ab-angle->ABCF B

Percentage Accurate: 53.6% → 66.7%

Time: 24.0s

Alternatives: 16

Speedup: 23.3×

• 32.6% 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: 53.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 66.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 2 \cdot \left(b\_m + a\right)\\ \mathbf{if}\;b\_m \leq 4.6 \cdot 10^{+218}:\\ \;\;\;\;\left(\left(b\_m - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right) \cdot t\_0\right)\right) \cdot \cos \left({\left(\sqrt{\pi}\right)}^{2} \cdot \frac{angle}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m - a\right) \cdot \left(t\_0 \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot {\left(\sqrt[3]{{\pi}^{1.5}}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* 2.0 (+ b_m a))))
   (if (<= b_m 4.6e+218)
     (*
      (*
       (- b_m a)
       (*
        (sin
         (* 0.005555555555555556 (* angle (* (cbrt PI) (pow (cbrt PI) 2.0)))))
        t_0))
      (cos (* (pow (sqrt PI) 2.0) (/ angle 180.0))))
     (*
      (* (- b_m a) (* t_0 (sin (* 0.005555555555555556 (* angle PI)))))
      (cos (* (/ angle 180.0) (pow (cbrt (pow PI 1.5)) 2.0)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = 2.0 * (b_m + a);
	double tmp;
	if (b_m <= 4.6e+218) {
		tmp = ((b_m - a) * (sin((0.005555555555555556 * (angle * (cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0))))) * t_0)) * cos((pow(sqrt(((double) M_PI)), 2.0) * (angle / 180.0)));
	} else {
		tmp = ((b_m - a) * (t_0 * sin((0.005555555555555556 * (angle * ((double) M_PI)))))) * cos(((angle / 180.0) * pow(cbrt(pow(((double) M_PI), 1.5)), 2.0)));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double t_0 = 2.0 * (b_m + a);
	double tmp;
	if (b_m <= 4.6e+218) {
		tmp = ((b_m - a) * (Math.sin((0.005555555555555556 * (angle * (Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.PI), 2.0))))) * t_0)) * Math.cos((Math.pow(Math.sqrt(Math.PI), 2.0) * (angle / 180.0)));
	} else {
		tmp = ((b_m - a) * (t_0 * Math.sin((0.005555555555555556 * (angle * Math.PI))))) * Math.cos(((angle / 180.0) * Math.pow(Math.cbrt(Math.pow(Math.PI, 1.5)), 2.0)));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(2.0 * Float64(b_m + a))
	tmp = 0.0
	if (b_m <= 4.6e+218)
		tmp = Float64(Float64(Float64(b_m - a) * Float64(sin(Float64(0.005555555555555556 * Float64(angle * Float64(cbrt(pi) * (cbrt(pi) ^ 2.0))))) * t_0)) * cos(Float64((sqrt(pi) ^ 2.0) * Float64(angle / 180.0))));
	else
		tmp = Float64(Float64(Float64(b_m - a) * Float64(t_0 * sin(Float64(0.005555555555555556 * Float64(angle * pi))))) * cos(Float64(Float64(angle / 180.0) * (cbrt((pi ^ 1.5)) ^ 2.0))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(2.0 * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 4.6e+218], N[(N[(N[(b$95$m - a), $MachinePrecision] * N[(N[Sin[N[(0.005555555555555556 * N[(angle * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision] * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$m - a), $MachinePrecision] * N[(t$95$0 * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * N[Power[N[Power[N[Power[Pi, 1.5], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.6000000000000002e218

   1. Initial program 57.2%

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

      1. add-exp-log 30.5%

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

      2. *-commutative 30.5%

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

      3. div-inv 30.1%

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

      4. metadata-eval 30.1%

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

   4. Applied egg-rr 30.1%

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

      1. rem-exp-log 55.9%

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

      2. add-sqr-sqrt 34.2%

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

      3. add-sqr-sqrt 55.9%

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

      4. unpow2 55.9%

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

      5. unpow2 55.9%

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

      6. difference-of-squares 58.0%

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

      7. associate-*r* 58.0%

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

      8. associate-*r* 68.0%

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

   6. Applied egg-rr 68.0%

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

      1. *-commutative 68.0%

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

      2. associate-*r* 68.5%

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

      3. *-commutative 68.5%

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

      4. *-commutative 68.5%

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

      5. +-commutative 68.5%

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

   8. Simplified 68.5%

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

      1. add-sqr-sqrt 68.3%

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

      2. pow2 68.3%

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

   10. Applied egg-rr 68.3%

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

       1. add-cube-cbrt 73.0%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)}\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\left(\sqrt{\pi}\right)}^{2} \cdot \frac{angle}{180}\right) \]

       2. pow2 73.0%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \sqrt[3]{\pi}\right)\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\left(\sqrt{\pi}\right)}^{2} \cdot \frac{angle}{180}\right) \]

   12. Applied egg-rr 73.0%

       \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right)}\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\left(\sqrt{\pi}\right)}^{2} \cdot \frac{angle}{180}\right) \]

   if 4.6000000000000002e218 < b

   1. Initial program 36.6%

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

      1. add-exp-log 14.6%

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

      2. *-commutative 14.6%

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

      3. div-inv 21.8%

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

      4. metadata-eval 21.8%

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

   4. Applied egg-rr 21.8%

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

      1. rem-exp-log 43.7%

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

      2. add-sqr-sqrt 43.7%

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

      3. add-sqr-sqrt 43.7%

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

      4. unpow2 43.7%

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

      5. unpow2 43.7%

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

      6. difference-of-squares 58.0%

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

      7. associate-*r* 58.0%

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

      8. associate-*r* 70.7%

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

   6. Applied egg-rr 70.7%

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

      1. *-commutative 70.7%

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

      2. associate-*r* 70.9%

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

      3. *-commutative 70.9%

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

      4. *-commutative 70.9%

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

      5. +-commutative 70.9%

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

   8. Simplified 70.9%

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

      1. add-sqr-sqrt 63.8%

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

      2. pow2 63.8%

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

   10. Applied egg-rr 63.8%

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

       1. add-cbrt-cube 63.8%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\color{blue}{\left(\sqrt[3]{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}}\right)}}^{2} \cdot \frac{angle}{180}\right) \]

       2. pow1/3 63.8%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\color{blue}{\left({\left(\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}^{0.3333333333333333}\right)}}^{2} \cdot \frac{angle}{180}\right) \]

       3. add-sqr-sqrt 63.8%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\left({\left(\color{blue}{\pi} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}\right)}^{2} \cdot \frac{angle}{180}\right) \]

       4. pow1 63.8%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\left({\left(\color{blue}{{\pi}^{1}} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}\right)}^{2} \cdot \frac{angle}{180}\right) \]

       5. pow1/2 63.8%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\left({\left({\pi}^{1} \cdot \color{blue}{{\pi}^{0.5}}\right)}^{0.3333333333333333}\right)}^{2} \cdot \frac{angle}{180}\right) \]

       6. pow-prod-up 63.8%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\left({\color{blue}{\left({\pi}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}\right)}^{2} \cdot \frac{angle}{180}\right) \]

       7. metadata-eval 63.8%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\left({\left({\pi}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}\right)}^{2} \cdot \frac{angle}{180}\right) \]

   12. Applied egg-rr 63.8%

       \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\color{blue}{\left({\left({\pi}^{1.5}\right)}^{0.3333333333333333}\right)}}^{2} \cdot \frac{angle}{180}\right) \]
   13. Step-by-step derivation

       1. unpow1/3 85.2%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\color{blue}{\left(\sqrt[3]{{\pi}^{1.5}}\right)}}^{2} \cdot \frac{angle}{180}\right) \]

   14. Simplified 85.2%

       \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\color{blue}{\left(\sqrt[3]{{\pi}^{1.5}}\right)}}^{2} \cdot \frac{angle}{180}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.6 \cdot 10^{+218}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right) \cdot \left(2 \cdot \left(b + a\right)\right)\right)\right) \cdot \cos \left({\left(\sqrt{\pi}\right)}^{2} \cdot \frac{angle}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(2 \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot {\left(\sqrt[3]{{\pi}^{1.5}}\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \left(b\_m - a\right) \cdot \left(\left(2 \cdot \left(b\_m + a\right)\right) \cdot \sin t\_0\right)\\ \mathbf{if}\;{b\_m}^{2} - {a}^{2} \leq -500000:\\ \;\;\;\;t\_1 \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot {\left(\sqrt[3]{\cos t\_0}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (* (- b_m a) (* (* 2.0 (+ b_m a)) (sin t_0)))))
   (if (<= (- (pow b_m 2.0) (pow a 2.0)) -500000.0)
     (* t_1 (cos (expm1 (log1p (* PI (* 0.005555555555555556 angle))))))
     (* t_1 (pow (cbrt (cos t_0)) 3.0)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = (b_m - a) * ((2.0 * (b_m + a)) * sin(t_0));
	double tmp;
	if ((pow(b_m, 2.0) - pow(a, 2.0)) <= -500000.0) {
		tmp = t_1 * cos(expm1(log1p((((double) M_PI) * (0.005555555555555556 * angle)))));
	} else {
		tmp = t_1 * pow(cbrt(cos(t_0)), 3.0);
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = (b_m - a) * ((2.0 * (b_m + a)) * Math.sin(t_0));
	double tmp;
	if ((Math.pow(b_m, 2.0) - Math.pow(a, 2.0)) <= -500000.0) {
		tmp = t_1 * Math.cos(Math.expm1(Math.log1p((Math.PI * (0.005555555555555556 * angle)))));
	} else {
		tmp = t_1 * Math.pow(Math.cbrt(Math.cos(t_0)), 3.0);
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = Float64(Float64(b_m - a) * Float64(Float64(2.0 * Float64(b_m + a)) * sin(t_0)))
	tmp = 0.0
	if (Float64((b_m ^ 2.0) - (a ^ 2.0)) <= -500000.0)
		tmp = Float64(t_1 * cos(expm1(log1p(Float64(pi * Float64(0.005555555555555556 * angle))))));
	else
		tmp = Float64(t_1 * (cbrt(cos(t_0)) ^ 3.0));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b$95$m - a), $MachinePrecision] * N[(N[(2.0 * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], -500000.0], N[(t$95$1 * N[Cos[N[(Exp[N[Log[1 + N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Power[N[Power[N[Cos[t$95$0], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $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))) < -5e5

   1. Initial program 48.0%

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

      1. add-exp-log 19.5%

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

      2. *-commutative 19.5%

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

      3. div-inv 17.0%

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

      4. metadata-eval 17.0%

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

   4. Applied egg-rr 17.0%

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

      1. rem-exp-log 46.7%

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

      2. add-sqr-sqrt 0.0%

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

      3. add-sqr-sqrt 46.7%

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

      4. unpow2 46.7%

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

      5. unpow2 46.7%

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

      6. difference-of-squares 46.7%

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

      7. associate-*r* 46.7%

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

      8. associate-*r* 61.6%

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

   6. Applied egg-rr 61.6%

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

      1. *-commutative 61.6%

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

      2. associate-*r* 64.0%

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

      3. *-commutative 64.0%

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

      4. *-commutative 64.0%

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

      5. +-commutative 64.0%

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

   8. Simplified 64.0%

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

      1. div-inv 66.5%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) \]

      2. metadata-eval 66.5%

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

      3. expm1-log1p-u 64.4%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \]

      4. expm1-undefine 63.2%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)} - 1\right)} \]

      5. *-commutative 63.2%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(e^{\mathsf{log1p}\left(\color{blue}{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}\right)} - 1\right) \]

      6. associate-*l* 61.9%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(e^{\mathsf{log1p}\left(\color{blue}{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)} - 1\right) \]

   10. Applied egg-rr 61.9%

       \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-define 63.2%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)} \]

       2. associate-*r* 64.4%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}\right)\right)\right) \]

       3. *-commutative 64.4%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(0.005555555555555556 \cdot angle\right)} \cdot \pi\right)\right)\right) \]

       4. *-commutative 64.4%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\pi \cdot \left(0.005555555555555556 \cdot angle\right)}\right)\right)\right) \]

   12. Simplified 64.4%

       \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)} \]

   if -5e5 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 59.7%

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

      1. add-exp-log 34.3%

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

      2. *-commutative 34.3%

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

      3. div-inv 35.4%

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

      4. metadata-eval 35.4%

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

   4. Applied egg-rr 35.4%

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

      1. rem-exp-log 59.2%

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

      2. add-sqr-sqrt 50.5%

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

      3. add-sqr-sqrt 59.2%

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

      4. unpow2 59.2%

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

      5. unpow2 59.2%

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

      6. difference-of-squares 63.2%

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

      7. associate-*r* 63.2%

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

      8. associate-*r* 71.1%

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

   6. Applied egg-rr 71.1%

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

      1. *-commutative 71.1%

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

      2. associate-*r* 70.7%

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

      3. *-commutative 70.7%

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

      4. *-commutative 70.7%

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

      5. +-commutative 70.7%

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

   8. Simplified 70.7%

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

      1. div-inv 70.0%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) \]

      2. metadata-eval 70.0%

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

      3. add-cube-cbrt 70.0%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left(\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right) \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)} \]

      4. pow3 71.2%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}\right)} \]

      5. *-commutative 71.2%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\left(\sqrt[3]{\color{blue}{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}}\right)}^{3}\right) \]

      6. associate-*l* 71.4%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\left(\sqrt[3]{\color{blue}{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}}\right)}^{3}\right) \]

   10. Applied egg-rr 71.4%

       \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left({\left(\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}^{3}\right)} \]
   11. Step-by-step derivation

       1. rem-cube-cbrt 71.5%

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

       2. associate-*r* 70.0%

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

       3. metadata-eval 70.0%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \pi\right) \]

       4. div-inv 70.7%

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

       5. *-commutative 70.7%

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

       6. add-sqr-sqrt 70.3%

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

       7. unpow2 70.3%

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

       8. add-cube-cbrt 70.3%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\cos \left({\left(\sqrt{\pi}\right)}^{2} \cdot \frac{angle}{180}\right)} \cdot \sqrt[3]{\cos \left({\left(\sqrt{\pi}\right)}^{2} \cdot \frac{angle}{180}\right)}\right) \cdot \sqrt[3]{\cos \left({\left(\sqrt{\pi}\right)}^{2} \cdot \frac{angle}{180}\right)}\right)} \]

       9. pow3 70.3%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{\cos \left({\left(\sqrt{\pi}\right)}^{2} \cdot \frac{angle}{180}\right)}\right)}^{3}} \]

   12. Applied egg-rr 73.0%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -500000:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(2 \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(2 \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot {\left(\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \left(b\_m - a\right) \cdot \left(\left(2 \cdot \left(b\_m + a\right)\right) \cdot \sin t\_0\right)\\ \mathbf{if}\;{b\_m}^{2} - {a}^{2} \leq -500000:\\ \;\;\;\;t\_1 \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \log \left(e^{\cos t\_0}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (* (- b_m a) (* (* 2.0 (+ b_m a)) (sin t_0)))))
   (if (<= (- (pow b_m 2.0) (pow a 2.0)) -500000.0)
     (* t_1 (cos (expm1 (log1p (* PI (* 0.005555555555555556 angle))))))
     (* t_1 (log (exp (cos t_0)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = (b_m - a) * ((2.0 * (b_m + a)) * sin(t_0));
	double tmp;
	if ((pow(b_m, 2.0) - pow(a, 2.0)) <= -500000.0) {
		tmp = t_1 * cos(expm1(log1p((((double) M_PI) * (0.005555555555555556 * angle)))));
	} else {
		tmp = t_1 * log(exp(cos(t_0)));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = (b_m - a) * ((2.0 * (b_m + a)) * Math.sin(t_0));
	double tmp;
	if ((Math.pow(b_m, 2.0) - Math.pow(a, 2.0)) <= -500000.0) {
		tmp = t_1 * Math.cos(Math.expm1(Math.log1p((Math.PI * (0.005555555555555556 * angle)))));
	} else {
		tmp = t_1 * Math.log(Math.exp(Math.cos(t_0)));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	t_1 = (b_m - a) * ((2.0 * (b_m + a)) * math.sin(t_0))
	tmp = 0
	if (math.pow(b_m, 2.0) - math.pow(a, 2.0)) <= -500000.0:
		tmp = t_1 * math.cos(math.expm1(math.log1p((math.pi * (0.005555555555555556 * angle)))))
	else:
		tmp = t_1 * math.log(math.exp(math.cos(t_0)))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = Float64(Float64(b_m - a) * Float64(Float64(2.0 * Float64(b_m + a)) * sin(t_0)))
	tmp = 0.0
	if (Float64((b_m ^ 2.0) - (a ^ 2.0)) <= -500000.0)
		tmp = Float64(t_1 * cos(expm1(log1p(Float64(pi * Float64(0.005555555555555556 * angle))))));
	else
		tmp = Float64(t_1 * log(exp(cos(t_0))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b$95$m - a), $MachinePrecision] * N[(N[(2.0 * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], -500000.0], N[(t$95$1 * N[Cos[N[(Exp[N[Log[1 + N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Log[N[Exp[N[Cos[t$95$0], $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))) < -5e5

   1. Initial program 48.0%

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

      1. add-exp-log 19.5%

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

      2. *-commutative 19.5%

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

      3. div-inv 17.0%

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

      4. metadata-eval 17.0%

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

   4. Applied egg-rr 17.0%

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

      1. rem-exp-log 46.7%

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

      2. add-sqr-sqrt 0.0%

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

      3. add-sqr-sqrt 46.7%

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

      4. unpow2 46.7%

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

      5. unpow2 46.7%

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

      6. difference-of-squares 46.7%

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

      7. associate-*r* 46.7%

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

      8. associate-*r* 61.6%

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

   6. Applied egg-rr 61.6%

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

      1. *-commutative 61.6%

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

      2. associate-*r* 64.0%

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

      3. *-commutative 64.0%

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

      4. *-commutative 64.0%

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

      5. +-commutative 64.0%

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

   8. Simplified 64.0%

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

      1. div-inv 66.5%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) \]

      2. metadata-eval 66.5%

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

      3. expm1-log1p-u 64.4%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \]

      4. expm1-undefine 63.2%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)} - 1\right)} \]

      5. *-commutative 63.2%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(e^{\mathsf{log1p}\left(\color{blue}{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}\right)} - 1\right) \]

      6. associate-*l* 61.9%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(e^{\mathsf{log1p}\left(\color{blue}{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)} - 1\right) \]

   10. Applied egg-rr 61.9%

       \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-define 63.2%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)} \]

       2. associate-*r* 64.4%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}\right)\right)\right) \]

       3. *-commutative 64.4%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(0.005555555555555556 \cdot angle\right)} \cdot \pi\right)\right)\right) \]

       4. *-commutative 64.4%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\pi \cdot \left(0.005555555555555556 \cdot angle\right)}\right)\right)\right) \]

   12. Simplified 64.4%

       \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)} \]

   if -5e5 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 59.7%

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

      1. add-exp-log 34.3%

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

      2. *-commutative 34.3%

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

      3. div-inv 35.4%

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

      4. metadata-eval 35.4%

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

   4. Applied egg-rr 35.4%

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

      1. rem-exp-log 59.2%

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

      2. add-sqr-sqrt 50.5%

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

      3. add-sqr-sqrt 59.2%

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

      4. unpow2 59.2%

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

      5. unpow2 59.2%

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

      6. difference-of-squares 63.2%

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

      7. associate-*r* 63.2%

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

      8. associate-*r* 71.1%

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

   6. Applied egg-rr 71.1%

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

      1. *-commutative 71.1%

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

      2. associate-*r* 70.7%

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

      3. *-commutative 70.7%

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

      4. *-commutative 70.7%

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

      5. +-commutative 70.7%

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

   8. Simplified 70.7%

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

      1. div-inv 70.0%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) \]

      2. metadata-eval 70.0%

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

      3. add-cube-cbrt 70.0%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left(\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right) \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)} \]

      4. pow3 71.2%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}\right)} \]

      5. *-commutative 71.2%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\left(\sqrt[3]{\color{blue}{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}}\right)}^{3}\right) \]

      6. associate-*l* 71.4%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\left(\sqrt[3]{\color{blue}{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}}\right)}^{3}\right) \]

   10. Applied egg-rr 71.4%

       \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left({\left(\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}^{3}\right)} \]
   11. Step-by-step derivation

       1. add-log-exp 71.4%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \color{blue}{\log \left(e^{\cos \left({\left(\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}^{3}\right)}\right)} \]

       2. rem-cube-cbrt 71.5%

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

       3. *-commutative 71.5%

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

       4. associate-*l* 73.0%

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

       5. *-commutative 73.0%

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

   12. Applied egg-rr 73.0%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -500000:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(2 \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(2 \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \log \left(e^{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(b\_m - a\right) \cdot \left(\left(2 \cdot \left(b\_m + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{if}\;{b\_m}^{2} - {a}^{2} \leq -5 \cdot 10^{+114}:\\ \;\;\;\;t\_0 \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \cos \left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0
         (*
          (- b_m a)
          (* (* 2.0 (+ b_m a)) (sin (* 0.005555555555555556 (* angle PI)))))))
   (if (<= (- (pow b_m 2.0) (pow a 2.0)) -5e+114)
     (* t_0 (cos (expm1 (log1p (* PI (* 0.005555555555555556 angle))))))
     (* t_0 (cos (/ 1.0 (/ 180.0 (* angle PI))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = (b_m - a) * ((2.0 * (b_m + a)) * sin((0.005555555555555556 * (angle * ((double) M_PI)))));
	double tmp;
	if ((pow(b_m, 2.0) - pow(a, 2.0)) <= -5e+114) {
		tmp = t_0 * cos(expm1(log1p((((double) M_PI) * (0.005555555555555556 * angle)))));
	} else {
		tmp = t_0 * cos((1.0 / (180.0 / (angle * ((double) M_PI)))));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double t_0 = (b_m - a) * ((2.0 * (b_m + a)) * Math.sin((0.005555555555555556 * (angle * Math.PI))));
	double tmp;
	if ((Math.pow(b_m, 2.0) - Math.pow(a, 2.0)) <= -5e+114) {
		tmp = t_0 * Math.cos(Math.expm1(Math.log1p((Math.PI * (0.005555555555555556 * angle)))));
	} else {
		tmp = t_0 * Math.cos((1.0 / (180.0 / (angle * Math.PI))));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	t_0 = (b_m - a) * ((2.0 * (b_m + a)) * math.sin((0.005555555555555556 * (angle * math.pi))))
	tmp = 0
	if (math.pow(b_m, 2.0) - math.pow(a, 2.0)) <= -5e+114:
		tmp = t_0 * math.cos(math.expm1(math.log1p((math.pi * (0.005555555555555556 * angle)))))
	else:
		tmp = t_0 * math.cos((1.0 / (180.0 / (angle * math.pi))))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(Float64(b_m - a) * Float64(Float64(2.0 * Float64(b_m + a)) * sin(Float64(0.005555555555555556 * Float64(angle * pi)))))
	tmp = 0.0
	if (Float64((b_m ^ 2.0) - (a ^ 2.0)) <= -5e+114)
		tmp = Float64(t_0 * cos(expm1(log1p(Float64(pi * Float64(0.005555555555555556 * angle))))));
	else
		tmp = Float64(t_0 * cos(Float64(1.0 / Float64(180.0 / Float64(angle * pi)))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(N[(b$95$m - a), $MachinePrecision] * N[(N[(2.0 * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], -5e+114], N[(t$95$0 * N[Cos[N[(Exp[N[Log[1 + N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Cos[N[(1.0 / N[(180.0 / N[(angle * Pi), $MachinePrecision]), $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))) < -5.0000000000000001e114

   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. Step-by-step derivation

      1. add-exp-log 20.2%

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

      2. *-commutative 20.2%

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

      3. div-inv 17.4%

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

      4. metadata-eval 17.4%

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

   4. Applied egg-rr 17.4%

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

      1. rem-exp-log 46.1%

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

      2. add-sqr-sqrt 0.0%

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

      3. add-sqr-sqrt 46.1%

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

      4. unpow2 46.1%

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

      5. unpow2 46.1%

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

      6. difference-of-squares 46.1%

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

      7. associate-*r* 46.1%

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

      8. associate-*r* 62.7%

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

   6. Applied egg-rr 62.7%

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

      1. *-commutative 62.7%

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

      2. associate-*r* 64.9%

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

      3. *-commutative 64.9%

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

      4. *-commutative 64.9%

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

      5. +-commutative 64.9%

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

   8. Simplified 64.9%

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

      1. div-inv 67.7%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) \]

      2. metadata-eval 67.7%

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

      3. expm1-log1p-u 65.6%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \]

      4. expm1-undefine 64.2%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)} - 1\right)} \]

      5. *-commutative 64.2%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(e^{\mathsf{log1p}\left(\color{blue}{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}\right)} - 1\right) \]

      6. associate-*l* 62.8%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(e^{\mathsf{log1p}\left(\color{blue}{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)} - 1\right) \]

   10. Applied egg-rr 62.8%

       \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-define 64.2%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)} \]

       2. associate-*r* 65.6%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}\right)\right)\right) \]

       3. *-commutative 65.6%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(0.005555555555555556 \cdot angle\right)} \cdot \pi\right)\right)\right) \]

       4. *-commutative 65.6%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\pi \cdot \left(0.005555555555555556 \cdot angle\right)}\right)\right)\right) \]

   12. Simplified 65.6%

       \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)} \]

   if -5.0000000000000001e114 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 59.4%

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

      1. add-exp-log 33.3%

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

      2. *-commutative 33.3%

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

      3. div-inv 34.5%

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

      4. metadata-eval 34.5%

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

   4. Applied egg-rr 34.5%

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

      1. rem-exp-log 58.9%

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

      2. add-sqr-sqrt 48.3%

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

      3. add-sqr-sqrt 58.9%

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

      4. unpow2 58.9%

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

      5. unpow2 58.9%

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

      6. difference-of-squares 62.7%

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

      7. associate-*r* 62.7%

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

      8. associate-*r* 70.3%

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

   6. Applied egg-rr 70.3%

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

      1. *-commutative 70.3%

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

      2. associate-*r* 70.0%

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

      3. *-commutative 70.0%

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

      4. *-commutative 70.0%

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

      5. +-commutative 70.0%

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

   8. Simplified 70.0%

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

      1. associate-*r/ 71.1%

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

      2. *-commutative 71.1%

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

      3. clear-num 71.7%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)} \]

   10. Applied egg-rr 71.7%

       \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -5 \cdot 10^{+114}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(2 \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(2 \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \cos \left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(b\_m - a\right) \cdot \left(\left(2 \cdot \left(b\_m + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{if}\;b\_m \leq 1.55 \cdot 10^{+148}:\\ \;\;\;\;t\_0 \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \cos \left(\frac{angle}{180} \cdot {\left(\sqrt[3]{{\pi}^{1.5}}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0
         (*
          (- b_m a)
          (* (* 2.0 (+ b_m a)) (sin (* 0.005555555555555556 (* angle PI)))))))
   (if (<= b_m 1.55e+148)
     (* t_0 (cos (expm1 (log1p (* PI (* 0.005555555555555556 angle))))))
     (* t_0 (cos (* (/ angle 180.0) (pow (cbrt (pow PI 1.5)) 2.0)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = (b_m - a) * ((2.0 * (b_m + a)) * sin((0.005555555555555556 * (angle * ((double) M_PI)))));
	double tmp;
	if (b_m <= 1.55e+148) {
		tmp = t_0 * cos(expm1(log1p((((double) M_PI) * (0.005555555555555556 * angle)))));
	} else {
		tmp = t_0 * cos(((angle / 180.0) * pow(cbrt(pow(((double) M_PI), 1.5)), 2.0)));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double t_0 = (b_m - a) * ((2.0 * (b_m + a)) * Math.sin((0.005555555555555556 * (angle * Math.PI))));
	double tmp;
	if (b_m <= 1.55e+148) {
		tmp = t_0 * Math.cos(Math.expm1(Math.log1p((Math.PI * (0.005555555555555556 * angle)))));
	} else {
		tmp = t_0 * Math.cos(((angle / 180.0) * Math.pow(Math.cbrt(Math.pow(Math.PI, 1.5)), 2.0)));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(Float64(b_m - a) * Float64(Float64(2.0 * Float64(b_m + a)) * sin(Float64(0.005555555555555556 * Float64(angle * pi)))))
	tmp = 0.0
	if (b_m <= 1.55e+148)
		tmp = Float64(t_0 * cos(expm1(log1p(Float64(pi * Float64(0.005555555555555556 * angle))))));
	else
		tmp = Float64(t_0 * cos(Float64(Float64(angle / 180.0) * (cbrt((pi ^ 1.5)) ^ 2.0))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(N[(b$95$m - a), $MachinePrecision] * N[(N[(2.0 * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 1.55e+148], N[(t$95$0 * N[Cos[N[(Exp[N[Log[1 + N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * N[Power[N[Power[N[Power[Pi, 1.5], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.54999999999999988e148

   1. Initial program 58.0%

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

      1. add-exp-log 31.8%

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

      2. *-commutative 31.8%

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

      3. div-inv 30.9%

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

      4. metadata-eval 30.9%

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

   4. Applied egg-rr 30.9%

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

      1. rem-exp-log 56.2%

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

      2. add-sqr-sqrt 33.0%

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

      3. add-sqr-sqrt 56.2%

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

      4. unpow2 56.2%

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

      5. unpow2 56.2%

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

      6. difference-of-squares 58.0%

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

      7. associate-*r* 58.0%

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

      8. associate-*r* 66.2%

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

   6. Applied egg-rr 66.2%

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

      1. *-commutative 66.2%

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

      2. associate-*r* 66.7%

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

      3. *-commutative 66.7%

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

      4. *-commutative 66.7%

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

      5. +-commutative 66.7%

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

   8. Simplified 66.7%

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

      1. div-inv 67.1%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) \]

      2. metadata-eval 67.1%

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

      3. expm1-log1p-u 61.2%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \]

      4. expm1-undefine 60.7%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)} - 1\right)} \]

      5. *-commutative 60.7%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(e^{\mathsf{log1p}\left(\color{blue}{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}\right)} - 1\right) \]

      6. associate-*l* 60.3%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(e^{\mathsf{log1p}\left(\color{blue}{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)} - 1\right) \]

   10. Applied egg-rr 60.3%

       \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-define 60.7%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)} \]

       2. associate-*r* 61.2%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}\right)\right)\right) \]

       3. *-commutative 61.2%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(0.005555555555555556 \cdot angle\right)} \cdot \pi\right)\right)\right) \]

       4. *-commutative 61.2%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\pi \cdot \left(0.005555555555555556 \cdot angle\right)}\right)\right)\right) \]

   12. Simplified 61.2%

       \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)} \]

   if 1.54999999999999988e148 < b

   1. Initial program 41.4%

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

      1. add-exp-log 13.6%

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

      2. *-commutative 13.6%

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

      3. div-inv 20.3%

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

      4. metadata-eval 20.3%

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

   4. Applied egg-rr 20.3%

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

      1. rem-exp-log 48.0%

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

      2. add-sqr-sqrt 48.0%

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

      3. add-sqr-sqrt 48.0%

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

      4. unpow2 48.0%

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

      5. unpow2 48.0%

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

      6. difference-of-squares 58.2%

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

      7. associate-*r* 58.2%

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

      8. associate-*r* 82.8%

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

   6. Applied egg-rr 82.8%

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

      1. *-commutative 82.8%

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

      2. associate-*r* 82.9%

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

      3. *-commutative 82.9%

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

      4. *-commutative 82.9%

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

      5. +-commutative 82.9%

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

   8. Simplified 82.9%

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

      1. add-sqr-sqrt 79.6%

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

      2. pow2 79.6%

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

   10. Applied egg-rr 79.6%

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

       1. add-cbrt-cube 79.6%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\color{blue}{\left(\sqrt[3]{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}}\right)}}^{2} \cdot \frac{angle}{180}\right) \]

       2. pow1/3 79.6%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\color{blue}{\left({\left(\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}^{0.3333333333333333}\right)}}^{2} \cdot \frac{angle}{180}\right) \]

       3. add-sqr-sqrt 79.6%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\left({\left(\color{blue}{\pi} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}\right)}^{2} \cdot \frac{angle}{180}\right) \]

       4. pow1 79.6%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\left({\left(\color{blue}{{\pi}^{1}} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}\right)}^{2} \cdot \frac{angle}{180}\right) \]

       5. pow1/2 79.6%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\left({\left({\pi}^{1} \cdot \color{blue}{{\pi}^{0.5}}\right)}^{0.3333333333333333}\right)}^{2} \cdot \frac{angle}{180}\right) \]

       6. pow-prod-up 79.6%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\left({\color{blue}{\left({\pi}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}\right)}^{2} \cdot \frac{angle}{180}\right) \]

       7. metadata-eval 79.6%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\left({\left({\pi}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}\right)}^{2} \cdot \frac{angle}{180}\right) \]

   12. Applied egg-rr 79.6%

       \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\color{blue}{\left({\left({\pi}^{1.5}\right)}^{0.3333333333333333}\right)}}^{2} \cdot \frac{angle}{180}\right) \]
   13. Step-by-step derivation

       1. unpow1/3 89.6%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\color{blue}{\left(\sqrt[3]{{\pi}^{1.5}}\right)}}^{2} \cdot \frac{angle}{180}\right) \]

   14. Simplified 89.6%

       \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\color{blue}{\left(\sqrt[3]{{\pi}^{1.5}}\right)}}^{2} \cdot \frac{angle}{180}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.55 \cdot 10^{+148}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(2 \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(2 \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot {\left(\sqrt[3]{{\pi}^{1.5}}\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(b\_m - a\right) \cdot \left(\left(2 \cdot \left(b\_m + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{if}\;{b\_m}^{2} - {a}^{2} \leq -2 \cdot 10^{+134}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \cos \left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0
         (*
          (- b_m a)
          (* (* 2.0 (+ b_m a)) (sin (* 0.005555555555555556 (* angle PI)))))))
   (if (<= (- (pow b_m 2.0) (pow a 2.0)) -2e+134)
     t_0
     (* t_0 (cos (/ 1.0 (/ 180.0 (* angle PI))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = (b_m - a) * ((2.0 * (b_m + a)) * sin((0.005555555555555556 * (angle * ((double) M_PI)))));
	double tmp;
	if ((pow(b_m, 2.0) - pow(a, 2.0)) <= -2e+134) {
		tmp = t_0;
	} else {
		tmp = t_0 * cos((1.0 / (180.0 / (angle * ((double) M_PI)))));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double t_0 = (b_m - a) * ((2.0 * (b_m + a)) * Math.sin((0.005555555555555556 * (angle * Math.PI))));
	double tmp;
	if ((Math.pow(b_m, 2.0) - Math.pow(a, 2.0)) <= -2e+134) {
		tmp = t_0;
	} else {
		tmp = t_0 * Math.cos((1.0 / (180.0 / (angle * Math.PI))));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	t_0 = (b_m - a) * ((2.0 * (b_m + a)) * math.sin((0.005555555555555556 * (angle * math.pi))))
	tmp = 0
	if (math.pow(b_m, 2.0) - math.pow(a, 2.0)) <= -2e+134:
		tmp = t_0
	else:
		tmp = t_0 * math.cos((1.0 / (180.0 / (angle * math.pi))))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(Float64(b_m - a) * Float64(Float64(2.0 * Float64(b_m + a)) * sin(Float64(0.005555555555555556 * Float64(angle * pi)))))
	tmp = 0.0
	if (Float64((b_m ^ 2.0) - (a ^ 2.0)) <= -2e+134)
		tmp = t_0;
	else
		tmp = Float64(t_0 * cos(Float64(1.0 / Float64(180.0 / Float64(angle * pi)))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	t_0 = (b_m - a) * ((2.0 * (b_m + a)) * sin((0.005555555555555556 * (angle * pi))));
	tmp = 0.0;
	if (((b_m ^ 2.0) - (a ^ 2.0)) <= -2e+134)
		tmp = t_0;
	else
		tmp = t_0 * cos((1.0 / (180.0 / (angle * pi))));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(N[(b$95$m - a), $MachinePrecision] * N[(N[(2.0 * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], -2e+134], t$95$0, N[(t$95$0 * N[Cos[N[(1.0 / N[(180.0 / N[(angle * Pi), $MachinePrecision]), $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))) < -1.99999999999999984e134

   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. Step-by-step derivation

      1. add-exp-log 19.7%

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

      2. *-commutative 19.7%

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

      3. div-inv 16.8%

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

      4. metadata-eval 16.8%

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

   4. Applied egg-rr 16.8%

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

      1. rem-exp-log 46.1%

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

      2. add-sqr-sqrt 0.0%

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

      3. add-sqr-sqrt 46.1%

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

      4. unpow2 46.1%

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

      5. unpow2 46.1%

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

      6. difference-of-squares 46.1%

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

      7. associate-*r* 46.1%

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

      8. associate-*r* 63.3%

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

   6. Applied egg-rr 63.3%

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

      1. *-commutative 63.3%

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

      2. associate-*r* 65.6%

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

      3. *-commutative 65.6%

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

      4. *-commutative 65.6%

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

      5. +-commutative 65.6%

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

   8. Simplified 65.6%

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

   9. Taylor expanded in angle around 0 76.0%

      \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \color{blue}{1} \]

   if -1.99999999999999984e134 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 59.2%

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

      1. add-exp-log 33.3%

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

      2. *-commutative 33.3%

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

      3. div-inv 34.4%

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

      4. metadata-eval 34.4%

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

   4. Applied egg-rr 34.4%

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

      1. rem-exp-log 58.7%

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

      2. add-sqr-sqrt 47.5%

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

      3. add-sqr-sqrt 58.7%

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

      4. unpow2 58.7%

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

      5. unpow2 58.7%

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

      6. difference-of-squares 62.4%

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

      7. associate-*r* 62.4%

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

      8. associate-*r* 69.9%

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

   6. Applied egg-rr 69.9%

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

      1. *-commutative 69.9%

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

      2. associate-*r* 69.7%

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

      3. *-commutative 69.7%

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

      4. *-commutative 69.7%

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

      5. +-commutative 69.7%

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

   8. Simplified 69.7%

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

      1. associate-*r/ 70.6%

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

      2. *-commutative 70.6%

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

      3. clear-num 71.2%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)} \]

   10. Applied egg-rr 71.2%

       \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -2 \cdot 10^{+134}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(2 \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(2 \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \cos \left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.4% accurate, 1.0× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0
         (*
          (- b_m a)
          (* (* 2.0 (+ b_m a)) (sin (* 0.005555555555555556 (* angle PI)))))))
   (if (<= (- (pow b_m 2.0) (pow a 2.0)) (- INFINITY))
     t_0
     (* t_0 (cos (* angle (/ PI 180.0)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = (b_m - a) * ((2.0 * (b_m + a)) * sin((0.005555555555555556 * (angle * ((double) M_PI)))));
	double tmp;
	if ((pow(b_m, 2.0) - pow(a, 2.0)) <= -((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = t_0 * cos((angle * (((double) M_PI) / 180.0)));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double t_0 = (b_m - a) * ((2.0 * (b_m + a)) * Math.sin((0.005555555555555556 * (angle * Math.PI))));
	double tmp;
	if ((Math.pow(b_m, 2.0) - Math.pow(a, 2.0)) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else {
		tmp = t_0 * Math.cos((angle * (Math.PI / 180.0)));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	t_0 = (b_m - a) * ((2.0 * (b_m + a)) * math.sin((0.005555555555555556 * (angle * math.pi))))
	tmp = 0
	if (math.pow(b_m, 2.0) - math.pow(a, 2.0)) <= -math.inf:
		tmp = t_0
	else:
		tmp = t_0 * math.cos((angle * (math.pi / 180.0)))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(Float64(b_m - a) * Float64(Float64(2.0 * Float64(b_m + a)) * sin(Float64(0.005555555555555556 * Float64(angle * pi)))))
	tmp = 0.0
	if (Float64((b_m ^ 2.0) - (a ^ 2.0)) <= Float64(-Inf))
		tmp = t_0;
	else
		tmp = Float64(t_0 * cos(Float64(angle * Float64(pi / 180.0))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	t_0 = (b_m - a) * ((2.0 * (b_m + a)) * sin((0.005555555555555556 * (angle * pi))));
	tmp = 0.0;
	if (((b_m ^ 2.0) - (a ^ 2.0)) <= -Inf)
		tmp = t_0;
	else
		tmp = t_0 * cos((angle * (pi / 180.0)));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(N[(b$95$m - a), $MachinePrecision] * N[(N[(2.0 * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], (-Infinity)], t$95$0, N[(t$95$0 * N[Cos[N[(angle * N[(Pi / 180.0), $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))) < -inf.0

   1. Initial program 43.3%

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

      1. add-exp-log 22.1%

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

      2. *-commutative 22.1%

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

      3. div-inv 18.4%

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

      4. metadata-eval 18.4%

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

   4. Applied egg-rr 18.4%

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

      1. rem-exp-log 41.4%

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

      2. add-sqr-sqrt 0.0%

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

      3. add-sqr-sqrt 41.4%

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

      4. unpow2 41.4%

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

      5. unpow2 41.4%

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

      6. difference-of-squares 41.4%

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

      7. associate-*r* 41.4%

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

      8. associate-*r* 64.0%

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

   6. Applied egg-rr 64.0%

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

      1. *-commutative 64.0%

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

      2. associate-*r* 67.8%

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

      3. *-commutative 67.8%

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

      4. *-commutative 67.8%

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

      5. +-commutative 67.8%

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

   8. Simplified 67.8%

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

   9. Taylor expanded in angle around 0 81.0%

      \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \color{blue}{1} \]

   if -inf.0 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 59.4%

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

      1. add-exp-log 31.6%

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

      2. *-commutative 31.6%

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

      3. div-inv 32.6%

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

      4. metadata-eval 32.6%

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

   4. Applied egg-rr 32.6%

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

      1. rem-exp-log 58.9%

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

      2. add-sqr-sqrt 43.8%

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

      3. add-sqr-sqrt 58.9%

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

      4. unpow2 58.9%

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

      5. unpow2 58.9%

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

      6. difference-of-squares 62.4%

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

      7. associate-*r* 62.4%

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

      8. associate-*r* 69.2%

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

   6. Applied egg-rr 69.2%

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

      1. *-commutative 69.2%

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

      2. associate-*r* 68.8%

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

      3. *-commutative 68.8%

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

      4. *-commutative 68.8%

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

      5. +-commutative 68.8%

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

   8. Simplified 68.8%

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

      1. div-inv 68.2%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) \]

      2. metadata-eval 68.2%

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

      3. add-cube-cbrt 68.3%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left(\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right) \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)} \]

      4. pow3 69.3%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}\right)} \]

      5. *-commutative 69.3%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\left(\sqrt[3]{\color{blue}{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}}\right)}^{3}\right) \]

      6. associate-*l* 69.6%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left({\left(\sqrt[3]{\color{blue}{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}}\right)}^{3}\right) \]

   10. Applied egg-rr 69.6%

       \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \color{blue}{\left({\left(\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}^{3}\right)} \]
   11. Step-by-step derivation

       1. rem-cube-cbrt 69.7%

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

       2. associate-*r* 68.2%

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

       3. metadata-eval 68.2%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \pi\right) \]

       4. div-inv 68.8%

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

       5. *-commutative 68.8%

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

       6. clear-num 68.9%

          \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \color{blue}{\frac{1}{\frac{180}{angle}}}\right) \]

       7. un-div-inv 69.9%

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

   12. Applied egg-rr 69.9%

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

       1. associate-/r/ 69.7%

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

   14. Simplified 69.7%

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

4. Final simplification 72.0%

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

Alternative 8: 66.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 2 \cdot \left(b\_m + a\right)\\ \mathbf{if}\;{b\_m}^{2} - {a}^{2} \leq -5 \cdot 10^{+299}:\\ \;\;\;\;\left(b\_m - a\right) \cdot \left(t\_0 \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \left(\left(b\_m - a\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* 2.0 (+ b_m a))))
   (if (<= (- (pow b_m 2.0) (pow a 2.0)) -5e+299)
     (* (- b_m a) (* t_0 (sin (* 0.005555555555555556 (* angle PI)))))
     (*
      (* t_0 (* (- b_m a) (sin (* PI (* 0.005555555555555556 angle)))))
      (cos (* PI (/ angle 180.0)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = 2.0 * (b_m + a);
	double tmp;
	if ((pow(b_m, 2.0) - pow(a, 2.0)) <= -5e+299) {
		tmp = (b_m - a) * (t_0 * sin((0.005555555555555556 * (angle * ((double) M_PI)))));
	} else {
		tmp = (t_0 * ((b_m - a) * sin((((double) M_PI) * (0.005555555555555556 * angle))))) * cos((((double) M_PI) * (angle / 180.0)));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double t_0 = 2.0 * (b_m + a);
	double tmp;
	if ((Math.pow(b_m, 2.0) - Math.pow(a, 2.0)) <= -5e+299) {
		tmp = (b_m - a) * (t_0 * Math.sin((0.005555555555555556 * (angle * Math.PI))));
	} else {
		tmp = (t_0 * ((b_m - a) * Math.sin((Math.PI * (0.005555555555555556 * angle))))) * Math.cos((Math.PI * (angle / 180.0)));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	t_0 = 2.0 * (b_m + a)
	tmp = 0
	if (math.pow(b_m, 2.0) - math.pow(a, 2.0)) <= -5e+299:
		tmp = (b_m - a) * (t_0 * math.sin((0.005555555555555556 * (angle * math.pi))))
	else:
		tmp = (t_0 * ((b_m - a) * math.sin((math.pi * (0.005555555555555556 * angle))))) * math.cos((math.pi * (angle / 180.0)))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(2.0 * Float64(b_m + a))
	tmp = 0.0
	if (Float64((b_m ^ 2.0) - (a ^ 2.0)) <= -5e+299)
		tmp = Float64(Float64(b_m - a) * Float64(t_0 * sin(Float64(0.005555555555555556 * Float64(angle * pi)))));
	else
		tmp = Float64(Float64(t_0 * Float64(Float64(b_m - a) * sin(Float64(pi * Float64(0.005555555555555556 * angle))))) * cos(Float64(pi * Float64(angle / 180.0))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	t_0 = 2.0 * (b_m + a);
	tmp = 0.0;
	if (((b_m ^ 2.0) - (a ^ 2.0)) <= -5e+299)
		tmp = (b_m - a) * (t_0 * sin((0.005555555555555556 * (angle * pi))));
	else
		tmp = (t_0 * ((b_m - a) * sin((pi * (0.005555555555555556 * angle))))) * cos((pi * (angle / 180.0)));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(2.0 * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], -5e+299], N[(N[(b$95$m - a), $MachinePrecision] * N[(t$95$0 * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(N[(b$95$m - a), $MachinePrecision] * N[Sin[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Pi * N[(angle / 180.0), $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))) < -5.0000000000000003e299

   1. Initial program 42.5%

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

      1. add-exp-log 21.7%

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

      2. *-commutative 21.7%

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

      3. div-inv 18.0%

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

      4. metadata-eval 18.0%

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

   4. Applied egg-rr 18.0%

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

      1. rem-exp-log 40.7%

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

      2. add-sqr-sqrt 0.0%

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

      3. add-sqr-sqrt 40.7%

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

      4. unpow2 40.7%

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

      5. unpow2 40.7%

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

      6. difference-of-squares 40.7%

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

      7. associate-*r* 40.7%

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

      8. associate-*r* 62.8%

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

   6. Applied egg-rr 62.8%

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

      1. *-commutative 62.8%

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

      2. associate-*r* 66.6%

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

      3. *-commutative 66.6%

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

      4. *-commutative 66.6%

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

      5. +-commutative 66.6%

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

   8. Simplified 66.6%

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

   9. Taylor expanded in angle around 0 79.5%

      \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \color{blue}{1} \]

   if -5.0000000000000003e299 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 59.7%

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

      1. add-exp-log 31.8%

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

      2. *-commutative 31.8%

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

      3. div-inv 32.8%

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

      4. metadata-eval 32.8%

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

   4. Applied egg-rr 32.8%

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

      1. rem-exp-log 55.5%

         \[\leadsto \color{blue}{\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \cdot 1 \]

      2. *-commutative 55.5%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot 1 \]

      3. add-sqr-sqrt 40.9%

         \[\leadsto \left(\color{blue}{\left(\sqrt{2 \cdot \left({b}^{2} - {a}^{2}\right)} \cdot \sqrt{2 \cdot \left({b}^{2} - {a}^{2}\right)}\right)} \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

      4. add-sqr-sqrt 55.5%

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

      5. unpow2 55.5%

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

      6. unpow2 55.5%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

      7. difference-of-squares 58.0%

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

      8. associate-*r* 58.0%

         \[\leadsto \left(\color{blue}{\left(\left(2 \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

      9. associate-*l* 64.9%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot 1 \]

   6. Applied egg-rr 69.6%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -5 \cdot 10^{+299}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(2 \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\\ \mathbf{if}\;{b\_m}^{2} - {a}^{2} \leq 5 \cdot 10^{-97}:\\ \;\;\;\;\left(2 \cdot \left(b\_m + a\right)\right) \cdot \left(\left(b\_m - a\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b\_m \cdot \left(b\_m \cdot t\_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (sin (* PI (* 0.005555555555555556 angle)))))
   (if (<= (- (pow b_m 2.0) (pow a 2.0)) 5e-97)
     (* (* 2.0 (+ b_m a)) (* (- b_m a) t_0))
     (* (cos (* PI (/ angle 180.0))) (* 2.0 (* b_m (* b_m t_0)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = sin((((double) M_PI) * (0.005555555555555556 * angle)));
	double tmp;
	if ((pow(b_m, 2.0) - pow(a, 2.0)) <= 5e-97) {
		tmp = (2.0 * (b_m + a)) * ((b_m - a) * t_0);
	} else {
		tmp = cos((((double) M_PI) * (angle / 180.0))) * (2.0 * (b_m * (b_m * t_0)));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double t_0 = Math.sin((Math.PI * (0.005555555555555556 * angle)));
	double tmp;
	if ((Math.pow(b_m, 2.0) - Math.pow(a, 2.0)) <= 5e-97) {
		tmp = (2.0 * (b_m + a)) * ((b_m - a) * t_0);
	} else {
		tmp = Math.cos((Math.PI * (angle / 180.0))) * (2.0 * (b_m * (b_m * t_0)));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	t_0 = math.sin((math.pi * (0.005555555555555556 * angle)))
	tmp = 0
	if (math.pow(b_m, 2.0) - math.pow(a, 2.0)) <= 5e-97:
		tmp = (2.0 * (b_m + a)) * ((b_m - a) * t_0)
	else:
		tmp = math.cos((math.pi * (angle / 180.0))) * (2.0 * (b_m * (b_m * t_0)))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = sin(Float64(pi * Float64(0.005555555555555556 * angle)))
	tmp = 0.0
	if (Float64((b_m ^ 2.0) - (a ^ 2.0)) <= 5e-97)
		tmp = Float64(Float64(2.0 * Float64(b_m + a)) * Float64(Float64(b_m - a) * t_0));
	else
		tmp = Float64(cos(Float64(pi * Float64(angle / 180.0))) * Float64(2.0 * Float64(b_m * Float64(b_m * t_0))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	t_0 = sin((pi * (0.005555555555555556 * angle)));
	tmp = 0.0;
	if (((b_m ^ 2.0) - (a ^ 2.0)) <= 5e-97)
		tmp = (2.0 * (b_m + a)) * ((b_m - a) * t_0);
	else
		tmp = cos((pi * (angle / 180.0))) * (2.0 * (b_m * (b_m * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[Sin[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], 5e-97], N[(N[(2.0 * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(b$95$m * N[(b$95$m * t$95$0), $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))) < 4.9999999999999995e-97

   1. Initial program 56.6%

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

      1. add-exp-log 36.3%

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

      2. *-commutative 36.3%

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

      3. div-inv 34.9%

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

      4. metadata-eval 34.9%

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

   4. Applied egg-rr 34.9%

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

   5. Taylor expanded in angle around 0 37.9%

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

      1. rem-exp-log 61.8%

         \[\leadsto \color{blue}{\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \cdot 1 \]

      2. *-commutative 61.8%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot 1 \]

      3. add-sqr-sqrt 18.8%

         \[\leadsto \left(\color{blue}{\left(\sqrt{2 \cdot \left({b}^{2} - {a}^{2}\right)} \cdot \sqrt{2 \cdot \left({b}^{2} - {a}^{2}\right)}\right)} \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

      4. add-sqr-sqrt 61.8%

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

      5. unpow2 61.8%

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

      6. unpow2 61.8%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

      7. difference-of-squares 61.8%

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

      8. associate-*r* 61.8%

         \[\leadsto \left(\color{blue}{\left(\left(2 \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

      9. associate-*l* 70.2%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot 1 \]

   7. Applied egg-rr 70.2%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot 1 \]

   if 4.9999999999999995e-97 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   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. Step-by-step derivation

      1. unpow2 55.3%

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

      2. unpow2 55.3%

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

      3. difference-of-squares 61.4%

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

   4. Applied egg-rr 61.4%

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

      1. add-log-exp 23.8%

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

      2. div-inv 22.9%

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

      3. metadata-eval 22.9%

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

   6. Applied egg-rr 22.9%

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

   7. Taylor expanded in a around 0 57.9%

      \[\leadsto \color{blue}{\left(2 \cdot \left(a \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + -1 \cdot b\right)\right)\right) + 2 \cdot \left({b}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   9. Simplified 70.9%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(b + 0\right)\right) \cdot b\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 5 \cdot 10^{-97}:\\ \;\;\;\;\left(2 \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot \left(b \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 2 \cdot \left(b\_m + a\right)\\ \mathbf{if}\;{b\_m}^{2} \leq 5 \cdot 10^{-97}:\\ \;\;\;\;t\_0 \cdot \left(\left(b\_m - a\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m - a\right) \cdot \left(t\_0 \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* 2.0 (+ b_m a))))
   (if (<= (pow b_m 2.0) 5e-97)
     (* t_0 (* (- b_m a) (sin (* PI (* 0.005555555555555556 angle)))))
     (*
      (* (- b_m a) (* t_0 (sin (* 0.005555555555555556 (* angle PI)))))
      (cos (/ PI (/ 180.0 angle)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = 2.0 * (b_m + a);
	double tmp;
	if (pow(b_m, 2.0) <= 5e-97) {
		tmp = t_0 * ((b_m - a) * sin((((double) M_PI) * (0.005555555555555556 * angle))));
	} else {
		tmp = ((b_m - a) * (t_0 * sin((0.005555555555555556 * (angle * ((double) M_PI)))))) * cos((((double) M_PI) / (180.0 / angle)));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double t_0 = 2.0 * (b_m + a);
	double tmp;
	if (Math.pow(b_m, 2.0) <= 5e-97) {
		tmp = t_0 * ((b_m - a) * Math.sin((Math.PI * (0.005555555555555556 * angle))));
	} else {
		tmp = ((b_m - a) * (t_0 * Math.sin((0.005555555555555556 * (angle * Math.PI))))) * Math.cos((Math.PI / (180.0 / angle)));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	t_0 = 2.0 * (b_m + a)
	tmp = 0
	if math.pow(b_m, 2.0) <= 5e-97:
		tmp = t_0 * ((b_m - a) * math.sin((math.pi * (0.005555555555555556 * angle))))
	else:
		tmp = ((b_m - a) * (t_0 * math.sin((0.005555555555555556 * (angle * math.pi))))) * math.cos((math.pi / (180.0 / angle)))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(2.0 * Float64(b_m + a))
	tmp = 0.0
	if ((b_m ^ 2.0) <= 5e-97)
		tmp = Float64(t_0 * Float64(Float64(b_m - a) * sin(Float64(pi * Float64(0.005555555555555556 * angle)))));
	else
		tmp = Float64(Float64(Float64(b_m - a) * Float64(t_0 * sin(Float64(0.005555555555555556 * Float64(angle * pi))))) * cos(Float64(pi / Float64(180.0 / angle))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	t_0 = 2.0 * (b_m + a);
	tmp = 0.0;
	if ((b_m ^ 2.0) <= 5e-97)
		tmp = t_0 * ((b_m - a) * sin((pi * (0.005555555555555556 * angle))));
	else
		tmp = ((b_m - a) * (t_0 * sin((0.005555555555555556 * (angle * pi))))) * cos((pi / (180.0 / angle)));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(2.0 * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[b$95$m, 2.0], $MachinePrecision], 5e-97], N[(t$95$0 * N[(N[(b$95$m - a), $MachinePrecision] * N[Sin[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$m - a), $MachinePrecision] * N[(t$95$0 * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b #s(literal 2 binary64)) < 4.9999999999999995e-97

   1. Initial program 57.1%

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

      1. add-exp-log 40.0%

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

      2. *-commutative 40.0%

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

      3. div-inv 39.2%

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

      4. metadata-eval 39.2%

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

   4. Applied egg-rr 39.2%

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

   5. Taylor expanded in angle around 0 41.9%

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

      1. rem-exp-log 62.9%

         \[\leadsto \color{blue}{\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \cdot 1 \]

      2. *-commutative 62.9%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot 1 \]

      3. add-sqr-sqrt 24.2%

         \[\leadsto \left(\color{blue}{\left(\sqrt{2 \cdot \left({b}^{2} - {a}^{2}\right)} \cdot \sqrt{2 \cdot \left({b}^{2} - {a}^{2}\right)}\right)} \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

      4. add-sqr-sqrt 62.9%

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

      5. unpow2 62.9%

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

      6. unpow2 62.9%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

      7. difference-of-squares 62.9%

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

      8. associate-*r* 62.9%

         \[\leadsto \left(\color{blue}{\left(\left(2 \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

      9. associate-*l* 71.3%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot 1 \]

   7. Applied egg-rr 71.3%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot 1 \]

   if 4.9999999999999995e-97 < (pow.f64 b #s(literal 2 binary64))

   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. Step-by-step derivation

      1. add-exp-log 21.8%

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

      2. *-commutative 21.8%

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

      3. div-inv 22.5%

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

      4. metadata-eval 22.5%

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

   4. Applied egg-rr 22.5%

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

      1. rem-exp-log 54.5%

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

      2. add-sqr-sqrt 42.8%

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

      3. add-sqr-sqrt 54.5%

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

      4. unpow2 54.5%

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

      5. unpow2 54.5%

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

      6. difference-of-squares 59.3%

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

      7. associate-*r* 59.3%

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

      8. associate-*r* 70.7%

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

   6. Applied egg-rr 70.7%

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

      1. *-commutative 70.7%

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

      2. associate-*r* 70.5%

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

      3. *-commutative 70.5%

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

      4. *-commutative 70.5%

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

      5. +-commutative 70.5%

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

   8. Simplified 70.5%

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

      1. clear-num 71.8%

         \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \color{blue}{\frac{1}{\frac{180}{angle}}}\right) \]

      2. un-div-inv 72.9%

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

   10. Applied egg-rr 72.9%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 5 \cdot 10^{-97}:\\ \;\;\;\;\left(2 \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(2 \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+58}:\\ \;\;\;\;\left(2 \cdot \left(b\_m + a\right)\right) \cdot \left(\left(b\_m - a\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m - a\right) \cdot \left(b\_m + a\right)\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= (/ angle 180.0) 2e+58)
   (*
    (* 2.0 (+ b_m a))
    (* (- b_m a) (sin (* PI (* 0.005555555555555556 angle)))))
   (* (* (- b_m a) (+ b_m a)) (* 2.0 (sin (* PI (/ angle 180.0)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if ((angle / 180.0) <= 2e+58) {
		tmp = (2.0 * (b_m + a)) * ((b_m - a) * sin((((double) M_PI) * (0.005555555555555556 * angle))));
	} else {
		tmp = ((b_m - a) * (b_m + a)) * (2.0 * sin((((double) M_PI) * (angle / 180.0))));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double tmp;
	if ((angle / 180.0) <= 2e+58) {
		tmp = (2.0 * (b_m + a)) * ((b_m - a) * Math.sin((Math.PI * (0.005555555555555556 * angle))));
	} else {
		tmp = ((b_m - a) * (b_m + a)) * (2.0 * Math.sin((Math.PI * (angle / 180.0))));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	tmp = 0
	if (angle / 180.0) <= 2e+58:
		tmp = (2.0 * (b_m + a)) * ((b_m - a) * math.sin((math.pi * (0.005555555555555556 * angle))))
	else:
		tmp = ((b_m - a) * (b_m + a)) * (2.0 * math.sin((math.pi * (angle / 180.0))))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 2e+58)
		tmp = Float64(Float64(2.0 * Float64(b_m + a)) * Float64(Float64(b_m - a) * sin(Float64(pi * Float64(0.005555555555555556 * angle)))));
	else
		tmp = Float64(Float64(Float64(b_m - a) * Float64(b_m + a)) * Float64(2.0 * sin(Float64(pi * Float64(angle / 180.0)))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	tmp = 0.0;
	if ((angle / 180.0) <= 2e+58)
		tmp = (2.0 * (b_m + a)) * ((b_m - a) * sin((pi * (0.005555555555555556 * angle))));
	else
		tmp = ((b_m - a) * (b_m + a)) * (2.0 * sin((pi * (angle / 180.0))));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e+58], N[(N[(2.0 * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * N[Sin[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$m - a), $MachinePrecision] * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1.99999999999999989e58

   1. Initial program 63.5%

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

      1. add-exp-log 33.4%

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

      2. *-commutative 33.4%

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

      3. div-inv 32.8%

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

      4. metadata-eval 32.8%

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

   4. Applied egg-rr 32.8%

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

   5. Taylor expanded in angle around 0 32.9%

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

      1. rem-exp-log 62.7%

         \[\leadsto \color{blue}{\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \cdot 1 \]

      2. *-commutative 62.7%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot 1 \]

      3. add-sqr-sqrt 36.0%

         \[\leadsto \left(\color{blue}{\left(\sqrt{2 \cdot \left({b}^{2} - {a}^{2}\right)} \cdot \sqrt{2 \cdot \left({b}^{2} - {a}^{2}\right)}\right)} \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

      4. add-sqr-sqrt 62.7%

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

      5. unpow2 62.7%

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

      6. unpow2 62.7%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

      7. difference-of-squares 64.2%

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

      8. associate-*r* 64.2%

         \[\leadsto \left(\color{blue}{\left(\left(2 \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

      9. associate-*l* 76.8%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot 1 \]

   7. Applied egg-rr 76.8%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot 1 \]

   if 1.99999999999999989e58 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 26.1%

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

      1. associate-*l* 26.1%

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

      2. *-commutative 26.1%

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

      3. associate-*l* 26.1%

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

   3. Simplified 26.1%

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

      1. unpow2 26.1%

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

      2. unpow2 26.1%

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

      3. difference-of-squares 30.0%

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

   6. Applied egg-rr 30.0%

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

   7. Taylor expanded in angle around 0 42.1%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+58}:\\ \;\;\;\;\left(2 \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.0% accurate, 3.6× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (*
  (- b_m a)
  (* (* 2.0 (+ b_m a)) (sin (* 0.005555555555555556 (* angle PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	return (b_m - a) * ((2.0 * (b_m + a)) * sin((0.005555555555555556 * (angle * ((double) M_PI)))));
}

Java

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

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	return (b_m - a) * ((2.0 * (b_m + a)) * math.sin((0.005555555555555556 * (angle * math.pi))))

Julia

b_m = abs(b)
function code(a, b_m, angle)
	return Float64(Float64(b_m - a) * Float64(Float64(2.0 * Float64(b_m + a)) * sin(Float64(0.005555555555555556 * Float64(angle * pi)))))
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle)
	tmp = (b_m - a) * ((2.0 * (b_m + a)) * sin((0.005555555555555556 * (angle * pi))));
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := N[(N[(b$95$m - a), $MachinePrecision] * N[(N[(2.0 * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. add-exp-log 29.6%

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

   2. *-commutative 29.6%

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

   3. div-inv 29.7%

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

   4. metadata-eval 29.7%

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

4. Applied egg-rr 29.7%

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

   1. rem-exp-log 55.3%

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

   2. add-sqr-sqrt 34.7%

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

   3. add-sqr-sqrt 55.3%

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

   4. unpow2 55.3%

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

   5. unpow2 55.3%

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

   6. difference-of-squares 58.0%

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

   7. associate-*r* 58.0%

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

   8. associate-*r* 68.1%

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

6. Applied egg-rr 68.1%

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

   1. *-commutative 68.1%

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

   2. associate-*r* 68.6%

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

   3. *-commutative 68.6%

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

   4. *-commutative 68.6%

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

   5. +-commutative 68.6%

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

8. Simplified 68.6%

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

9. Taylor expanded in angle around 0 68.8%

   \[\leadsto \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(a + b\right)\right)\right)\right) \cdot \color{blue}{1} \]

10. Final simplification 68.8%

    \[\leadsto \left(b - a\right) \cdot \left(\left(2 \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]
11. Add Preprocessing

Alternative 13: 65.4% accurate, 3.6× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (*
  (* 2.0 (+ b_m a))
  (* (- b_m a) (sin (* PI (* 0.005555555555555556 angle))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	return (2.0 * (b_m + a)) * ((b_m - a) * sin((((double) M_PI) * (0.005555555555555556 * angle))));
}

Java

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

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	return (2.0 * (b_m + a)) * ((b_m - a) * math.sin((math.pi * (0.005555555555555556 * angle))))

Julia

b_m = abs(b)
function code(a, b_m, angle)
	return Float64(Float64(2.0 * Float64(b_m + a)) * Float64(Float64(b_m - a) * sin(Float64(pi * Float64(0.005555555555555556 * angle)))))
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle)
	tmp = (2.0 * (b_m + a)) * ((b_m - a) * sin((pi * (0.005555555555555556 * angle))));
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := N[(N[(2.0 * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * N[Sin[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. add-exp-log 29.6%

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

   2. *-commutative 29.6%

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

   3. div-inv 29.7%

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

   4. metadata-eval 29.7%

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

4. Applied egg-rr 29.7%

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

5. Taylor expanded in angle around 0 29.9%

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

   1. rem-exp-log 55.9%

      \[\leadsto \color{blue}{\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \cdot 1 \]

   2. *-commutative 55.9%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot 1 \]

   3. add-sqr-sqrt 32.2%

      \[\leadsto \left(\color{blue}{\left(\sqrt{2 \cdot \left({b}^{2} - {a}^{2}\right)} \cdot \sqrt{2 \cdot \left({b}^{2} - {a}^{2}\right)}\right)} \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

   4. add-sqr-sqrt 55.9%

      \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

   5. unpow2 55.9%

      \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

   6. unpow2 55.9%

      \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

   7. difference-of-squares 57.9%

      \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

   8. associate-*r* 57.9%

      \[\leadsto \left(\color{blue}{\left(\left(2 \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 1 \]

   9. associate-*l* 68.0%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot 1 \]

7. Applied egg-rr 68.0%

   \[\leadsto \color{blue}{\left(\left(2 \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot 1 \]

8. Final simplification 68.0%

   \[\leadsto \left(2 \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \]
9. Add Preprocessing

Alternative 14: 58.1% accurate, 23.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.4 \cdot 10^{+148}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b\_m - a\right) \cdot \left(b\_m + a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b\_m \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(b\_m \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= b_m 1.4e+148)
   (* 0.011111111111111112 (* angle (* PI (* (- b_m a) (+ b_m a)))))
   (* b_m (* (* angle 0.011111111111111112) (* b_m PI)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (b_m <= 1.4e+148) {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * ((b_m - a) * (b_m + a))));
	} else {
		tmp = b_m * ((angle * 0.011111111111111112) * (b_m * ((double) M_PI)));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double tmp;
	if (b_m <= 1.4e+148) {
		tmp = 0.011111111111111112 * (angle * (Math.PI * ((b_m - a) * (b_m + a))));
	} else {
		tmp = b_m * ((angle * 0.011111111111111112) * (b_m * Math.PI));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	tmp = 0
	if b_m <= 1.4e+148:
		tmp = 0.011111111111111112 * (angle * (math.pi * ((b_m - a) * (b_m + a))))
	else:
		tmp = b_m * ((angle * 0.011111111111111112) * (b_m * math.pi))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (b_m <= 1.4e+148)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(Float64(b_m - a) * Float64(b_m + a)))));
	else
		tmp = Float64(b_m * Float64(Float64(angle * 0.011111111111111112) * Float64(b_m * pi)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	tmp = 0.0;
	if (b_m <= 1.4e+148)
		tmp = 0.011111111111111112 * (angle * (pi * ((b_m - a) * (b_m + a))));
	else
		tmp = b_m * ((angle * 0.011111111111111112) * (b_m * pi));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[b$95$m, 1.4e+148], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(N[(b$95$m - a), $MachinePrecision] * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b$95$m * N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(b$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3999999999999999e148

   1. Initial program 58.0%

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

      1. associate-*l* 58.0%

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

      2. *-commutative 58.0%

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

      3. associate-*l* 58.0%

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

   3. Simplified 58.0%

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

      1. unpow2 58.0%

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

      2. unpow2 58.0%

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

      3. difference-of-squares 59.8%

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

   6. Applied egg-rr 59.8%

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

   7. Taylor expanded in angle around 0 56.9%

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

   if 1.3999999999999999e148 < b

   1. Initial program 41.4%

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

      1. associate-*l* 41.4%

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

      2. *-commutative 41.4%

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

      3. associate-*l* 41.4%

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

   3. Simplified 41.4%

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

      1. unpow2 41.4%

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

      2. unpow2 41.4%

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

      3. difference-of-squares 51.5%

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

   6. Applied egg-rr 51.5%

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

   7. Taylor expanded in angle around 0 38.2%

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

   8. Taylor expanded in a around 0 34.9%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(\pi \cdot \left(b + -1 \cdot b\right)\right)\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]

   10. Simplified 59.5%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{+148}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(b \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 40.0% accurate, 29.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 10^{+61}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(b\_m \cdot \left(b\_m \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b\_m \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(b\_m \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= b_m 1e+61)
   (* 0.011111111111111112 (* angle (* b_m (* b_m PI))))
   (* b_m (* (* angle 0.011111111111111112) (* b_m PI)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (b_m <= 1e+61) {
		tmp = 0.011111111111111112 * (angle * (b_m * (b_m * ((double) M_PI))));
	} else {
		tmp = b_m * ((angle * 0.011111111111111112) * (b_m * ((double) M_PI)));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double tmp;
	if (b_m <= 1e+61) {
		tmp = 0.011111111111111112 * (angle * (b_m * (b_m * Math.PI)));
	} else {
		tmp = b_m * ((angle * 0.011111111111111112) * (b_m * Math.PI));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	tmp = 0
	if b_m <= 1e+61:
		tmp = 0.011111111111111112 * (angle * (b_m * (b_m * math.pi)))
	else:
		tmp = b_m * ((angle * 0.011111111111111112) * (b_m * math.pi))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (b_m <= 1e+61)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(b_m * Float64(b_m * pi))));
	else
		tmp = Float64(b_m * Float64(Float64(angle * 0.011111111111111112) * Float64(b_m * pi)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	tmp = 0.0;
	if (b_m <= 1e+61)
		tmp = 0.011111111111111112 * (angle * (b_m * (b_m * pi)));
	else
		tmp = b_m * ((angle * 0.011111111111111112) * (b_m * pi));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[b$95$m, 1e+61], N[(0.011111111111111112 * N[(angle * N[(b$95$m * N[(b$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b$95$m * N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(b$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.99999999999999949e60

   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. Step-by-step derivation

      1. associate-*l* 57.5%

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

      2. *-commutative 57.5%

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

      3. associate-*l* 57.5%

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

   3. Simplified 57.5%

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

      1. unpow2 57.5%

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

      2. unpow2 57.5%

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

      3. difference-of-squares 59.4%

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

   6. Applied egg-rr 59.4%

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

   7. Taylor expanded in angle around 0 57.7%

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

   8. Taylor expanded in a around 0 38.4%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(a \cdot \left(angle \cdot \left(\pi \cdot \left(b + -1 \cdot b\right)\right)\right) + angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]

   10. Simplified 38.3%

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

   11. Taylor expanded in b around 0 38.3%

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

       1. *-commutative 38.3%

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

   13. Simplified 38.3%

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

   if 9.99999999999999949e60 < b

   1. Initial program 50.6%

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

      1. associate-*l* 50.6%

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

      2. *-commutative 50.6%

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

      3. associate-*l* 50.6%

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

   3. Simplified 50.6%

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

      1. unpow2 50.6%

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

      2. unpow2 50.6%

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

      3. difference-of-squares 56.3%

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

   6. Applied egg-rr 56.3%

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

   7. Taylor expanded in angle around 0 43.1%

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

   8. Taylor expanded in a around 0 39.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(\pi \cdot \left(b + -1 \cdot b\right)\right)\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]

   10. Simplified 53.3%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{+61}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(b \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 35.1% accurate, 46.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := N[(0.011111111111111112 * N[(angle * N[(b$95$m * N[(b$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. associate-*l* 56.0%

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

   2. *-commutative 56.0%

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

   3. associate-*l* 56.0%

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

3. Simplified 56.0%

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

   1. unpow2 56.0%

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

   2. unpow2 56.0%

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

   3. difference-of-squares 58.8%

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

6. Applied egg-rr 58.8%

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

7. Taylor expanded in angle around 0 54.7%

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

8. Taylor expanded in a around 0 38.5%

   \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(a \cdot \left(angle \cdot \left(\pi \cdot \left(b + -1 \cdot b\right)\right)\right) + angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]

10. Simplified 38.5%

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

11. Taylor expanded in b around 0 38.5%

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

    1. *-commutative 38.5%

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

13. Simplified 38.5%

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

14. Final simplification 38.5%

    \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \left(b \cdot \pi\right)\right)\right) \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024108 
(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)))))