ab-angle->ABCF B

Percentage Accurate: 53.9% → 65.6%

Time: 17.7s

Alternatives: 11

Speedup: 3.5×

• 33.3% 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: 53.9% 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: 65.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\right)\\ t_1 := \cos \left(e^{\log \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b\_m \leq 2.2 \cdot 10^{+140}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \left(a \cdot \left(a \cdot t\_0\right) - t\_0 \cdot {b\_m}^{2}\right)\right)\\ \mathbf{elif}\;b\_m \leq 2.5 \cdot 10^{+248}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \mathsf{fma}\left(b\_m, b\_m \cdot \left(-t\_0\right), {a}^{2} \cdot t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \left(\sin \left(angle\_m \cdot \frac{\sqrt[3]{{\pi}^{3}}}{-180}\right) \cdot \left(\left(a + b\_m\right) \cdot \left(a - b\_m\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (sin (* -0.005555555555555556 (* PI angle_m))))
        (t_1 (cos (exp (log (* PI (* angle_m 0.005555555555555556)))))))
   (*
    angle_s
    (if (<= b_m 2.2e+140)
      (* t_1 (* 2.0 (- (* a (* a t_0)) (* t_0 (pow b_m 2.0)))))
      (if (<= b_m 2.5e+248)
        (* t_1 (* 2.0 (fma b_m (* b_m (- t_0)) (* (pow a 2.0) t_0))))
        (*
         t_1
         (*
          2.0
          (*
           (sin (* angle_m (/ (cbrt (pow PI 3.0)) -180.0)))
           (* (+ a b_m) (- a b_m))))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = sin((-0.005555555555555556 * (((double) M_PI) * angle_m)));
	double t_1 = cos(exp(log((((double) M_PI) * (angle_m * 0.005555555555555556)))));
	double tmp;
	if (b_m <= 2.2e+140) {
		tmp = t_1 * (2.0 * ((a * (a * t_0)) - (t_0 * pow(b_m, 2.0))));
	} else if (b_m <= 2.5e+248) {
		tmp = t_1 * (2.0 * fma(b_m, (b_m * -t_0), (pow(a, 2.0) * t_0)));
	} else {
		tmp = t_1 * (2.0 * (sin((angle_m * (cbrt(pow(((double) M_PI), 3.0)) / -180.0))) * ((a + b_m) * (a - b_m))));
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = sin(Float64(-0.005555555555555556 * Float64(pi * angle_m)))
	t_1 = cos(exp(log(Float64(pi * Float64(angle_m * 0.005555555555555556)))))
	tmp = 0.0
	if (b_m <= 2.2e+140)
		tmp = Float64(t_1 * Float64(2.0 * Float64(Float64(a * Float64(a * t_0)) - Float64(t_0 * (b_m ^ 2.0)))));
	elseif (b_m <= 2.5e+248)
		tmp = Float64(t_1 * Float64(2.0 * fma(b_m, Float64(b_m * Float64(-t_0)), Float64((a ^ 2.0) * t_0))));
	else
		tmp = Float64(t_1 * Float64(2.0 * Float64(sin(Float64(angle_m * Float64(cbrt((pi ^ 3.0)) / -180.0))) * Float64(Float64(a + b_m) * Float64(a - b_m)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(-0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[Exp[N[Log[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[b$95$m, 2.2e+140], N[(t$95$1 * N[(2.0 * N[(N[(a * N[(a * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Power[b$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 2.5e+248], N[(t$95$1 * N[(2.0 * N[(b$95$m * N[(b$95$m * (-t$95$0)), $MachinePrecision] + N[(N[Power[a, 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[(N[Sin[N[(angle$95$m * N[(N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(a + b$95$m), $MachinePrecision] * N[(a - b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.1999999999999998e140

   1. Initial program 62.8%

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

   3. Simplified 62.8%

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

      1. add-exp-log 33.1%

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

      2. add-sqr-sqrt 33.1%

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

      3. sqrt-unprod 56.0%

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

      4. associate-*r/ 56.0%

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

      5. associate-*r/ 56.0%

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

      6. frac-times 55.6%

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

      7. *-commutative 55.6%

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

      8. *-commutative 55.6%

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

      9. metadata-eval 55.6%

         \[\leadsto \cos \left(e^{\log \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)}\right) \cdot \left(2 \cdot \left(\sin \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)\right) \]

      10. metadata-eval 55.6%

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

      11. frac-times 56.0%

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

      12. associate-*r/ 56.0%

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

      13. associate-*r/ 56.0%

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

      14. sqrt-unprod 31.2%

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

      15. add-sqr-sqrt 31.2%

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

      16. div-inv 31.2%

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

      17. metadata-eval 31.2%

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

   6. Applied egg-rr 31.2%

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

      1. unpow2 31.2%

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

      2. unpow2 31.2%

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

      3. difference-of-squares 32.1%

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

   8. Applied egg-rr 32.1%

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

   9. Taylor expanded in a around 0 34.5%

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

       1. +-commutative 34.5%

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

       2. mul-1-neg 34.5%

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

       3. unsub-neg 34.5%

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

   11. Simplified 34.5%

       \[\leadsto \cos \left(e^{\log \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \left(2 \cdot \color{blue}{\left(\left(a \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(a + 0\right) - \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {b}^{2}\right)}\right) \]

   if 2.1999999999999998e140 < b < 2.4999999999999998e248

   1. Initial program 33.9%

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

   3. Simplified 38.4%

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

      1. add-exp-log 23.0%

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

      2. add-sqr-sqrt 23.0%

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

      3. sqrt-unprod 37.6%

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

      4. associate-*r/ 37.6%

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

      5. associate-*r/ 37.6%

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

      6. frac-times 37.6%

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

      7. *-commutative 37.6%

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

      8. *-commutative 37.6%

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

      9. metadata-eval 37.6%

         \[\leadsto \cos \left(e^{\log \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)}\right) \cdot \left(2 \cdot \left(\sin \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)\right) \]

      10. metadata-eval 37.6%

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

      11. frac-times 37.6%

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

      12. associate-*r/ 37.6%

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

      13. associate-*r/ 37.6%

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

      14. sqrt-unprod 24.8%

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

      15. add-sqr-sqrt 24.8%

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

      16. div-inv 24.8%

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

      17. metadata-eval 24.8%

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

   6. Applied egg-rr 24.8%

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

      1. unpow2 24.8%

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

      2. unpow2 24.8%

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

      3. difference-of-squares 24.8%

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

   8. Applied egg-rr 24.8%

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

   9. Taylor expanded in b around 0 45.0%

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

       1. fma-define 49.6%

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

   11. Simplified 49.6%

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

   if 2.4999999999999998e248 < b

   1. Initial program 50.0%

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

   3. Simplified 50.0%

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

      1. add-exp-log 50.0%

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

      2. add-sqr-sqrt 50.0%

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

      3. sqrt-unprod 50.0%

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

      4. associate-*r/ 50.0%

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

      5. associate-*r/ 50.0%

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

      6. frac-times 50.0%

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

      7. *-commutative 50.0%

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

      8. *-commutative 50.0%

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

      9. metadata-eval 50.0%

         \[\leadsto \cos \left(e^{\log \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)}\right) \cdot \left(2 \cdot \left(\sin \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)\right) \]

      10. metadata-eval 50.0%

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

      11. frac-times 50.0%

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

      12. associate-*r/ 50.0%

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

      13. associate-*r/ 50.0%

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

      14. sqrt-unprod 8.3%

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

      15. add-sqr-sqrt 8.3%

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

      16. div-inv 8.3%

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

      17. metadata-eval 8.3%

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

   6. Applied egg-rr 8.3%

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

      1. unpow2 8.3%

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

      2. unpow2 8.3%

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

      3. difference-of-squares 16.7%

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

   8. Applied egg-rr 16.7%

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

      1. add-cbrt-cube 33.3%

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

      2. pow3 33.3%

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

   10. Applied egg-rr 33.3%

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

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{+140}:\\ \;\;\;\;\cos \left(e^{\log \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \left(2 \cdot \left(a \cdot \left(a \cdot \sin \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right) - \sin \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot {b}^{2}\right)\right)\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{+248}:\\ \;\;\;\;\cos \left(e^{\log \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \left(2 \cdot \mathsf{fma}\left(b, b \cdot \left(-\sin \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right), {a}^{2} \cdot \sin \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(e^{\log \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \left(2 \cdot \left(\sin \left(angle \cdot \frac{\sqrt[3]{{\pi}^{3}}}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 60.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(angle\_m \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{a}^{2} \leq \infty:\\ \;\;\;\;\cos \left(e^{\log \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)}\right) \cdot \left(2 \cdot \mathsf{fma}\left(b\_m, \mathsf{fma}\left(-1, b\_m \cdot t\_0, t\_0 \cdot \left(a \cdot 0\right)\right), {a}^{2} \cdot t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot {\left(a \cdot \sqrt{\pi \cdot angle\_m}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (sin (* angle_m (* PI -0.005555555555555556)))))
   (*
    angle_s
    (if (<= (pow a 2.0) INFINITY)
      (*
       (cos (exp (log (* PI (* angle_m 0.005555555555555556)))))
       (*
        2.0
        (fma
         b_m
         (fma -1.0 (* b_m t_0) (* t_0 (* a 0.0)))
         (* (pow a 2.0) t_0))))
      (* -0.011111111111111112 (pow (* a (sqrt (* PI angle_m))) 2.0))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = sin((angle_m * (((double) M_PI) * -0.005555555555555556)));
	double tmp;
	if (pow(a, 2.0) <= ((double) INFINITY)) {
		tmp = cos(exp(log((((double) M_PI) * (angle_m * 0.005555555555555556))))) * (2.0 * fma(b_m, fma(-1.0, (b_m * t_0), (t_0 * (a * 0.0))), (pow(a, 2.0) * t_0)));
	} else {
		tmp = -0.011111111111111112 * pow((a * sqrt((((double) M_PI) * angle_m))), 2.0);
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = sin(Float64(angle_m * Float64(pi * -0.005555555555555556)))
	tmp = 0.0
	if ((a ^ 2.0) <= Inf)
		tmp = Float64(cos(exp(log(Float64(pi * Float64(angle_m * 0.005555555555555556))))) * Float64(2.0 * fma(b_m, fma(-1.0, Float64(b_m * t_0), Float64(t_0 * Float64(a * 0.0))), Float64((a ^ 2.0) * t_0))));
	else
		tmp = Float64(-0.011111111111111112 * (Float64(a * sqrt(Float64(pi * angle_m))) ^ 2.0));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(angle$95$m * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[Power[a, 2.0], $MachinePrecision], Infinity], N[(N[Cos[N[Exp[N[Log[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(b$95$m * N[(-1.0 * N[(b$95$m * t$95$0), $MachinePrecision] + N[(t$95$0 * N[(a * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[a, 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[Power[N[(a * N[Sqrt[N[(Pi * angle$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 59.7%

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

   3. Simplified 60.1%

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

      1. add-exp-log 33.0%

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

      2. add-sqr-sqrt 33.0%

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

      3. sqrt-unprod 54.1%

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

      4. associate-*r/ 54.1%

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

      5. associate-*r/ 54.1%

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

      6. frac-times 53.8%

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

      7. *-commutative 53.8%

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

      8. *-commutative 53.8%

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

      9. metadata-eval 53.8%

         \[\leadsto \cos \left(e^{\log \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)}\right) \cdot \left(2 \cdot \left(\sin \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)\right) \]

      10. metadata-eval 53.8%

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

      11. frac-times 54.1%

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

      12. associate-*r/ 54.1%

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

      13. associate-*r/ 54.1%

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

      14. sqrt-unprod 29.5%

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

      15. add-sqr-sqrt 29.5%

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

      16. div-inv 29.5%

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

      17. metadata-eval 29.5%

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

   6. Applied egg-rr 29.5%

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

      1. unpow2 29.5%

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

      2. unpow2 29.5%

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

      3. difference-of-squares 30.7%

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

   8. Applied egg-rr 30.7%

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

   9. Taylor expanded in b around 0 32.0%

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

       1. fma-define 33.2%

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

   11. Simplified 32.9%

       \[\leadsto \cos \left(e^{\log \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \left(2 \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(-1, b \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right), \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot \left(0 \cdot a\right)\right), \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot {a}^{2}\right)}\right) \]

   if +inf.0 < (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. Step-by-step derivation

      1. associate-*l* 59.7%

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

      2. associate-*l* 59.7%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in angle around 0 57.8%

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

   6. Taylor expanded in b around 0 37.5%

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

      1. add-sqr-sqrt 27.5%

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

      2. pow2 27.5%

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

      3. sqrt-prod 20.0%

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

      4. sqrt-pow1 21.0%

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

      5. metadata-eval 21.0%

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

      6. pow1 21.0%

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

      7. *-commutative 21.0%

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

   8. Applied egg-rr 21.0%

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

4. Final simplification 32.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq \infty:\\ \;\;\;\;\cos \left(e^{\log \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \left(2 \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(-1, b \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right), \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot \left(a \cdot 0\right)\right), {a}^{2} \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot {\left(a \cdot \sqrt{\pi \cdot angle}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(angle\_m \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\\ t_1 := \sin \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\right)\\ t_2 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b\_m \leq 1.25 \cdot 10^{+135}:\\ \;\;\;\;\cos \left(e^{\log t\_2}\right) \cdot \left(2 \cdot \left(a \cdot \left(a \cdot t\_1\right) - t\_1 \cdot {b\_m}^{2}\right)\right)\\ \mathbf{elif}\;b\_m \leq 1.4 \cdot 10^{+202}:\\ \;\;\;\;\left(2 \cdot \cos t\_2\right) \cdot \mathsf{fma}\left(b\_m, b\_m \cdot \left(-t\_0\right), {a}^{2} \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(e^{\log \left(\pi \cdot angle\_m\right) - \log 180}\right) \cdot \left(2 \cdot \left(\left(\left(a + b\_m\right) \cdot \left(a - b\_m\right)\right) \cdot \sin \left(angle\_m \cdot \frac{\pi}{-180}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (sin (* angle_m (* PI -0.005555555555555556))))
        (t_1 (sin (* -0.005555555555555556 (* PI angle_m))))
        (t_2 (* PI (* angle_m 0.005555555555555556))))
   (*
    angle_s
    (if (<= b_m 1.25e+135)
      (*
       (cos (exp (log t_2)))
       (* 2.0 (- (* a (* a t_1)) (* t_1 (pow b_m 2.0)))))
      (if (<= b_m 1.4e+202)
        (* (* 2.0 (cos t_2)) (fma b_m (* b_m (- t_0)) (* (pow a 2.0) t_0)))
        (*
         (cos (exp (- (log (* PI angle_m)) (log 180.0))))
         (*
          2.0
          (* (* (+ a b_m) (- a b_m)) (sin (* angle_m (/ PI -180.0)))))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = sin((angle_m * (((double) M_PI) * -0.005555555555555556)));
	double t_1 = sin((-0.005555555555555556 * (((double) M_PI) * angle_m)));
	double t_2 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double tmp;
	if (b_m <= 1.25e+135) {
		tmp = cos(exp(log(t_2))) * (2.0 * ((a * (a * t_1)) - (t_1 * pow(b_m, 2.0))));
	} else if (b_m <= 1.4e+202) {
		tmp = (2.0 * cos(t_2)) * fma(b_m, (b_m * -t_0), (pow(a, 2.0) * t_0));
	} else {
		tmp = cos(exp((log((((double) M_PI) * angle_m)) - log(180.0)))) * (2.0 * (((a + b_m) * (a - b_m)) * sin((angle_m * (((double) M_PI) / -180.0)))));
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = sin(Float64(angle_m * Float64(pi * -0.005555555555555556)))
	t_1 = sin(Float64(-0.005555555555555556 * Float64(pi * angle_m)))
	t_2 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	tmp = 0.0
	if (b_m <= 1.25e+135)
		tmp = Float64(cos(exp(log(t_2))) * Float64(2.0 * Float64(Float64(a * Float64(a * t_1)) - Float64(t_1 * (b_m ^ 2.0)))));
	elseif (b_m <= 1.4e+202)
		tmp = Float64(Float64(2.0 * cos(t_2)) * fma(b_m, Float64(b_m * Float64(-t_0)), Float64((a ^ 2.0) * t_0)));
	else
		tmp = Float64(cos(exp(Float64(log(Float64(pi * angle_m)) - log(180.0)))) * Float64(2.0 * Float64(Float64(Float64(a + b_m) * Float64(a - b_m)) * sin(Float64(angle_m * Float64(pi / -180.0))))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(angle$95$m * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(-0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[b$95$m, 1.25e+135], N[(N[Cos[N[Exp[N[Log[t$95$2], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(N[(a * N[(a * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Power[b$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.4e+202], N[(N[(2.0 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] * N[(b$95$m * N[(b$95$m * (-t$95$0)), $MachinePrecision] + N[(N[Power[a, 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[Exp[N[(N[Log[N[(Pi * angle$95$m), $MachinePrecision]], $MachinePrecision] - N[Log[180.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(N[(N[(a + b$95$m), $MachinePrecision] * N[(a - b$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.25000000000000007e135

   1. Initial program 63.1%

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

   3. Simplified 63.1%

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

      1. add-exp-log 33.2%

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

      2. add-sqr-sqrt 33.2%

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

      3. sqrt-unprod 56.3%

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

      4. associate-*r/ 56.3%

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

      5. associate-*r/ 56.3%

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

      6. frac-times 55.8%

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

      7. *-commutative 55.8%

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

      8. *-commutative 55.8%

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

      9. metadata-eval 55.8%

         \[\leadsto \cos \left(e^{\log \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)}\right) \cdot \left(2 \cdot \left(\sin \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)\right) \]

      10. metadata-eval 55.8%

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

      11. frac-times 56.3%

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

      12. associate-*r/ 56.3%

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

      13. associate-*r/ 56.3%

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

      14. sqrt-unprod 31.3%

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

      15. add-sqr-sqrt 31.3%

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

      16. div-inv 31.3%

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

      17. metadata-eval 31.3%

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

   6. Applied egg-rr 31.3%

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

      1. unpow2 31.3%

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

      2. unpow2 31.3%

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

      3. difference-of-squares 32.2%

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

   8. Applied egg-rr 32.2%

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

   9. Taylor expanded in a around 0 34.6%

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

       1. +-commutative 34.6%

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

       2. mul-1-neg 34.6%

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

       3. unsub-neg 34.6%

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

   11. Simplified 34.6%

       \[\leadsto \cos \left(e^{\log \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \left(2 \cdot \color{blue}{\left(\left(a \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(a + 0\right) - \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {b}^{2}\right)}\right) \]

   if 1.25000000000000007e135 < b < 1.40000000000000008e202

   1. Initial program 34.2%

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

   3. Simplified 40.3%

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

      1. add-exp-log 19.1%

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

      2. add-sqr-sqrt 19.1%

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

      3. sqrt-unprod 33.0%

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

      4. associate-*r/ 33.0%

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

      5. associate-*r/ 33.0%

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

      6. frac-times 33.0%

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

      7. *-commutative 33.0%

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

      8. *-commutative 33.0%

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

      9. metadata-eval 33.0%

         \[\leadsto \cos \left(e^{\log \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)}\right) \cdot \left(2 \cdot \left(\sin \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)\right) \]

      10. metadata-eval 33.0%

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

      11. frac-times 33.0%

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

      12. associate-*r/ 33.0%

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

      13. associate-*r/ 33.0%

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

      14. sqrt-unprod 21.6%

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

      15. add-sqr-sqrt 21.6%

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

      16. div-inv 21.6%

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

      17. metadata-eval 21.6%

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

   6. Applied egg-rr 21.6%

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

      1. unpow2 21.6%

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

      2. unpow2 21.6%

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

      3. difference-of-squares 21.6%

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

   8. Applied egg-rr 21.6%

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

   9. Taylor expanded in b around 0 49.4%

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

       1. fma-define 49.4%

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

   11. Simplified 51.0%

       \[\leadsto \cos \left(e^{\log \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \left(2 \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(-1, b \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right), \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot \left(0 \cdot a\right)\right), \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot {a}^{2}\right)}\right) \]
   12. Step-by-step derivation

       1. pow1 51.0%

          \[\leadsto \color{blue}{{\left(\cos \left(e^{\log \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \left(2 \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(-1, b \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right), \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot \left(0 \cdot a\right)\right), \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot {a}^{2}\right)\right)\right)}^{1}} \]

   13. Applied egg-rr 75.5%

       \[\leadsto \color{blue}{{\left(\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right) \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(-1, b \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right), 0\right), \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot {a}^{2}\right)\right)}^{1}} \]
   14. Step-by-step derivation

       1. unpow1 75.5%

          \[\leadsto \color{blue}{\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right) \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(-1, b \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right), 0\right), \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot {a}^{2}\right)} \]

       2. *-commutative 75.5%

          \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(-1, b \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right), 0\right), \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot {a}^{2}\right) \]

   15. Simplified 75.5%

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

   if 1.40000000000000008e202 < b

   1. Initial program 42.1%

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

   3. Simplified 42.1%

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

      1. add-exp-log 42.1%

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

      2. add-sqr-sqrt 42.1%

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

      3. sqrt-unprod 47.4%

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

      4. associate-*r/ 47.4%

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

      5. associate-*r/ 47.4%

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

      6. frac-times 47.4%

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

      7. *-commutative 47.4%

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

      8. *-commutative 47.4%

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

      9. metadata-eval 47.4%

         \[\leadsto \cos \left(e^{\log \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)}\right) \cdot \left(2 \cdot \left(\sin \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)\right) \]

      10. metadata-eval 47.4%

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

      11. frac-times 47.4%

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

      12. associate-*r/ 47.4%

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

      13. associate-*r/ 47.4%

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

      14. sqrt-unprod 15.8%

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

      15. add-sqr-sqrt 15.8%

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

      16. div-inv 15.8%

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

      17. metadata-eval 15.8%

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

   6. Applied egg-rr 15.8%

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

      1. unpow2 15.8%

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

      2. unpow2 15.8%

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

      3. difference-of-squares 21.1%

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

   8. Applied egg-rr 21.1%

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

      1. log-prod 21.1%

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

      2. metadata-eval 21.1%

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

      3. div-inv 21.1%

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

      4. sum-log 21.1%

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

      5. associate-*r/ 21.1%

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

      6. *-commutative 21.1%

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

      7. log-div 15.8%

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

      8. *-commutative 15.8%

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

   10. Applied egg-rr 15.8%

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

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{+135}:\\ \;\;\;\;\cos \left(e^{\log \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \left(2 \cdot \left(a \cdot \left(a \cdot \sin \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right) - \sin \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot {b}^{2}\right)\right)\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{+202}:\\ \;\;\;\;\left(2 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \mathsf{fma}\left(b, b \cdot \left(-\sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right), {a}^{2} \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(e^{\log \left(\pi \cdot angle\right) - \log 180}\right) \cdot \left(2 \cdot \left(\left(\left(a + b\right) \cdot \left(a - b\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(angle\_m \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{a}^{2} \leq \infty:\\ \;\;\;\;\left(2 \cdot \cos \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right) \cdot \mathsf{fma}\left(b\_m, b\_m \cdot \left(-t\_0\right), {a}^{2} \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot {\left(a \cdot \sqrt{\pi \cdot angle\_m}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (sin (* angle_m (* PI -0.005555555555555556)))))
   (*
    angle_s
    (if (<= (pow a 2.0) INFINITY)
      (*
       (* 2.0 (cos (* PI (* angle_m 0.005555555555555556))))
       (fma b_m (* b_m (- t_0)) (* (pow a 2.0) t_0)))
      (* -0.011111111111111112 (pow (* a (sqrt (* PI angle_m))) 2.0))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = sin((angle_m * (((double) M_PI) * -0.005555555555555556)));
	double tmp;
	if (pow(a, 2.0) <= ((double) INFINITY)) {
		tmp = (2.0 * cos((((double) M_PI) * (angle_m * 0.005555555555555556)))) * fma(b_m, (b_m * -t_0), (pow(a, 2.0) * t_0));
	} else {
		tmp = -0.011111111111111112 * pow((a * sqrt((((double) M_PI) * angle_m))), 2.0);
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = sin(Float64(angle_m * Float64(pi * -0.005555555555555556)))
	tmp = 0.0
	if ((a ^ 2.0) <= Inf)
		tmp = Float64(Float64(2.0 * cos(Float64(pi * Float64(angle_m * 0.005555555555555556)))) * fma(b_m, Float64(b_m * Float64(-t_0)), Float64((a ^ 2.0) * t_0)));
	else
		tmp = Float64(-0.011111111111111112 * (Float64(a * sqrt(Float64(pi * angle_m))) ^ 2.0));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(angle$95$m * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[Power[a, 2.0], $MachinePrecision], Infinity], N[(N[(2.0 * N[Cos[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b$95$m * N[(b$95$m * (-t$95$0)), $MachinePrecision] + N[(N[Power[a, 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[Power[N[(a * N[Sqrt[N[(Pi * angle$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 59.7%

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

   3. Simplified 60.1%

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

      1. add-exp-log 33.0%

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

      2. add-sqr-sqrt 33.0%

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

      3. sqrt-unprod 54.1%

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

      4. associate-*r/ 54.1%

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

      5. associate-*r/ 54.1%

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

      6. frac-times 53.8%

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

      7. *-commutative 53.8%

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

      8. *-commutative 53.8%

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

      9. metadata-eval 53.8%

         \[\leadsto \cos \left(e^{\log \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)}\right) \cdot \left(2 \cdot \left(\sin \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)\right) \]

      10. metadata-eval 53.8%

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

      11. frac-times 54.1%

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

      12. associate-*r/ 54.1%

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

      13. associate-*r/ 54.1%

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

      14. sqrt-unprod 29.5%

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

      15. add-sqr-sqrt 29.5%

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

      16. div-inv 29.5%

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

      17. metadata-eval 29.5%

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

   6. Applied egg-rr 29.5%

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

      1. unpow2 29.5%

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

      2. unpow2 29.5%

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

      3. difference-of-squares 30.7%

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

   8. Applied egg-rr 30.7%

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

   9. Taylor expanded in b around 0 32.0%

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

       1. fma-define 33.2%

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

   11. Simplified 32.9%

       \[\leadsto \cos \left(e^{\log \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \left(2 \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(-1, b \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right), \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot \left(0 \cdot a\right)\right), \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot {a}^{2}\right)}\right) \]
   12. Step-by-step derivation

       1. pow1 32.9%

          \[\leadsto \color{blue}{{\left(\cos \left(e^{\log \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \left(2 \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(-1, b \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right), \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot \left(0 \cdot a\right)\right), \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot {a}^{2}\right)\right)\right)}^{1}} \]

   13. Applied egg-rr 64.8%

       \[\leadsto \color{blue}{{\left(\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right) \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(-1, b \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right), 0\right), \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot {a}^{2}\right)\right)}^{1}} \]
   14. Step-by-step derivation

       1. unpow1 64.8%

          \[\leadsto \color{blue}{\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right) \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(-1, b \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right), 0\right), \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot {a}^{2}\right)} \]

       2. *-commutative 64.8%

          \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(-1, b \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right), 0\right), \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot {a}^{2}\right) \]

   15. Simplified 64.8%

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

   if +inf.0 < (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. Step-by-step derivation

      1. associate-*l* 59.7%

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

      2. associate-*l* 59.7%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in angle around 0 57.8%

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

   6. Taylor expanded in b around 0 37.5%

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

      1. add-sqr-sqrt 27.5%

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

      2. pow2 27.5%

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

      3. sqrt-prod 20.0%

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

      4. sqrt-pow1 21.0%

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

      5. metadata-eval 21.0%

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

      6. pow1 21.0%

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

      7. *-commutative 21.0%

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

   8. Applied egg-rr 21.0%

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

4. Final simplification 64.8%

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

Alternative 5: 58.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 2.1 \cdot 10^{+185}:\\ \;\;\;\;\cos \left(e^{\log \left(\pi \cdot angle\_m\right) - \log 180}\right) \cdot \left(2 \cdot \left(\left(\left(a + b\_m\right) \cdot \left(a - b\_m\right)\right) \cdot \sin \left(angle\_m \cdot \frac{\pi}{-180}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot {\left(a \cdot \sqrt{\pi \cdot angle\_m}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 2.1e+185)
    (*
     (cos (exp (- (log (* PI angle_m)) (log 180.0))))
     (* 2.0 (* (* (+ a b_m) (- a b_m)) (sin (* angle_m (/ PI -180.0))))))
    (* -0.011111111111111112 (pow (* a (sqrt (* PI angle_m))) 2.0)))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (a <= 2.1e+185) {
		tmp = cos(exp((log((((double) M_PI) * angle_m)) - log(180.0)))) * (2.0 * (((a + b_m) * (a - b_m)) * sin((angle_m * (((double) M_PI) / -180.0)))));
	} else {
		tmp = -0.011111111111111112 * pow((a * sqrt((((double) M_PI) * angle_m))), 2.0);
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (a <= 2.1e+185) {
		tmp = Math.cos(Math.exp((Math.log((Math.PI * angle_m)) - Math.log(180.0)))) * (2.0 * (((a + b_m) * (a - b_m)) * Math.sin((angle_m * (Math.PI / -180.0)))));
	} else {
		tmp = -0.011111111111111112 * Math.pow((a * Math.sqrt((Math.PI * angle_m))), 2.0);
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	tmp = 0
	if a <= 2.1e+185:
		tmp = math.cos(math.exp((math.log((math.pi * angle_m)) - math.log(180.0)))) * (2.0 * (((a + b_m) * (a - b_m)) * math.sin((angle_m * (math.pi / -180.0)))))
	else:
		tmp = -0.011111111111111112 * math.pow((a * math.sqrt((math.pi * angle_m))), 2.0)
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (a <= 2.1e+185)
		tmp = Float64(cos(exp(Float64(log(Float64(pi * angle_m)) - log(180.0)))) * Float64(2.0 * Float64(Float64(Float64(a + b_m) * Float64(a - b_m)) * sin(Float64(angle_m * Float64(pi / -180.0))))));
	else
		tmp = Float64(-0.011111111111111112 * (Float64(a * sqrt(Float64(pi * angle_m))) ^ 2.0));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	tmp = 0.0;
	if (a <= 2.1e+185)
		tmp = cos(exp((log((pi * angle_m)) - log(180.0)))) * (2.0 * (((a + b_m) * (a - b_m)) * sin((angle_m * (pi / -180.0)))));
	else
		tmp = -0.011111111111111112 * ((a * sqrt((pi * angle_m))) ^ 2.0);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 2.1e+185], N[(N[Cos[N[Exp[N[(N[Log[N[(Pi * angle$95$m), $MachinePrecision]], $MachinePrecision] - N[Log[180.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(N[(N[(a + b$95$m), $MachinePrecision] * N[(a - b$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[Power[N[(a * N[Sqrt[N[(Pi * angle$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 2.1e185

   1. Initial program 60.6%

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

   3. Simplified 60.6%

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

      1. add-exp-log 33.1%

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

      2. add-sqr-sqrt 33.1%

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

      3. sqrt-unprod 55.4%

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

      4. associate-*r/ 55.4%

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

      5. associate-*r/ 55.4%

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

      6. frac-times 55.0%

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

      7. *-commutative 55.0%

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

      8. *-commutative 55.0%

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

      9. metadata-eval 55.0%

         \[\leadsto \cos \left(e^{\log \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)}\right) \cdot \left(2 \cdot \left(\sin \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)\right) \]

      10. metadata-eval 55.0%

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

      11. frac-times 55.4%

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

      12. associate-*r/ 55.4%

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

      13. associate-*r/ 55.4%

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

      14. sqrt-unprod 30.6%

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

      15. add-sqr-sqrt 30.6%

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

      16. div-inv 30.6%

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

      17. metadata-eval 30.6%

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

   6. Applied egg-rr 30.6%

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

      1. unpow2 30.6%

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

      2. unpow2 30.6%

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

      3. difference-of-squares 31.4%

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

   8. Applied egg-rr 31.4%

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

      1. log-prod 30.9%

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

      2. metadata-eval 30.9%

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

      3. div-inv 30.9%

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

      4. sum-log 31.4%

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

      5. associate-*r/ 31.4%

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

      6. *-commutative 31.4%

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

      7. log-div 30.3%

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

      8. *-commutative 30.3%

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

   10. Applied egg-rr 30.3%

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

   if 2.1e185 < a

   1. Initial program 48.4%

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

      1. associate-*l* 48.4%

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

      2. associate-*l* 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in angle around 0 48.4%

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

   6. Taylor expanded in b around 0 64.2%

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

      1. add-sqr-sqrt 27.0%

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

      2. pow2 27.0%

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

      3. sqrt-prod 27.0%

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

      4. sqrt-pow1 36.6%

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

      5. metadata-eval 36.6%

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

      6. pow1 36.6%

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

      7. *-commutative 36.6%

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

   8. Applied egg-rr 36.6%

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

4. Final simplification 30.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.1 \cdot 10^{+185}:\\ \;\;\;\;\cos \left(e^{\log \left(\pi \cdot angle\right) - \log 180}\right) \cdot \left(2 \cdot \left(\left(\left(a + b\right) \cdot \left(a - b\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot {\left(a \cdot \sqrt{\pi \cdot angle}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 1.95 \cdot 10^{+185}:\\ \;\;\;\;\cos \left(e^{\log \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)}\right) \cdot \left(2 \cdot \left(\left(\left(a + b\_m\right) \cdot \left(a - b\_m\right)\right) \cdot \sin \left(angle\_m \cdot \frac{\pi}{-180}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot {\left(a \cdot \sqrt{\pi \cdot angle\_m}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 1.95e+185)
    (*
     (cos (exp (log (* PI (* angle_m 0.005555555555555556)))))
     (* 2.0 (* (* (+ a b_m) (- a b_m)) (sin (* angle_m (/ PI -180.0))))))
    (* -0.011111111111111112 (pow (* a (sqrt (* PI angle_m))) 2.0)))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (a <= 1.95e+185) {
		tmp = cos(exp(log((((double) M_PI) * (angle_m * 0.005555555555555556))))) * (2.0 * (((a + b_m) * (a - b_m)) * sin((angle_m * (((double) M_PI) / -180.0)))));
	} else {
		tmp = -0.011111111111111112 * pow((a * sqrt((((double) M_PI) * angle_m))), 2.0);
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (a <= 1.95e+185) {
		tmp = Math.cos(Math.exp(Math.log((Math.PI * (angle_m * 0.005555555555555556))))) * (2.0 * (((a + b_m) * (a - b_m)) * Math.sin((angle_m * (Math.PI / -180.0)))));
	} else {
		tmp = -0.011111111111111112 * Math.pow((a * Math.sqrt((Math.PI * angle_m))), 2.0);
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	tmp = 0
	if a <= 1.95e+185:
		tmp = math.cos(math.exp(math.log((math.pi * (angle_m * 0.005555555555555556))))) * (2.0 * (((a + b_m) * (a - b_m)) * math.sin((angle_m * (math.pi / -180.0)))))
	else:
		tmp = -0.011111111111111112 * math.pow((a * math.sqrt((math.pi * angle_m))), 2.0)
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (a <= 1.95e+185)
		tmp = Float64(cos(exp(log(Float64(pi * Float64(angle_m * 0.005555555555555556))))) * Float64(2.0 * Float64(Float64(Float64(a + b_m) * Float64(a - b_m)) * sin(Float64(angle_m * Float64(pi / -180.0))))));
	else
		tmp = Float64(-0.011111111111111112 * (Float64(a * sqrt(Float64(pi * angle_m))) ^ 2.0));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	tmp = 0.0;
	if (a <= 1.95e+185)
		tmp = cos(exp(log((pi * (angle_m * 0.005555555555555556))))) * (2.0 * (((a + b_m) * (a - b_m)) * sin((angle_m * (pi / -180.0)))));
	else
		tmp = -0.011111111111111112 * ((a * sqrt((pi * angle_m))) ^ 2.0);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 1.95e+185], N[(N[Cos[N[Exp[N[Log[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(N[(N[(a + b$95$m), $MachinePrecision] * N[(a - b$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[Power[N[(a * N[Sqrt[N[(Pi * angle$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 1.9499999999999999e185

   1. Initial program 60.6%

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

   3. Simplified 60.6%

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

      1. add-exp-log 33.1%

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

      2. add-sqr-sqrt 33.1%

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

      3. sqrt-unprod 55.4%

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

      4. associate-*r/ 55.4%

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

      5. associate-*r/ 55.4%

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

      6. frac-times 55.0%

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

      7. *-commutative 55.0%

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

      8. *-commutative 55.0%

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

      9. metadata-eval 55.0%

         \[\leadsto \cos \left(e^{\log \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)}\right) \cdot \left(2 \cdot \left(\sin \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)\right) \]

      10. metadata-eval 55.0%

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

      11. frac-times 55.4%

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

      12. associate-*r/ 55.4%

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

      13. associate-*r/ 55.4%

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

      14. sqrt-unprod 30.6%

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

      15. add-sqr-sqrt 30.6%

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

      16. div-inv 30.6%

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

      17. metadata-eval 30.6%

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

   6. Applied egg-rr 30.6%

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

      1. unpow2 30.6%

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

      2. unpow2 30.6%

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

      3. difference-of-squares 31.4%

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

   8. Applied egg-rr 31.4%

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

   if 1.9499999999999999e185 < a

   1. Initial program 48.4%

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

      1. associate-*l* 48.4%

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

      2. associate-*l* 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in angle around 0 48.4%

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

   6. Taylor expanded in b around 0 64.2%

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

      1. add-sqr-sqrt 27.0%

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

      2. pow2 27.0%

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

      3. sqrt-prod 27.0%

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

      4. sqrt-pow1 36.6%

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

      5. metadata-eval 36.6%

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

      6. pow1 36.6%

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

      7. *-commutative 36.6%

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

   8. Applied egg-rr 36.6%

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

4. Final simplification 31.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.95 \cdot 10^{+185}:\\ \;\;\;\;\cos \left(e^{\log \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \left(2 \cdot \left(\left(\left(a + b\right) \cdot \left(a - b\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot {\left(a \cdot \sqrt{\pi \cdot angle}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\left({b\_m}^{2} - {a}^{2}\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot {\left(a \cdot \sqrt{\pi \cdot angle\_m}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 1.4e+154)
    (*
     (- (pow b_m 2.0) (pow a 2.0))
     (sin (* 2.0 (* PI (* angle_m 0.005555555555555556)))))
    (* -0.011111111111111112 (pow (* a (sqrt (* PI angle_m))) 2.0)))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (a <= 1.4e+154) {
		tmp = (pow(b_m, 2.0) - pow(a, 2.0)) * sin((2.0 * (((double) M_PI) * (angle_m * 0.005555555555555556))));
	} else {
		tmp = -0.011111111111111112 * pow((a * sqrt((((double) M_PI) * angle_m))), 2.0);
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (a <= 1.4e+154) {
		tmp = (Math.pow(b_m, 2.0) - Math.pow(a, 2.0)) * Math.sin((2.0 * (Math.PI * (angle_m * 0.005555555555555556))));
	} else {
		tmp = -0.011111111111111112 * Math.pow((a * Math.sqrt((Math.PI * angle_m))), 2.0);
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	tmp = 0
	if a <= 1.4e+154:
		tmp = (math.pow(b_m, 2.0) - math.pow(a, 2.0)) * math.sin((2.0 * (math.pi * (angle_m * 0.005555555555555556))))
	else:
		tmp = -0.011111111111111112 * math.pow((a * math.sqrt((math.pi * angle_m))), 2.0)
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (a <= 1.4e+154)
		tmp = Float64(Float64((b_m ^ 2.0) - (a ^ 2.0)) * sin(Float64(2.0 * Float64(pi * Float64(angle_m * 0.005555555555555556)))));
	else
		tmp = Float64(-0.011111111111111112 * (Float64(a * sqrt(Float64(pi * angle_m))) ^ 2.0));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	tmp = 0.0;
	if (a <= 1.4e+154)
		tmp = ((b_m ^ 2.0) - (a ^ 2.0)) * sin((2.0 * (pi * (angle_m * 0.005555555555555556))));
	else
		tmp = -0.011111111111111112 * ((a * sqrt((pi * angle_m))) ^ 2.0);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 1.4e+154], N[(N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(2.0 * N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[Power[N[(a * N[Sqrt[N[(Pi * angle$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 1.4e154

   1. Initial program 61.6%

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

      1. associate-*l* 61.6%

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

      2. associate-*l* 61.6%

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

   3. Simplified 61.6%

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

      1. associate-*l* 61.6%

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

      2. associate-*r* 61.6%

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

      3. *-commutative 61.6%

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

      4. add-exp-log 31.1%

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

      5. associate-*r* 31.1%

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

      6. 2-sin 31.1%

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

   6. Applied egg-rr 32.1%

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

      1. rem-exp-log 62.7%

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

      2. *-commutative 62.7%

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

   8. Applied egg-rr 62.7%

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

   if 1.4e154 < a

   1. Initial program 40.8%

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

      1. associate-*l* 40.8%

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

      2. associate-*l* 40.8%

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

   3. Simplified 40.8%

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

   5. Taylor expanded in angle around 0 40.8%

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

   6. Taylor expanded in b around 0 53.8%

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

      1. add-sqr-sqrt 23.1%

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

      2. pow2 23.1%

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

      3. sqrt-prod 23.1%

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

      4. sqrt-pow1 38.9%

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

      5. metadata-eval 38.9%

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

      6. pow1 38.9%

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

      7. *-commutative 38.9%

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

   8. Applied egg-rr 38.9%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot {\left(a \cdot \sqrt{\pi \cdot angle}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot \left(\left(\left(a + b\_m\right) \cdot \left(a - b\_m\right)\right) \cdot \sin \left(angle\_m \cdot \frac{\pi}{-180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot {\left(a \cdot \sqrt{\pi \cdot angle\_m}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 1.4e+154)
    (* 2.0 (* (* (+ a b_m) (- a b_m)) (sin (* angle_m (/ PI -180.0)))))
    (* -0.011111111111111112 (pow (* a (sqrt (* PI angle_m))) 2.0)))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (a <= 1.4e+154) {
		tmp = 2.0 * (((a + b_m) * (a - b_m)) * sin((angle_m * (((double) M_PI) / -180.0))));
	} else {
		tmp = -0.011111111111111112 * pow((a * sqrt((((double) M_PI) * angle_m))), 2.0);
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (a <= 1.4e+154) {
		tmp = 2.0 * (((a + b_m) * (a - b_m)) * Math.sin((angle_m * (Math.PI / -180.0))));
	} else {
		tmp = -0.011111111111111112 * Math.pow((a * Math.sqrt((Math.PI * angle_m))), 2.0);
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	tmp = 0
	if a <= 1.4e+154:
		tmp = 2.0 * (((a + b_m) * (a - b_m)) * math.sin((angle_m * (math.pi / -180.0))))
	else:
		tmp = -0.011111111111111112 * math.pow((a * math.sqrt((math.pi * angle_m))), 2.0)
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (a <= 1.4e+154)
		tmp = Float64(2.0 * Float64(Float64(Float64(a + b_m) * Float64(a - b_m)) * sin(Float64(angle_m * Float64(pi / -180.0)))));
	else
		tmp = Float64(-0.011111111111111112 * (Float64(a * sqrt(Float64(pi * angle_m))) ^ 2.0));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	tmp = 0.0;
	if (a <= 1.4e+154)
		tmp = 2.0 * (((a + b_m) * (a - b_m)) * sin((angle_m * (pi / -180.0))));
	else
		tmp = -0.011111111111111112 * ((a * sqrt((pi * angle_m))) ^ 2.0);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 1.4e+154], N[(2.0 * N[(N[(N[(a + b$95$m), $MachinePrecision] * N[(a - b$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[Power[N[(a * N[Sqrt[N[(Pi * angle$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 1.4e154

   1. Initial program 61.6%

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

   3. Simplified 61.6%

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

      1. add-exp-log 33.7%

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

      2. add-sqr-sqrt 33.7%

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

      3. sqrt-unprod 55.9%

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

      4. associate-*r/ 55.9%

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

      5. associate-*r/ 55.9%

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

      6. frac-times 55.5%

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

      7. *-commutative 55.5%

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

      8. *-commutative 55.5%

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

      9. metadata-eval 55.5%

         \[\leadsto \cos \left(e^{\log \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)}\right) \cdot \left(2 \cdot \left(\sin \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)\right) \]

      10. metadata-eval 55.5%

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

      11. frac-times 55.9%

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

      12. associate-*r/ 55.9%

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

      13. associate-*r/ 55.9%

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

      14. sqrt-unprod 30.6%

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

      15. add-sqr-sqrt 30.6%

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

      16. div-inv 30.6%

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

      17. metadata-eval 30.6%

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

   6. Applied egg-rr 30.6%

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

      1. unpow2 30.6%

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

      2. unpow2 30.6%

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

      3. difference-of-squares 31.5%

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

   8. Applied egg-rr 31.5%

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

   9. Taylor expanded in angle around 0 61.7%

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

   if 1.4e154 < a

   1. Initial program 40.8%

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

      1. associate-*l* 40.8%

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

      2. associate-*l* 40.8%

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

   3. Simplified 40.8%

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

   5. Taylor expanded in angle around 0 40.8%

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

   6. Taylor expanded in b around 0 53.8%

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

      1. add-sqr-sqrt 23.1%

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

      2. pow2 23.1%

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

      3. sqrt-prod 23.1%

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

      4. sqrt-pow1 38.9%

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

      5. metadata-eval 38.9%

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

      6. pow1 38.9%

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

      7. *-commutative 38.9%

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

   8. Applied egg-rr 38.9%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot \left(\left(\left(a + b\right) \cdot \left(a - b\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot {\left(a \cdot \sqrt{\pi \cdot angle}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{+187}:\\ \;\;\;\;2 \cdot \left(\left(\left(a + b\_m\right) \cdot \left(a - b\_m\right)\right) \cdot \sin \left(angle\_m \cdot \frac{\pi}{-180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(\pi \cdot angle\_m\right) \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 5e+187)
    (* 2.0 (* (* (+ a b_m) (- a b_m)) (sin (* angle_m (/ PI -180.0)))))
    (* -0.011111111111111112 (* (* PI angle_m) (* a a))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (a <= 5e+187) {
		tmp = 2.0 * (((a + b_m) * (a - b_m)) * sin((angle_m * (((double) M_PI) / -180.0))));
	} else {
		tmp = -0.011111111111111112 * ((((double) M_PI) * angle_m) * (a * a));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (a <= 5e+187) {
		tmp = 2.0 * (((a + b_m) * (a - b_m)) * Math.sin((angle_m * (Math.PI / -180.0))));
	} else {
		tmp = -0.011111111111111112 * ((Math.PI * angle_m) * (a * a));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	tmp = 0
	if a <= 5e+187:
		tmp = 2.0 * (((a + b_m) * (a - b_m)) * math.sin((angle_m * (math.pi / -180.0))))
	else:
		tmp = -0.011111111111111112 * ((math.pi * angle_m) * (a * a))
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (a <= 5e+187)
		tmp = Float64(2.0 * Float64(Float64(Float64(a + b_m) * Float64(a - b_m)) * sin(Float64(angle_m * Float64(pi / -180.0)))));
	else
		tmp = Float64(-0.011111111111111112 * Float64(Float64(pi * angle_m) * Float64(a * a)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	tmp = 0.0;
	if (a <= 5e+187)
		tmp = 2.0 * (((a + b_m) * (a - b_m)) * sin((angle_m * (pi / -180.0))));
	else
		tmp = -0.011111111111111112 * ((pi * angle_m) * (a * a));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 5e+187], N[(2.0 * N[(N[(N[(a + b$95$m), $MachinePrecision] * N[(a - b$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 5.0000000000000001e187

   1. Initial program 60.1%

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

   3. Simplified 60.1%

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

      1. add-exp-log 32.8%

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

      2. add-sqr-sqrt 32.8%

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

      3. sqrt-unprod 55.0%

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

      4. associate-*r/ 55.0%

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

      5. associate-*r/ 55.0%

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

      6. frac-times 54.6%

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

      7. *-commutative 54.6%

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

      8. *-commutative 54.6%

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

      9. metadata-eval 54.6%

         \[\leadsto \cos \left(e^{\log \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)}\right) \cdot \left(2 \cdot \left(\sin \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)\right) \]

      10. metadata-eval 54.6%

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

      11. frac-times 55.0%

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

      12. associate-*r/ 55.0%

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

      13. associate-*r/ 55.0%

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

      14. sqrt-unprod 30.4%

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

      15. add-sqr-sqrt 30.4%

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

      16. div-inv 30.4%

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

      17. metadata-eval 30.4%

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

   6. Applied egg-rr 30.4%

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

      1. unpow2 30.4%

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

      2. unpow2 30.4%

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

      3. difference-of-squares 31.2%

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

   8. Applied egg-rr 31.2%

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

   9. Taylor expanded in angle around 0 60.7%

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

   if 5.0000000000000001e187 < a

   1. Initial program 53.7%

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

      1. associate-*l* 53.7%

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

      2. associate-*l* 53.7%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in angle around 0 53.7%

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

   6. Taylor expanded in b around 0 71.3%

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

      1. unpow2 71.3%

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

   8. Applied egg-rr 71.3%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{+187}:\\ \;\;\;\;2 \cdot \left(\left(\left(a + b\right) \cdot \left(a - b\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(\pi \cdot angle\right) \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.9% accurate, 32.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(-0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(\left(a + b\_m\right) \cdot \left(a - b\_m\right)\right)\right)\right)\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (* -0.011111111111111112 (* angle_m (* PI (* (+ a b_m) (- a b_m)))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	return angle_s * (-0.011111111111111112 * (angle_m * (((double) M_PI) * ((a + b_m) * (a - b_m)))));
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	return angle_s * (-0.011111111111111112 * (angle_m * (Math.PI * ((a + b_m) * (a - b_m)))));
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	return angle_s * (-0.011111111111111112 * (angle_m * (math.pi * ((a + b_m) * (a - b_m)))))

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	return Float64(angle_s * Float64(-0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(Float64(a + b_m) * Float64(a - b_m))))))
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b_m, angle_m)
	tmp = angle_s * (-0.011111111111111112 * (angle_m * (pi * ((a + b_m) * (a - b_m)))));
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * N[(-0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(N[(a + b$95$m), $MachinePrecision] * N[(a - b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.7%

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

3. Simplified 60.1%

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

   1. add-exp-log 33.0%

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

   2. add-sqr-sqrt 33.0%

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

   3. sqrt-unprod 54.1%

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

   4. associate-*r/ 54.1%

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

   5. associate-*r/ 54.1%

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

   6. frac-times 53.8%

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

   7. *-commutative 53.8%

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

   8. *-commutative 53.8%

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

   9. metadata-eval 53.8%

      \[\leadsto \cos \left(e^{\log \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)}\right) \cdot \left(2 \cdot \left(\sin \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)\right) \]

   10. metadata-eval 53.8%

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

   11. frac-times 54.1%

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

   12. associate-*r/ 54.1%

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

   13. associate-*r/ 54.1%

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

   14. sqrt-unprod 29.5%

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

   15. add-sqr-sqrt 29.5%

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

   16. div-inv 29.5%

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

   17. metadata-eval 29.5%

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

6. Applied egg-rr 29.5%

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

   1. unpow2 29.5%

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

   2. unpow2 29.5%

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

   3. difference-of-squares 30.7%

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

8. Applied egg-rr 30.7%

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

9. Taylor expanded in angle around 0 60.1%

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

Alternative 11: 35.7% accurate, 46.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(-0.011111111111111112 \cdot \left(\left(\pi \cdot angle\_m\right) \cdot \left(a \cdot a\right)\right)\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (* angle_s (* -0.011111111111111112 (* (* PI angle_m) (* a a)))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	return angle_s * (-0.011111111111111112 * ((((double) M_PI) * angle_m) * (a * a)));
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	return angle_s * (-0.011111111111111112 * ((Math.PI * angle_m) * (a * a)));
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	return angle_s * (-0.011111111111111112 * ((math.pi * angle_m) * (a * a)))

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	return Float64(angle_s * Float64(-0.011111111111111112 * Float64(Float64(pi * angle_m) * Float64(a * a))))
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b_m, angle_m)
	tmp = angle_s * (-0.011111111111111112 * ((pi * angle_m) * (a * a)));
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * N[(-0.011111111111111112 * N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. associate-*l* 59.7%

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

   2. associate-*l* 59.7%

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

3. Simplified 59.7%

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

5. Taylor expanded in angle around 0 57.8%

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

6. Taylor expanded in b around 0 37.5%

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

   1. unpow2 37.5%

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

8. Applied egg-rr 37.5%

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

9. Final simplification 37.5%

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

Reproduce

herbie shell --seed 2024188 
(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)))))