ab-angle->ABCF B

Percentage Accurate: 54.4% → 67.2%

Time: 39.1s

Alternatives: 17

Speedup: 5.4×

• 32.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ t_1 := \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t_0\right) \cdot \cos t_0\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{+141}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(angle \cdot \frac{\pi}{180}\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;2 \cdot \left(\left(\left(b + a\right) \cdot \sin \left(\left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right)\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \left(\sqrt{\pi} \cdot \sqrt{\pi}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0)))
        (t_1 (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))))
   (if (<= t_1 2e+141)
     (*
      2.0
      (*
       (* (sin (* angle (/ PI 180.0))) (+ b a))
       (* (- b a) (cos (/ PI (/ 180.0 angle))))))
     (if (<= t_1 INFINITY)
       (*
        2.0
        (*
         (*
          (+ b a)
          (sin
           (*
            (* (cbrt PI) (* (cbrt PI) (cbrt PI)))
            (* angle 0.005555555555555556))))
         (* (- b a) (cos (* PI (* angle 0.005555555555555556))))))
       (*
        2.0
        (*
         (- b a)
         (*
          (+ b a)
          (sin
           (* (* angle 0.005555555555555556) (* (sqrt PI) (sqrt PI)))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	double t_1 = ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0);
	double tmp;
	if (t_1 <= 2e+141) {
		tmp = 2.0 * ((sin((angle * (((double) M_PI) / 180.0))) * (b + a)) * ((b - a) * cos((((double) M_PI) / (180.0 / angle)))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 2.0 * (((b + a) * sin(((cbrt(((double) M_PI)) * (cbrt(((double) M_PI)) * cbrt(((double) M_PI)))) * (angle * 0.005555555555555556)))) * ((b - a) * cos((((double) M_PI) * (angle * 0.005555555555555556)))));
	} else {
		tmp = 2.0 * ((b - a) * ((b + a) * sin(((angle * 0.005555555555555556) * (sqrt(((double) M_PI)) * sqrt(((double) M_PI)))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	double t_1 = ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_0)) * Math.cos(t_0);
	double tmp;
	if (t_1 <= 2e+141) {
		tmp = 2.0 * ((Math.sin((angle * (Math.PI / 180.0))) * (b + a)) * ((b - a) * Math.cos((Math.PI / (180.0 / angle)))));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 2.0 * (((b + a) * Math.sin(((Math.cbrt(Math.PI) * (Math.cbrt(Math.PI) * Math.cbrt(Math.PI))) * (angle * 0.005555555555555556)))) * ((b - a) * Math.cos((Math.PI * (angle * 0.005555555555555556)))));
	} else {
		tmp = 2.0 * ((b - a) * ((b + a) * Math.sin(((angle * 0.005555555555555556) * (Math.sqrt(Math.PI) * Math.sqrt(Math.PI))))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	t_1 = Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0))
	tmp = 0.0
	if (t_1 <= 2e+141)
		tmp = Float64(2.0 * Float64(Float64(sin(Float64(angle * Float64(pi / 180.0))) * Float64(b + a)) * Float64(Float64(b - a) * cos(Float64(pi / Float64(180.0 / angle))))));
	elseif (t_1 <= Inf)
		tmp = Float64(2.0 * Float64(Float64(Float64(b + a) * sin(Float64(Float64(cbrt(pi) * Float64(cbrt(pi) * cbrt(pi))) * Float64(angle * 0.005555555555555556)))) * Float64(Float64(b - a) * cos(Float64(pi * Float64(angle * 0.005555555555555556))))));
	else
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(Float64(angle * 0.005555555555555556) * Float64(sqrt(pi) * sqrt(pi)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, 2e+141], N[(2.0 * N[(N[(N[Sin[N[(angle * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Cos[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(2.0 * N[(N[(N[(b + a), $MachinePrecision] * N[Sin[N[(N[(N[Power[Pi, 1/3], $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[Pi, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Cos[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(N[(angle * 0.005555555555555556), $MachinePrecision] * N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) < 2.00000000000000003e141

   1. Initial program 59.3%

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

      1. associate-*l* 59.3%

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

      2. unpow2 59.3%

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

      3. unpow2 59.3%

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

      4. difference-of-squares 59.3%

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

   3. Simplified 59.3%

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

   4. Taylor expanded in angle around inf 65.1%

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

      1. associate-*r* 65.1%

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

      2. *-commutative 65.1%

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

      3. *-commutative 65.1%

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

      4. *-commutative 65.1%

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

      5. associate-*r* 67.0%

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

      6. +-commutative 67.0%

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

      7. *-commutative 67.0%

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

      8. *-commutative 67.0%

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

      9. associate-*r* 67.6%

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

   6. Simplified 67.6%

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

      1. metadata-eval 67.6%

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

      2. div-inv 67.0%

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

      3. associate-*r/ 65.8%

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

   8. Applied egg-rr 65.8%

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

      1. associate-/l* 68.0%

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

   10. Simplified 68.0%

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

       1. metadata-eval 67.6%

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

       2. div-inv 67.0%

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

       3. associate-*r/ 65.8%

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

   12. Applied egg-rr 66.2%

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

       1. associate-/l* 67.2%

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

       2. associate-/r/ 69.5%

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

   14. Simplified 69.5%

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

   if 2.00000000000000003e141 < (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) < +inf.0

   1. Initial program 53.8%

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

      1. associate-*l* 53.8%

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

      2. unpow2 53.8%

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

      3. unpow2 53.8%

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

      4. difference-of-squares 53.8%

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

   3. Simplified 53.8%

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

   4. Taylor expanded in angle around inf 72.2%

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

      1. associate-*r* 72.2%

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

      2. *-commutative 72.2%

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

      3. *-commutative 72.2%

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

      4. *-commutative 72.2%

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

      5. associate-*r* 71.8%

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

      6. +-commutative 71.8%

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

      7. *-commutative 71.8%

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

      8. *-commutative 71.8%

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

      9. associate-*r* 71.8%

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

   6. Simplified 71.8%

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

      1. add-cube-cbrt 79.3%

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

   8. Applied egg-rr 79.3%

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

   if +inf.0 < (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))

   1. Initial program 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. unpow2 0.0%

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

      4. difference-of-squares 48.1%

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

   3. Simplified 48.1%

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

   4. Taylor expanded in angle around inf 66.6%

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

      1. associate-*r* 66.6%

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

      2. *-commutative 66.6%

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

      3. *-commutative 66.6%

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

      4. *-commutative 66.6%

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

      5. associate-*r* 73.1%

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

      6. +-commutative 73.1%

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

      7. *-commutative 73.1%

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

      8. *-commutative 73.1%

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

      9. associate-*r* 66.5%

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

   6. Simplified 66.5%

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

      1. add-sqr-sqrt 79.7%

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

   8. Applied egg-rr 79.7%

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

   9. Taylor expanded in angle around 0 99.7%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \leq 2 \cdot 10^{+141}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(angle \cdot \frac{\pi}{180}\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)\\ \mathbf{elif}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \leq \infty:\\ \;\;\;\;2 \cdot \left(\left(\left(b + a\right) \cdot \sin \left(\left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right)\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \left(\sqrt{\pi} \cdot \sqrt{\pi}\right)\right)\right)\right)\\ \end{array} \]

Alternative 2: 67.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(angle \cdot \frac{\pi}{180}\right) \cdot \left(b + a\right)\\ t_1 := \pi \cdot \frac{angle}{180}\\ \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t_1\right) \cdot \cos t_1 \leq 5 \cdot 10^{+282}:\\ \;\;\;\;2 \cdot \left(t_0 \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_0 \cdot \left(b - a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (sin (* angle (/ PI 180.0))) (+ b a)))
        (t_1 (* PI (/ angle 180.0))))
   (if (<=
        (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_1)) (cos t_1))
        5e+282)
     (* 2.0 (* t_0 (* (- b a) (cos (/ PI (/ 180.0 angle))))))
     (* 2.0 (* t_0 (- b a))))))

C

double code(double a, double b, double angle) {
	double t_0 = sin((angle * (((double) M_PI) / 180.0))) * (b + a);
	double t_1 = ((double) M_PI) * (angle / 180.0);
	double tmp;
	if ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_1)) * cos(t_1)) <= 5e+282) {
		tmp = 2.0 * (t_0 * ((b - a) * cos((((double) M_PI) / (180.0 / angle)))));
	} else {
		tmp = 2.0 * (t_0 * (b - a));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.sin((angle * (Math.PI / 180.0))) * (b + a);
	double t_1 = Math.PI * (angle / 180.0);
	double tmp;
	if ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_1)) * Math.cos(t_1)) <= 5e+282) {
		tmp = 2.0 * (t_0 * ((b - a) * Math.cos((Math.PI / (180.0 / angle)))));
	} else {
		tmp = 2.0 * (t_0 * (b - a));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.sin((angle * (math.pi / 180.0))) * (b + a)
	t_1 = math.pi * (angle / 180.0)
	tmp = 0
	if (((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin(t_1)) * math.cos(t_1)) <= 5e+282:
		tmp = 2.0 * (t_0 * ((b - a) * math.cos((math.pi / (180.0 / angle)))))
	else:
		tmp = 2.0 * (t_0 * (b - a))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(sin(Float64(angle * Float64(pi / 180.0))) * Float64(b + a))
	t_1 = Float64(pi * Float64(angle / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_1)) * cos(t_1)) <= 5e+282)
		tmp = Float64(2.0 * Float64(t_0 * Float64(Float64(b - a) * cos(Float64(pi / Float64(180.0 / angle))))));
	else
		tmp = Float64(2.0 * Float64(t_0 * Float64(b - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = sin((angle * (pi / 180.0))) * (b + a);
	t_1 = pi * (angle / 180.0);
	tmp = 0.0;
	if ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(t_1)) * cos(t_1)) <= 5e+282)
		tmp = 2.0 * (t_0 * ((b - a) * cos((pi / (180.0 / angle)))));
	else
		tmp = 2.0 * (t_0 * (b - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Sin[N[(angle * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision], 5e+282], N[(2.0 * N[(t$95$0 * N[(N[(b - a), $MachinePrecision] * N[Cos[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$0 * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) < 4.99999999999999978e282

   1. Initial program 57.7%

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

      1. associate-*l* 57.7%

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

      2. unpow2 57.7%

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

      3. unpow2 57.7%

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

      4. difference-of-squares 57.7%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in angle around inf 63.3%

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

      1. associate-*r* 63.3%

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

      2. *-commutative 63.3%

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

      3. *-commutative 63.3%

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

      4. *-commutative 63.3%

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

      5. associate-*r* 65.0%

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

      6. +-commutative 65.0%

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

      7. *-commutative 65.0%

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

      8. *-commutative 65.0%

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

      9. associate-*r* 65.5%

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

   6. Simplified 65.5%

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

      1. metadata-eval 65.5%

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

      2. div-inv 65.0%

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

      3. associate-*r/ 63.9%

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

   8. Applied egg-rr 63.9%

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

      1. associate-/l* 65.9%

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

   10. Simplified 65.9%

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

       1. metadata-eval 65.5%

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

       2. div-inv 65.0%

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

       3. associate-*r/ 63.9%

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

   12. Applied egg-rr 64.3%

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

       1. associate-/l* 65.4%

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

       2. associate-/r/ 67.4%

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

   14. Simplified 67.4%

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

   if 4.99999999999999978e282 < (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))

   1. Initial program 44.4%

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

      1. associate-*l* 44.4%

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

      2. unpow2 44.4%

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

      3. unpow2 44.4%

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

      4. difference-of-squares 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in angle around inf 78.4%

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

      1. associate-*r* 78.4%

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

      2. *-commutative 78.4%

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

      3. *-commutative 78.4%

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

      4. *-commutative 78.4%

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

      5. associate-*r* 80.1%

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

      6. +-commutative 80.1%

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

      7. *-commutative 80.1%

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

      8. *-commutative 80.1%

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

      9. associate-*r* 78.3%

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

   6. Simplified 78.3%

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

      1. metadata-eval 78.3%

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

      2. div-inv 78.3%

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

      3. associate-*r/ 76.5%

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

   8. Applied egg-rr 76.5%

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

      1. associate-/l* 78.3%

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

   10. Simplified 78.3%

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

       1. metadata-eval 78.3%

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

       2. div-inv 78.3%

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

       3. associate-*r/ 76.5%

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

   12. Applied egg-rr 71.3%

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

       1. associate-/l* 80.0%

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

       2. associate-/r/ 74.8%

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

   14. Simplified 74.8%

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

   15. Taylor expanded in angle around 0 83.7%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \leq 5 \cdot 10^{+282}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(angle \cdot \frac{\pi}{180}\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(angle \cdot \frac{\pi}{180}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)\\ \end{array} \]

Alternative 3: 67.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_1 := \left(b + a\right) \cdot \sin t_0\\ t_2 := {\left(\sqrt[3]{t_0}\right)}^{3}\\ \mathbf{if}\;a \leq -5.7 \cdot 10^{+202}:\\ \;\;\;\;2 \cdot \left(\left(\left(b + a\right) \cdot \sin t_2\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{-225}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \left(\left(b - a\right) \cdot \cos t_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* angle 0.005555555555555556)))
        (t_1 (* (+ b a) (sin t_0)))
        (t_2 (pow (cbrt t_0) 3.0)))
   (if (<= a -5.7e+202)
     (* 2.0 (* (* (+ b a) (sin t_2)) (* (- b a) (cos (/ (* PI angle) 180.0)))))
     (if (<= a 2.3e-225)
       (* 2.0 (* (* (- b a) (cos (/ PI (/ 180.0 angle)))) t_1))
       (* 2.0 (* t_1 (* (- b a) (cos t_2))))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle * 0.005555555555555556);
	double t_1 = (b + a) * sin(t_0);
	double t_2 = pow(cbrt(t_0), 3.0);
	double tmp;
	if (a <= -5.7e+202) {
		tmp = 2.0 * (((b + a) * sin(t_2)) * ((b - a) * cos(((((double) M_PI) * angle) / 180.0))));
	} else if (a <= 2.3e-225) {
		tmp = 2.0 * (((b - a) * cos((((double) M_PI) / (180.0 / angle)))) * t_1);
	} else {
		tmp = 2.0 * (t_1 * ((b - a) * cos(t_2)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle * 0.005555555555555556);
	double t_1 = (b + a) * Math.sin(t_0);
	double t_2 = Math.pow(Math.cbrt(t_0), 3.0);
	double tmp;
	if (a <= -5.7e+202) {
		tmp = 2.0 * (((b + a) * Math.sin(t_2)) * ((b - a) * Math.cos(((Math.PI * angle) / 180.0))));
	} else if (a <= 2.3e-225) {
		tmp = 2.0 * (((b - a) * Math.cos((Math.PI / (180.0 / angle)))) * t_1);
	} else {
		tmp = 2.0 * (t_1 * ((b - a) * Math.cos(t_2)));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle * 0.005555555555555556))
	t_1 = Float64(Float64(b + a) * sin(t_0))
	t_2 = cbrt(t_0) ^ 3.0
	tmp = 0.0
	if (a <= -5.7e+202)
		tmp = Float64(2.0 * Float64(Float64(Float64(b + a) * sin(t_2)) * Float64(Float64(b - a) * cos(Float64(Float64(pi * angle) / 180.0)))));
	elseif (a <= 2.3e-225)
		tmp = Float64(2.0 * Float64(Float64(Float64(b - a) * cos(Float64(pi / Float64(180.0 / angle)))) * t_1));
	else
		tmp = Float64(2.0 * Float64(t_1 * Float64(Float64(b - a) * cos(t_2))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b + a), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]}, If[LessEqual[a, -5.7e+202], N[(2.0 * N[(N[(N[(b + a), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Cos[N[(N[(Pi * angle), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.3e-225], N[(2.0 * N[(N[(N[(b - a), $MachinePrecision] * N[Cos[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$1 * N[(N[(b - a), $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.6999999999999997e202

   1. Initial program 54.2%

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

      1. associate-*l* 54.2%

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

      2. unpow2 54.2%

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

      3. unpow2 54.2%

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

      4. difference-of-squares 62.2%

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

   3. Simplified 62.2%

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

   4. Taylor expanded in angle around inf 73.0%

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

      1. associate-*r* 73.0%

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

      2. *-commutative 73.0%

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

      3. *-commutative 73.0%

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

      4. *-commutative 73.0%

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

      5. associate-*r* 76.9%

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

      6. +-commutative 76.9%

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

      7. *-commutative 76.9%

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

      8. *-commutative 76.9%

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

      9. associate-*r* 69.2%

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

   6. Simplified 69.2%

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

      1. metadata-eval 69.2%

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

      2. div-inv 69.2%

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

      3. associate-*r/ 69.2%

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

   8. Applied egg-rr 69.2%

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

      1. expm1-log1p-u 53.8%

         \[\leadsto 2 \cdot \left(\left(\sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right) \]

   10. Applied egg-rr 53.8%

       \[\leadsto 2 \cdot \left(\left(\sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right) \]
   11. Step-by-step derivation

       1. add-cube-cbrt 49.8%

          \[\leadsto 2 \cdot \left(\left(\sin \color{blue}{\left(\left(\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right) \cdot \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)} \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right) \]

       2. pow3 53.7%

          \[\leadsto 2 \cdot \left(\left(\sin \color{blue}{\left({\left(\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3}\right)} \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right) \]

       3. expm1-log1p-u 88.3%

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

   12. Applied egg-rr 88.3%

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

   if -5.6999999999999997e202 < a < 2.2999999999999999e-225

   1. Initial program 59.0%

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

      1. associate-*l* 59.0%

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

      2. unpow2 59.0%

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

      3. unpow2 59.0%

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

      4. difference-of-squares 59.8%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in angle around inf 65.9%

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

      1. associate-*r* 65.9%

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

      2. *-commutative 65.9%

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

      3. *-commutative 65.9%

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

      4. *-commutative 65.9%

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

      5. associate-*r* 67.5%

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

      6. +-commutative 67.5%

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

      7. *-commutative 67.5%

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

      8. *-commutative 67.5%

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

      9. associate-*r* 69.3%

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

   6. Simplified 69.3%

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

      1. metadata-eval 69.3%

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

      2. div-inv 69.8%

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

      3. associate-*r/ 67.2%

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

   8. Applied egg-rr 67.2%

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

      1. associate-/l* 70.6%

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

   10. Simplified 70.6%

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

   if 2.2999999999999999e-225 < a

   1. Initial program 50.1%

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

      1. associate-*l* 50.1%

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

      2. unpow2 50.1%

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

      3. unpow2 50.1%

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

      4. difference-of-squares 54.0%

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

   3. Simplified 54.0%

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

   4. Taylor expanded in angle around inf 65.8%

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

      1. associate-*r* 65.8%

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

      2. *-commutative 65.8%

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

      3. *-commutative 65.8%

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

      4. *-commutative 65.8%

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

      5. associate-*r* 67.1%

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

      6. +-commutative 67.1%

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

      7. *-commutative 67.1%

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

      8. *-commutative 67.1%

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

      9. associate-*r* 66.9%

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

   6. Simplified 66.9%

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

      1. metadata-eval 66.9%

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

      2. div-inv 65.5%

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

      3. associate-*r/ 65.5%

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

   8. Applied egg-rr 65.5%

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

      1. associate-/l* 67.1%

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

   10. Simplified 67.1%

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

       1. associate-/l* 65.5%

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

       2. div-inv 67.1%

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

       3. metadata-eval 67.1%

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

       4. associate-*r* 66.9%

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

       5. add-cube-cbrt 68.5%

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

       6. pow3 71.4%

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

   12. Applied egg-rr 71.4%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.7 \cdot 10^{+202}:\\ \;\;\;\;2 \cdot \left(\left(\left(b + a\right) \cdot \sin \left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{-225}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}\right)\right)\right)\\ \end{array} \]

Alternative 4: 66.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 5 \cdot 10^{-245}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(angle \cdot \frac{\pi}{180}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (pow b 2.0) 5e-245)
   (*
    2.0
    (*
     (* (- b a) (cos (* PI (* angle 0.005555555555555556))))
     (* (+ b a) (sin (* 0.005555555555555556 (* PI angle))))))
   (* 2.0 (* (* (sin (* angle (/ PI 180.0))) (+ b a)) (- b a)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (pow(b, 2.0) <= 5e-245) {
		tmp = 2.0 * (((b - a) * cos((((double) M_PI) * (angle * 0.005555555555555556)))) * ((b + a) * sin((0.005555555555555556 * (((double) M_PI) * angle)))));
	} else {
		tmp = 2.0 * ((sin((angle * (((double) M_PI) / 180.0))) * (b + a)) * (b - a));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.pow(b, 2.0) <= 5e-245) {
		tmp = 2.0 * (((b - a) * Math.cos((Math.PI * (angle * 0.005555555555555556)))) * ((b + a) * Math.sin((0.005555555555555556 * (Math.PI * angle)))));
	} else {
		tmp = 2.0 * ((Math.sin((angle * (Math.PI / 180.0))) * (b + a)) * (b - a));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.pow(b, 2.0) <= 5e-245:
		tmp = 2.0 * (((b - a) * math.cos((math.pi * (angle * 0.005555555555555556)))) * ((b + a) * math.sin((0.005555555555555556 * (math.pi * angle)))))
	else:
		tmp = 2.0 * ((math.sin((angle * (math.pi / 180.0))) * (b + a)) * (b - a))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if ((b ^ 2.0) <= 5e-245)
		tmp = Float64(2.0 * Float64(Float64(Float64(b - a) * cos(Float64(pi * Float64(angle * 0.005555555555555556)))) * Float64(Float64(b + a) * sin(Float64(0.005555555555555556 * Float64(pi * angle))))));
	else
		tmp = Float64(2.0 * Float64(Float64(sin(Float64(angle * Float64(pi / 180.0))) * Float64(b + a)) * Float64(b - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((b ^ 2.0) <= 5e-245)
		tmp = 2.0 * (((b - a) * cos((pi * (angle * 0.005555555555555556)))) * ((b + a) * sin((0.005555555555555556 * (pi * angle)))));
	else
		tmp = 2.0 * ((sin((angle * (pi / 180.0))) * (b + a)) * (b - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[Power[b, 2.0], $MachinePrecision], 5e-245], N[(2.0 * N[(N[(N[(b - a), $MachinePrecision] * N[Cos[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Sin[N[(angle * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b 2) < 4.9999999999999997e-245

   1. Initial program 66.9%

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

      1. associate-*l* 66.9%

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

      2. unpow2 66.9%

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

      3. unpow2 66.9%

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

      4. difference-of-squares 66.9%

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

   3. Simplified 66.9%

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

   4. Taylor expanded in angle around inf 69.7%

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

      1. associate-*r* 69.7%

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

      2. *-commutative 69.7%

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

      3. *-commutative 69.7%

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

      4. *-commutative 69.7%

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

      5. associate-*r* 70.7%

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

      6. +-commutative 70.7%

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

      7. *-commutative 70.7%

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

      8. *-commutative 70.7%

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

      9. associate-*r* 71.9%

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

   6. Simplified 71.9%

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

   7. Taylor expanded in angle around inf 73.0%

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

   if 4.9999999999999997e-245 < (pow.f64 b 2)

   1. Initial program 48.1%

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

      1. associate-*l* 48.1%

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

      2. unpow2 48.1%

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

      3. unpow2 48.1%

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

      4. difference-of-squares 52.5%

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

   3. Simplified 52.5%

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

   4. Taylor expanded in angle around inf 64.9%

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

      1. associate-*r* 64.9%

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

      2. *-commutative 64.9%

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

      3. *-commutative 64.9%

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

      4. *-commutative 64.9%

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

      5. associate-*r* 67.0%

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

      6. +-commutative 67.0%

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

      7. *-commutative 67.0%

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

      8. *-commutative 67.0%

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

      9. associate-*r* 66.3%

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

   6. Simplified 66.3%

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

      1. metadata-eval 66.3%

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

      2. div-inv 65.2%

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

      3. associate-*r/ 64.6%

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

   8. Applied egg-rr 64.6%

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

      1. associate-/l* 67.4%

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

   10. Simplified 67.4%

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

       1. metadata-eval 66.3%

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

       2. div-inv 65.2%

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

       3. associate-*r/ 64.6%

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

   12. Applied egg-rr 63.4%

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

       1. associate-/l* 67.3%

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

       2. associate-/r/ 68.2%

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

   14. Simplified 68.2%

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

   15. Taylor expanded in angle around 0 67.7%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 5 \cdot 10^{-245}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(angle \cdot \frac{\pi}{180}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)\\ \end{array} \]

Alternative 5: 67.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ 2 \cdot \left(\left(\left(b - a\right) \cdot \cos t_0\right) \cdot \left(\left(b + a\right) \cdot \sin t_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* angle 0.005555555555555556))))
   (* 2.0 (* (* (- b a) (cos t_0)) (* (+ b a) (sin t_0))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle * 0.005555555555555556);
	return 2.0 * (((b - a) * cos(t_0)) * ((b + a) * sin(t_0)));
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle * 0.005555555555555556);
	return 2.0 * (((b - a) * Math.cos(t_0)) * ((b + a) * Math.sin(t_0)));
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle * 0.005555555555555556)
	return 2.0 * (((b - a) * math.cos(t_0)) * ((b + a) * math.sin(t_0)))

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle * 0.005555555555555556))
	return Float64(2.0 * Float64(Float64(Float64(b - a) * cos(t_0)) * Float64(Float64(b + a) * sin(t_0))))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = pi * (angle * 0.005555555555555556);
	tmp = 2.0 * (((b - a) * cos(t_0)) * ((b + a) * sin(t_0)));
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[(2.0 * N[(N[(N[(b - a), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 54.8%

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

   1. associate-*l* 54.8%

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

   2. unpow2 54.8%

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

   3. unpow2 54.8%

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

   4. difference-of-squares 57.6%

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

3. Simplified 57.6%

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

4. Taylor expanded in angle around inf 66.6%

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

   1. associate-*r* 66.6%

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

   2. *-commutative 66.6%

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

   3. *-commutative 66.6%

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

   4. *-commutative 66.6%

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

   5. associate-*r* 68.3%

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

   6. +-commutative 68.3%

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

   7. *-commutative 68.3%

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

   8. *-commutative 68.3%

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

   9. associate-*r* 68.3%

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

6. Simplified 68.3%

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

7. Final simplification 68.3%

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

Alternative 6: 67.6% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (*
  2.0
  (*
   (* (- b a) (cos (/ PI (/ 180.0 angle))))
   (* (+ b a) (sin (* PI (* angle 0.005555555555555556)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[(2.0 * N[(N[(N[(b - a), $MachinePrecision] * N[Cos[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.8%

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

   1. associate-*l* 54.8%

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

   2. unpow2 54.8%

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

   3. unpow2 54.8%

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

   4. difference-of-squares 57.6%

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

3. Simplified 57.6%

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

4. Taylor expanded in angle around inf 66.6%

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

   1. associate-*r* 66.6%

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

   2. *-commutative 66.6%

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

   3. *-commutative 66.6%

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

   4. *-commutative 66.6%

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

   5. associate-*r* 68.3%

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

   6. +-commutative 68.3%

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

   7. *-commutative 68.3%

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

   8. *-commutative 68.3%

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

   9. associate-*r* 68.3%

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

6. Simplified 68.3%

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

   1. metadata-eval 68.3%

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

   2. div-inv 67.9%

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

   3. associate-*r/ 66.7%

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

8. Applied egg-rr 66.7%

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

   1. associate-/l* 68.6%

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

10. Simplified 68.6%

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

11. Final simplification 68.6%

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

Alternative 7: 66.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -1:\\ \;\;\;\;\left(\sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 0.5\right) \cdot \left(2 \cdot {\left(b + a\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(angle \cdot \frac{\pi}{180}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) -1.0)
   (*
    (* (sin (* 2.0 (* PI (* angle 0.005555555555555556)))) 0.5)
    (* 2.0 (pow (+ b a) 2.0)))
   (* 2.0 (* (* (sin (* angle (/ PI 180.0))) (+ b a)) (- b a)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= -1.0) {
		tmp = (sin((2.0 * (((double) M_PI) * (angle * 0.005555555555555556)))) * 0.5) * (2.0 * pow((b + a), 2.0));
	} else {
		tmp = 2.0 * ((sin((angle * (((double) M_PI) / 180.0))) * (b + a)) * (b - a));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= -1.0) {
		tmp = (Math.sin((2.0 * (Math.PI * (angle * 0.005555555555555556)))) * 0.5) * (2.0 * Math.pow((b + a), 2.0));
	} else {
		tmp = 2.0 * ((Math.sin((angle * (Math.PI / 180.0))) * (b + a)) * (b - a));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (angle / 180.0) <= -1.0:
		tmp = (math.sin((2.0 * (math.pi * (angle * 0.005555555555555556)))) * 0.5) * (2.0 * math.pow((b + a), 2.0))
	else:
		tmp = 2.0 * ((math.sin((angle * (math.pi / 180.0))) * (b + a)) * (b - a))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= -1.0)
		tmp = Float64(Float64(sin(Float64(2.0 * Float64(pi * Float64(angle * 0.005555555555555556)))) * 0.5) * Float64(2.0 * (Float64(b + a) ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(sin(Float64(angle * Float64(pi / 180.0))) * Float64(b + a)) * Float64(b - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((angle / 180.0) <= -1.0)
		tmp = (sin((2.0 * (pi * (angle * 0.005555555555555556)))) * 0.5) * (2.0 * ((b + a) ^ 2.0));
	else
		tmp = 2.0 * ((sin((angle * (pi / 180.0))) * (b + a)) * (b - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], -1.0], N[(N[(N[Sin[N[(2.0 * N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 * N[Power[N[(b + a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Sin[N[(angle * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < -1

   1. Initial program 36.5%

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

      1. *-commutative 36.5%

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

      2. associate-*l* 36.5%

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

      3. unpow2 36.5%

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

      4. fma-neg 38.1%

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

      5. unpow2 38.1%

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

   3. Simplified 38.1%

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

      1. *-commutative 38.1%

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

      2. associate-*r* 38.1%

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

      3. fma-neg 36.5%

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

      4. difference-of-squares 36.5%

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

      5. *-commutative 36.5%

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

      6. expm1-log1p-u 18.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)} \]

      7. expm1-udef 13.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 15.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(2 \cdot \left({\left(b + a\right)}^{2} \cdot \left(\left(\sin 0 + \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right) \cdot 0.5\right)\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 17.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \left({\left(b + a\right)}^{2} \cdot \left(\left(\sin 0 + \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right) \cdot 0.5\right)\right)\right)\right)} \]

      2. expm1-log1p 36.6%

         \[\leadsto \color{blue}{2 \cdot \left({\left(b + a\right)}^{2} \cdot \left(\left(\sin 0 + \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right) \cdot 0.5\right)\right)} \]

      3. associate-*r* 36.6%

         \[\leadsto \color{blue}{\left(2 \cdot {\left(b + a\right)}^{2}\right) \cdot \left(\left(\sin 0 + \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right) \cdot 0.5\right)} \]

      4. *-commutative 36.6%

         \[\leadsto \color{blue}{\left(\left(\sin 0 + \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right) \cdot 0.5\right) \cdot \left(2 \cdot {\left(b + a\right)}^{2}\right)} \]

      5. sin-0 36.6%

         \[\leadsto \left(\left(\color{blue}{0} + \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right) \cdot 0.5\right) \cdot \left(2 \cdot {\left(b + a\right)}^{2}\right) \]

      6. +-lft-identity 36.6%

         \[\leadsto \left(\color{blue}{\sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)} \cdot 0.5\right) \cdot \left(2 \cdot {\left(b + a\right)}^{2}\right) \]

      7. *-commutative 36.6%

         \[\leadsto \left(\sin \color{blue}{\left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot 0.5\right) \cdot \left(2 \cdot {\left(b + a\right)}^{2}\right) \]

   7. Simplified 36.6%

      \[\leadsto \color{blue}{\left(\sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 0.5\right) \cdot \left(2 \cdot {\left(b + a\right)}^{2}\right)} \]

   if -1 < (/.f64 angle 180)

   1. Initial program 60.6%

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

      1. associate-*l* 60.6%

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

      2. unpow2 60.6%

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

      3. unpow2 60.6%

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

      4. difference-of-squares 64.4%

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

   3. Simplified 64.4%

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

   4. Taylor expanded in angle around inf 77.3%

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

      1. associate-*r* 77.3%

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

      2. *-commutative 77.3%

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

      3. *-commutative 77.3%

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

      4. *-commutative 77.3%

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

      5. associate-*r* 77.4%

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

      6. +-commutative 77.4%

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

      7. *-commutative 77.4%

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

      8. *-commutative 77.4%

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

      9. associate-*r* 78.4%

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

   6. Simplified 78.4%

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

      1. metadata-eval 78.4%

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

      2. div-inv 77.9%

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

      3. associate-*r/ 76.3%

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

   8. Applied egg-rr 76.3%

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

      1. associate-/l* 77.7%

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

   10. Simplified 77.7%

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

       1. metadata-eval 78.4%

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

       2. div-inv 77.9%

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

       3. associate-*r/ 76.3%

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

   12. Applied egg-rr 75.5%

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

       1. associate-/l* 77.6%

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

       2. associate-/r/ 78.3%

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

   14. Simplified 78.3%

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

   15. Taylor expanded in angle around 0 78.9%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -1:\\ \;\;\;\;\left(\sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 0.5\right) \cdot \left(2 \cdot {\left(b + a\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(angle \cdot \frac{\pi}{180}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)\\ \end{array} \]

Alternative 8: 65.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.2 \cdot 10^{-115}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a -6.2e-115)
   (* 0.011111111111111112 (* (* PI (+ b a)) (* angle (- b a))))
   (*
    2.0
    (* (- b a) (* (+ b a) (sin (* PI (* angle 0.005555555555555556))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= -6.2e-115) {
		tmp = 0.011111111111111112 * ((((double) M_PI) * (b + a)) * (angle * (b - a)));
	} else {
		tmp = 2.0 * ((b - a) * ((b + a) * sin((((double) M_PI) * (angle * 0.005555555555555556)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= -6.2e-115) {
		tmp = 0.011111111111111112 * ((Math.PI * (b + a)) * (angle * (b - a)));
	} else {
		tmp = 2.0 * ((b - a) * ((b + a) * Math.sin((Math.PI * (angle * 0.005555555555555556)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= -6.2e-115:
		tmp = 0.011111111111111112 * ((math.pi * (b + a)) * (angle * (b - a)))
	else:
		tmp = 2.0 * ((b - a) * ((b + a) * math.sin((math.pi * (angle * 0.005555555555555556)))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= -6.2e-115)
		tmp = Float64(0.011111111111111112 * Float64(Float64(pi * Float64(b + a)) * Float64(angle * Float64(b - a))));
	else
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(pi * Float64(angle * 0.005555555555555556))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= -6.2e-115)
		tmp = 0.011111111111111112 * ((pi * (b + a)) * (angle * (b - a)));
	else
		tmp = 2.0 * ((b - a) * ((b + a) * sin((pi * (angle * 0.005555555555555556)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, -6.2e-115], N[(0.011111111111111112 * N[(N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.20000000000000013e-115

   1. Initial program 58.1%

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

      1. associate-*l* 58.1%

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

      2. unpow2 58.1%

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

      3. unpow2 58.1%

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

      4. difference-of-squares 61.6%

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

   3. Simplified 61.6%

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

   4. Taylor expanded in angle around 0 61.0%

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

      1. associate-*r* 72.3%

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

      2. *-commutative 72.3%

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

      3. +-commutative 72.3%

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

   6. Simplified 72.3%

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

   if -6.20000000000000013e-115 < a

   1. Initial program 53.0%

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

      1. associate-*l* 53.0%

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

      2. unpow2 53.0%

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

      3. unpow2 53.0%

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

      4. difference-of-squares 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in angle around inf 64.4%

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

      1. associate-*r* 64.4%

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

      2. *-commutative 64.4%

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

      3. *-commutative 64.4%

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

      4. *-commutative 64.4%

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

      5. associate-*r* 65.2%

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

      6. +-commutative 65.2%

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

      7. *-commutative 65.2%

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

      8. *-commutative 65.2%

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

      9. associate-*r* 65.8%

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

   6. Simplified 65.8%

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

   7. Taylor expanded in angle around 0 64.2%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.2 \cdot 10^{-115}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\\ \end{array} \]

Alternative 9: 65.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\left(\sin \left(angle \cdot \frac{\pi}{180}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (* 2.0 (* (* (sin (* angle (/ PI 180.0))) (+ b a)) (- b a))))

C

double code(double a, double b, double angle) {
	return 2.0 * ((sin((angle * (((double) M_PI) / 180.0))) * (b + a)) * (b - a));
}

Java

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

Python

def code(a, b, angle):
	return 2.0 * ((math.sin((angle * (math.pi / 180.0))) * (b + a)) * (b - a))

Julia

function code(a, b, angle)
	return Float64(2.0 * Float64(Float64(sin(Float64(angle * Float64(pi / 180.0))) * Float64(b + a)) * Float64(b - a)))
end

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[(2.0 * N[(N[(N[Sin[N[(angle * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.8%

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

   1. associate-*l* 54.8%

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

   2. unpow2 54.8%

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

   3. unpow2 54.8%

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

   4. difference-of-squares 57.6%

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

3. Simplified 57.6%

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

4. Taylor expanded in angle around inf 66.6%

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

   1. associate-*r* 66.6%

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

   2. *-commutative 66.6%

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

   3. *-commutative 66.6%

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

   4. *-commutative 66.6%

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

   5. associate-*r* 68.3%

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

   6. +-commutative 68.3%

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

   7. *-commutative 68.3%

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

   8. *-commutative 68.3%

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

   9. associate-*r* 68.3%

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

6. Simplified 68.3%

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

   1. metadata-eval 68.3%

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

   2. div-inv 67.9%

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

   3. associate-*r/ 66.7%

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

8. Applied egg-rr 66.7%

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

   1. associate-/l* 68.6%

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

10. Simplified 68.6%

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

    1. metadata-eval 68.3%

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

    2. div-inv 67.9%

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

    3. associate-*r/ 66.7%

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

12. Applied egg-rr 65.8%

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

    1. associate-/l* 68.6%

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

    2. associate-/r/ 69.0%

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

14. Simplified 69.0%

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

15. Taylor expanded in angle around 0 66.9%

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

16. Final simplification 66.9%

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

Alternative 10: 62.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 3.9 \cdot 10^{+124}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot angle\right) \cdot {\left(b + a\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 3.9e+124)
   (* 0.011111111111111112 (* (* PI (+ b a)) (* angle (- b a))))
   (* 0.011111111111111112 (* (* PI angle) (pow (+ b a) 2.0)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 3.9e+124) {
		tmp = 0.011111111111111112 * ((((double) M_PI) * (b + a)) * (angle * (b - a)));
	} else {
		tmp = 0.011111111111111112 * ((((double) M_PI) * angle) * pow((b + a), 2.0));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 3.9e+124) {
		tmp = 0.011111111111111112 * ((Math.PI * (b + a)) * (angle * (b - a)));
	} else {
		tmp = 0.011111111111111112 * ((Math.PI * angle) * Math.pow((b + a), 2.0));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 3.9e+124:
		tmp = 0.011111111111111112 * ((math.pi * (b + a)) * (angle * (b - a)))
	else:
		tmp = 0.011111111111111112 * ((math.pi * angle) * math.pow((b + a), 2.0))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 3.9e+124)
		tmp = Float64(0.011111111111111112 * Float64(Float64(pi * Float64(b + a)) * Float64(angle * Float64(b - a))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(pi * angle) * (Float64(b + a) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 3.9e+124)
		tmp = 0.011111111111111112 * ((pi * (b + a)) * (angle * (b - a)));
	else
		tmp = 0.011111111111111112 * ((pi * angle) * ((b + a) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 3.9e+124], N[(0.011111111111111112 * N[(N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(Pi * angle), $MachinePrecision] * N[Power[N[(b + a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 3.9e124

   1. Initial program 58.1%

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

      1. associate-*l* 58.1%

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

      2. unpow2 58.1%

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

      3. unpow2 58.1%

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

      4. difference-of-squares 61.5%

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

   3. Simplified 61.5%

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

   4. Taylor expanded in angle around 0 59.2%

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

      1. associate-*r* 71.2%

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

      2. *-commutative 71.2%

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

      3. +-commutative 71.2%

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

   6. Simplified 71.2%

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

   if 3.9e124 < angle

   1. Initial program 37.3%

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

      1. *-commutative 37.3%

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

      2. associate-*l* 37.3%

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

      3. unpow2 37.3%

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

      4. fma-neg 39.7%

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

      5. unpow2 39.7%

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

   3. Simplified 39.7%

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

      1. *-commutative 39.7%

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

      2. associate-*r* 39.7%

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

      3. fma-neg 37.3%

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

      4. difference-of-squares 37.3%

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

      5. *-commutative 37.3%

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

      6. add-log-exp 32.9%

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

      7. *-commutative 32.9%

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

      8. difference-of-squares 32.9%

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

      9. fma-neg 35.3%

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

   5. Applied egg-rr 25.4%

      \[\leadsto \color{blue}{\log \left({\left(\sqrt{e^{\sin 0 + \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)}}\right)}^{\left(2 \cdot {\left(b + a\right)}^{2}\right)}\right)} \]
   6. Step-by-step derivation

      1. log-pow 29.5%

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

      2. sin-0 29.5%

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

      3. +-lft-identity 29.5%

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

      4. *-commutative 29.5%

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

   7. Simplified 29.5%

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

   8. Taylor expanded in angle around 0 32.9%

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

      1. associate-*r* 32.9%

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

      2. +-commutative 32.9%

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

   10. Simplified 32.9%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 3.9 \cdot 10^{+124}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot angle\right) \cdot {\left(b + a\right)}^{2}\right)\\ \end{array} \]

Alternative 11: 62.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 3.9 \cdot 10^{+124}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot {\left(b + a\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 3.9e+124)
   (* 0.011111111111111112 (* (* PI (+ b a)) (* angle (- b a))))
   (* angle (* 0.011111111111111112 (* PI (pow (+ b a) 2.0))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 3.9e+124) {
		tmp = 0.011111111111111112 * ((((double) M_PI) * (b + a)) * (angle * (b - a)));
	} else {
		tmp = angle * (0.011111111111111112 * (((double) M_PI) * pow((b + a), 2.0)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 3.9e+124) {
		tmp = 0.011111111111111112 * ((Math.PI * (b + a)) * (angle * (b - a)));
	} else {
		tmp = angle * (0.011111111111111112 * (Math.PI * Math.pow((b + a), 2.0)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 3.9e+124:
		tmp = 0.011111111111111112 * ((math.pi * (b + a)) * (angle * (b - a)))
	else:
		tmp = angle * (0.011111111111111112 * (math.pi * math.pow((b + a), 2.0)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 3.9e+124)
		tmp = Float64(0.011111111111111112 * Float64(Float64(pi * Float64(b + a)) * Float64(angle * Float64(b - a))));
	else
		tmp = Float64(angle * Float64(0.011111111111111112 * Float64(pi * (Float64(b + a) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 3.9e+124)
		tmp = 0.011111111111111112 * ((pi * (b + a)) * (angle * (b - a)));
	else
		tmp = angle * (0.011111111111111112 * (pi * ((b + a) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 3.9e+124], N[(0.011111111111111112 * N[(N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(angle * N[(0.011111111111111112 * N[(Pi * N[Power[N[(b + a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 3.9e124

   1. Initial program 58.1%

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

      1. associate-*l* 58.1%

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

      2. unpow2 58.1%

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

      3. unpow2 58.1%

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

      4. difference-of-squares 61.5%

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

   3. Simplified 61.5%

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

   4. Taylor expanded in angle around 0 59.2%

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

      1. associate-*r* 71.2%

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

      2. *-commutative 71.2%

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

      3. +-commutative 71.2%

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

   6. Simplified 71.2%

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

   if 3.9e124 < angle

   1. Initial program 37.3%

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

      1. *-commutative 37.3%

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

      2. associate-*l* 37.3%

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

      3. unpow2 37.3%

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

      4. fma-neg 39.7%

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

      5. unpow2 39.7%

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

   3. Simplified 39.7%

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

      1. *-commutative 39.7%

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

      2. associate-*r* 39.7%

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

      3. fma-neg 37.3%

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

      4. difference-of-squares 37.3%

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

      5. *-commutative 37.3%

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

      6. add-log-exp 32.9%

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

      7. *-commutative 32.9%

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

      8. difference-of-squares 32.9%

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

      9. fma-neg 35.3%

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

   5. Applied egg-rr 25.4%

      \[\leadsto \color{blue}{\log \left({\left(\sqrt{e^{\sin 0 + \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)}}\right)}^{\left(2 \cdot {\left(b + a\right)}^{2}\right)}\right)} \]
   6. Step-by-step derivation

      1. log-pow 29.5%

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

      2. sin-0 29.5%

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

      3. +-lft-identity 29.5%

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

      4. *-commutative 29.5%

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

   7. Simplified 29.5%

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

   8. Taylor expanded in angle around 0 32.9%

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

      1. *-commutative 32.9%

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

      2. associate-*l* 32.9%

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

      3. +-commutative 32.9%

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

   10. Simplified 32.9%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 3.9 \cdot 10^{+124}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot {\left(b + a\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 12: 44.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot -0.011111111111111112\\ \mathbf{if}\;a \leq -2.8 \cdot 10^{+203}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq -3 \cdot 10^{+130}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{elif}\;a \leq -5 \cdot 10^{-23} \lor \neg \left(a \leq 1.7 \cdot 10^{+154}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* angle (* PI (* a a))) -0.011111111111111112)))
   (if (<= a -2.8e+203)
     t_0
     (if (<= a -3e+130)
       (* 0.011111111111111112 (* angle (* PI (* b b))))
       (if (or (<= a -5e-23) (not (<= a 1.7e+154)))
         t_0
         (* 0.011111111111111112 (* PI (* angle (* b b)))))))))

C

double code(double a, double b, double angle) {
	double t_0 = (angle * (((double) M_PI) * (a * a))) * -0.011111111111111112;
	double tmp;
	if (a <= -2.8e+203) {
		tmp = t_0;
	} else if (a <= -3e+130) {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (b * b)));
	} else if ((a <= -5e-23) || !(a <= 1.7e+154)) {
		tmp = t_0;
	} else {
		tmp = 0.011111111111111112 * (((double) M_PI) * (angle * (b * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (angle * (Math.PI * (a * a))) * -0.011111111111111112;
	double tmp;
	if (a <= -2.8e+203) {
		tmp = t_0;
	} else if (a <= -3e+130) {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b * b)));
	} else if ((a <= -5e-23) || !(a <= 1.7e+154)) {
		tmp = t_0;
	} else {
		tmp = 0.011111111111111112 * (Math.PI * (angle * (b * b)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = (angle * (math.pi * (a * a))) * -0.011111111111111112
	tmp = 0
	if a <= -2.8e+203:
		tmp = t_0
	elif a <= -3e+130:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b * b)))
	elif (a <= -5e-23) or not (a <= 1.7e+154):
		tmp = t_0
	else:
		tmp = 0.011111111111111112 * (math.pi * (angle * (b * b)))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(angle * Float64(pi * Float64(a * a))) * -0.011111111111111112)
	tmp = 0.0
	if (a <= -2.8e+203)
		tmp = t_0;
	elseif (a <= -3e+130)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b * b))));
	elseif ((a <= -5e-23) || !(a <= 1.7e+154))
		tmp = t_0;
	else
		tmp = Float64(0.011111111111111112 * Float64(pi * Float64(angle * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = (angle * (pi * (a * a))) * -0.011111111111111112;
	tmp = 0.0;
	if (a <= -2.8e+203)
		tmp = t_0;
	elseif (a <= -3e+130)
		tmp = 0.011111111111111112 * (angle * (pi * (b * b)));
	elseif ((a <= -5e-23) || ~((a <= 1.7e+154)))
		tmp = t_0;
	else
		tmp = 0.011111111111111112 * (pi * (angle * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(angle * N[(Pi * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]}, If[LessEqual[a, -2.8e+203], t$95$0, If[LessEqual[a, -3e+130], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[a, -5e-23], N[Not[LessEqual[a, 1.7e+154]], $MachinePrecision]], t$95$0, N[(0.011111111111111112 * N[(Pi * N[(angle * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.7999999999999999e203 or -2.9999999999999999e130 < a < -5.0000000000000002e-23 or 1.69999999999999987e154 < a

   1. Initial program 51.8%

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

      1. associate-*l* 51.8%

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

      2. unpow2 51.8%

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

      3. unpow2 51.8%

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

      4. difference-of-squares 58.1%

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

   3. Simplified 58.1%

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

   4. Taylor expanded in angle around 0 54.4%

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

   5. Taylor expanded in b around 0 53.2%

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

      1. *-commutative 53.2%

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

      2. *-commutative 53.2%

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

      3. unpow2 53.2%

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

   7. Simplified 53.2%

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

   if -2.7999999999999999e203 < a < -2.9999999999999999e130

   1. Initial program 34.1%

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

      1. associate-*l* 34.1%

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

      2. unpow2 34.1%

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

      3. unpow2 34.1%

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

      4. difference-of-squares 41.8%

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

   3. Simplified 41.8%

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

   4. Taylor expanded in angle around 0 57.7%

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

   5. Taylor expanded in b around inf 46.8%

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

      1. *-commutative 46.8%

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

      2. unpow2 46.8%

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

   7. Simplified 46.8%

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

   if -5.0000000000000002e-23 < a < 1.69999999999999987e154

   1. Initial program 58.0%

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

      1. associate-*l* 58.0%

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

      2. unpow2 58.0%

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

      3. unpow2 58.0%

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

      4. difference-of-squares 58.7%

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

   3. Simplified 58.7%

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

   4. Taylor expanded in angle around 0 52.7%

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

   5. Taylor expanded in b around inf 48.0%

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

      1. *-commutative 48.0%

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

      2. unpow2 48.0%

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

   7. Simplified 48.0%

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

   8. Taylor expanded in angle around 0 48.0%

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

      1. associate-*r* 48.0%

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

      2. *-commutative 48.0%

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

      3. unpow2 48.0%

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

   10. Simplified 48.0%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.8 \cdot 10^{+203}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot -0.011111111111111112\\ \mathbf{elif}\;a \leq -3 \cdot 10^{+130}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{elif}\;a \leq -5 \cdot 10^{-23} \lor \neg \left(a \leq 1.7 \cdot 10^{+154}\right):\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot -0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 13: 61.9% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 4.8 \cdot 10^{+129}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 4.8e+129)
   (* 0.011111111111111112 (* (* PI (+ b a)) (* angle (- b a))))
   (* 0.011111111111111112 (* angle (* (- b a) (* b PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 4.8e+129) {
		tmp = 0.011111111111111112 * ((((double) M_PI) * (b + a)) * (angle * (b - a)));
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 4.8e+129) {
		tmp = 0.011111111111111112 * ((Math.PI * (b + a)) * (angle * (b - a)));
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 4.8e+129:
		tmp = 0.011111111111111112 * ((math.pi * (b + a)) * (angle * (b - a)))
	else:
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 4.8e+129)
		tmp = Float64(0.011111111111111112 * Float64(Float64(pi * Float64(b + a)) * Float64(angle * Float64(b - a))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(b * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 4.8e+129)
		tmp = 0.011111111111111112 * ((pi * (b + a)) * (angle * (b - a)));
	else
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 4.8e+129], N[(0.011111111111111112 * N[(N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 4.7999999999999997e129

   1. Initial program 58.2%

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

      1. associate-*l* 58.2%

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

      2. unpow2 58.2%

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

      3. unpow2 58.2%

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

      4. difference-of-squares 61.5%

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

   3. Simplified 61.5%

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

   4. Taylor expanded in angle around 0 58.7%

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

      1. associate-*r* 70.6%

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

      2. *-commutative 70.6%

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

      3. +-commutative 70.6%

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

   6. Simplified 70.6%

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

   if 4.7999999999999997e129 < angle

   1. Initial program 36.0%

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

      1. associate-*l* 36.0%

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

      2. unpow2 36.0%

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

      3. unpow2 36.0%

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

      4. difference-of-squares 36.0%

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

   3. Simplified 36.0%

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

   4. Taylor expanded in angle around 0 24.6%

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

   5. Taylor expanded in a around 0 29.6%

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

      1. *-commutative 29.6%

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

   7. Simplified 29.6%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 4.8 \cdot 10^{+129}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 14: 46.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{-23}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+154}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot -0.011111111111111112\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a -2.7e-23)
   (* 0.011111111111111112 (* angle (* (- b a) (* a PI))))
   (if (<= a 1.8e+154)
     (* 0.011111111111111112 (* PI (* angle (* b b))))
     (* (* angle (* PI (* a a))) -0.011111111111111112))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= -2.7e-23) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * ((double) M_PI))));
	} else if (a <= 1.8e+154) {
		tmp = 0.011111111111111112 * (((double) M_PI) * (angle * (b * b)));
	} else {
		tmp = (angle * (((double) M_PI) * (a * a))) * -0.011111111111111112;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= -2.7e-23) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * Math.PI)));
	} else if (a <= 1.8e+154) {
		tmp = 0.011111111111111112 * (Math.PI * (angle * (b * b)));
	} else {
		tmp = (angle * (Math.PI * (a * a))) * -0.011111111111111112;
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= -2.7e-23:
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * math.pi)))
	elif a <= 1.8e+154:
		tmp = 0.011111111111111112 * (math.pi * (angle * (b * b)))
	else:
		tmp = (angle * (math.pi * (a * a))) * -0.011111111111111112
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= -2.7e-23)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(a * pi))));
	elseif (a <= 1.8e+154)
		tmp = Float64(0.011111111111111112 * Float64(pi * Float64(angle * Float64(b * b))));
	else
		tmp = Float64(Float64(angle * Float64(pi * Float64(a * a))) * -0.011111111111111112);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= -2.7e-23)
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * pi)));
	elseif (a <= 1.8e+154)
		tmp = 0.011111111111111112 * (pi * (angle * (b * b)));
	else
		tmp = (angle * (pi * (a * a))) * -0.011111111111111112;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, -2.7e-23], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.8e+154], N[(0.011111111111111112 * N[(Pi * N[(angle * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(angle * N[(Pi * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.69999999999999985e-23

   1. Initial program 52.8%

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

      1. associate-*l* 52.8%

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

      2. unpow2 52.8%

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

      3. unpow2 52.8%

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

      4. difference-of-squares 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in angle around 0 58.8%

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

   5. Taylor expanded in a around inf 50.3%

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

      1. *-commutative 50.3%

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

   7. Simplified 50.3%

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

   if -2.69999999999999985e-23 < a < 1.8e154

   1. Initial program 58.0%

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

      1. associate-*l* 58.0%

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

      2. unpow2 58.0%

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

      3. unpow2 58.0%

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

      4. difference-of-squares 58.7%

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

   3. Simplified 58.7%

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

   4. Taylor expanded in angle around 0 52.7%

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

   5. Taylor expanded in b around inf 48.0%

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

      1. *-commutative 48.0%

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

      2. unpow2 48.0%

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

   7. Simplified 48.0%

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

   8. Taylor expanded in angle around 0 48.0%

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

      1. associate-*r* 48.0%

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

      2. *-commutative 48.0%

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

      3. unpow2 48.0%

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

   10. Simplified 48.0%

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

   if 1.8e154 < a

   1. Initial program 41.2%

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

      1. associate-*l* 41.2%

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

      2. unpow2 41.2%

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

      3. unpow2 41.2%

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

      4. difference-of-squares 52.3%

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

   3. Simplified 52.3%

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

   4. Taylor expanded in angle around 0 45.2%

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

   5. Taylor expanded in b around 0 52.1%

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

      1. *-commutative 52.1%

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

      2. *-commutative 52.1%

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

      3. unpow2 52.1%

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

   7. Simplified 52.1%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{-23}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+154}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot -0.011111111111111112\\ \end{array} \]

Alternative 15: 54.2% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

   1. associate-*l* 54.8%

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

   2. unpow2 54.8%

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

   3. unpow2 54.8%

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

   4. difference-of-squares 57.6%

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

3. Simplified 57.6%

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

4. Taylor expanded in angle around 0 53.5%

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

5. Final simplification 53.5%

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

Alternative 16: 34.3% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

   1. associate-*l* 54.8%

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

   2. unpow2 54.8%

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

   3. unpow2 54.8%

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

   4. difference-of-squares 57.6%

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

3. Simplified 57.6%

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

4. Taylor expanded in angle around 0 53.5%

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

5. Taylor expanded in b around inf 35.0%

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

   1. *-commutative 35.0%

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

   2. unpow2 35.0%

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

7. Simplified 35.0%

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

8. Final simplification 35.0%

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

Alternative 17: 34.4% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

   1. associate-*l* 54.8%

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

   2. unpow2 54.8%

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

   3. unpow2 54.8%

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

   4. difference-of-squares 57.6%

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

3. Simplified 57.6%

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

4. Taylor expanded in angle around 0 53.5%

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

5. Taylor expanded in b around inf 35.0%

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

   1. *-commutative 35.0%

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

   2. unpow2 35.0%

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

7. Simplified 35.0%

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

8. Taylor expanded in angle around 0 35.0%

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

   1. associate-*r* 35.0%

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

   2. *-commutative 35.0%

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

   3. unpow2 35.0%

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

10. Simplified 35.0%

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

11. Final simplification 35.0%

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

Reproduce

herbie shell --seed 2023181 
(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)))))