a_m * Math.PI) * (a_m * angle));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if a_m <= 2e+95:
		tmp = -0.011111111111111112 * (math.pi * (angle * (a_m * a_m)))
	else:
		tmp = -0.011111111111111112 * ((a_m * math.pi) * (a_m * angle))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (a_m <= 2e+95)
		tmp = Float64(-0.011111111111111112 * Float64(pi * Float64(angle * Float64(a_m * a_m))));
	else
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a_m * pi) * Float64(a_m * angle)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if (a_m <= 2e+95)
		tmp = -0.011111111111111112 * (pi * (angle * (a_m * a_m)));
	else
		tmp = -0.011111111111111112 * ((a_m * pi) * (a_m * angle));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[a$95$m, 2e+95], N[(-0.011111111111111112 * N[(Pi * N[(angle * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(N[(a$95$m * Pi), $MachinePrecision] * N[(a$95$m * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.00000000000000004e95

   1. Initial program 55.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. 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. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      12. --lowering--.f64 51.7

          \[\leadsto \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Simplified 51.7%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left({a}^{2} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left({a}^{2} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\color{blue}{\left({a}^{2} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \]

      7. PI-lowering-PI.f64 31.0

         \[\leadsto -0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \color{blue}{\pi}\right) \]

   8. Simplified 31.0%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]

   if 2.00000000000000004e95 < a

   1. Initial program 40.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. 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. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      12. --lowering--.f64 48.7

          \[\leadsto \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   5. Simplified 48.7%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left({a}^{2} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left({a}^{2} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\color{blue}{\left({a}^{2} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \]

      7. PI-lowering-PI.f64 38.3

         \[\leadsto -0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \color{blue}{\pi}\right) \]

   8. Simplified 38.3%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\left(a \cdot a\right) \cdot angle\right)\right)} \]

      2. associate-*l* N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(a \cdot \left(a \cdot angle\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(a \cdot angle\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(a \cdot angle\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot a\right)} \cdot \left(a \cdot angle\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot a\right) \cdot \left(a \cdot angle\right)\right) \]

      7. *-lowering-*.f64 52.7

         \[\leadsto -0.011111111111111112 \cdot \left(\left(\pi \cdot a\right) \cdot \color{blue}{\left(a \cdot angle\right)}\right) \]

   10. Applied egg-rr 52.7%

       \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(\pi \cdot a\right) \cdot \left(a \cdot angle\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2 \cdot 10^{+95}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a \cdot \pi\right) \cdot \left(a \cdot angle\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 38.9% accurate, 21.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ -0.011111111111111112 \cdot \left(\left(a\_m \cdot \pi\right) \cdot \left(a\_m \cdot angle\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (* -0.011111111111111112 (* (* a_m PI) (* a_m angle))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	return -0.011111111111111112 * ((a_m * ((double) M_PI)) * (a_m * angle));
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	return -0.011111111111111112 * ((a_m * Math.PI) * (a_m * angle));
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	return -0.011111111111111112 * ((a_m * math.pi) * (a_m * angle))

Julia

a_m = abs(a)
function code(a_m, b, angle)
	return Float64(-0.011111111111111112 * Float64(Float64(a_m * pi) * Float64(a_m * angle)))
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b, angle)
	tmp = -0.011111111111111112 * ((a_m * pi) * (a_m * angle));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := N[(-0.011111111111111112 * N[(N[(a$95$m * Pi), $MachinePrecision] * N[(a$95$m * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.9%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
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. associate-*r* N/A

      \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

   6. PI-lowering-PI.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

   7. unpow2 N/A

      \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

   8. unpow2 N/A

      \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

   9. difference-of-squares N/A

      \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

   12. --lowering--.f64 51.0

       \[\leadsto \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

5. Simplified 51.0%

   \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

6. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   2. associate-*r* N/A

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left({a}^{2} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left({a}^{2} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \frac{-1}{90} \cdot \left(\color{blue}{\left({a}^{2} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \]

   7. PI-lowering-PI.f64 32.7

      \[\leadsto -0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \color{blue}{\pi}\right) \]

8. Simplified 32.7%

   \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\left(a \cdot a\right) \cdot angle\right)\right)} \]

   2. associate-*l* N/A

      \[\leadsto \frac{-1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(a \cdot \left(a \cdot angle\right)\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(a \cdot angle\right)\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(a \cdot angle\right)\right)} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{-1}{90} \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot a\right)} \cdot \left(a \cdot angle\right)\right) \]

   6. PI-lowering-PI.f64 N/A

      \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot a\right) \cdot \left(a \cdot angle\right)\right) \]

   7. *-lowering-*.f64 35.7

      \[\leadsto -0.011111111111111112 \cdot \left(\left(\pi \cdot a\right) \cdot \color{blue}{\left(a \cdot angle\right)}\right) \]

10. Applied egg-rr 35.7%

    \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(\pi \cdot a\right) \cdot \left(a \cdot angle\right)\right)} \]

11. Final simplification 35.7%

    \[\leadsto -0.011111111111111112 \cdot \left(\left(a \cdot \pi\right) \cdot \left(a \cdot angle\right)\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(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)))))