ab-angle->ABCF C

Percentage Accurate: 79.2% → 79.2%

Time: 13.6s

Alternatives: 7

Speedup: 1.3×

• Could not converge on a ground truth

  1. a = 2.2241883011935595e-218
  2. b = 2.3726789498391433e+78
  3. angle = 2.7752268299740955e+46

• 35.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{hypot}\left(a, b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (pow (hypot a (* b (sin (* angle (* PI 0.005555555555555556))))) 2.0))

C

double code(double a, double b, double angle) {
	return pow(hypot(a, (b * sin((angle * (((double) M_PI) * 0.005555555555555556))))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow(Math.hypot(a, (b * Math.sin((angle * (Math.PI * 0.005555555555555556))))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow(math.hypot(a, (b * math.sin((angle * (math.pi * 0.005555555555555556))))), 2.0)

Julia

function code(a, b, angle)
	return hypot(a, Float64(b * sin(Float64(angle * Float64(pi * 0.005555555555555556))))) ^ 2.0
end

MATLAB

function tmp = code(a, b, angle)
	tmp = hypot(a, (b * sin((angle * (pi * 0.005555555555555556))))) ^ 2.0;
end

Wolfram

code[a_, b_, angle_] := N[Power[N[Sqrt[a ^ 2 + N[(b * N[Sin[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $MachinePrecision]

Derivation

1. Initial program 78.4%

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

   1. associate-*r/ 78.4%

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

   2. metadata-eval 78.4%

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

   3. metadata-eval 78.4%

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

   4. distribute-neg-frac2 78.4%

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

   5. distribute-frac-neg 78.4%

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

   6. distribute-rgt-neg-out 78.4%

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

   7. associate-/l* 78.4%

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

   8. neg-mul-1 78.4%

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

   9. *-commutative 78.4%

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

   10. associate-/l* 78.4%

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

   11. metadata-eval 78.4%

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

   12. metadata-eval 78.4%

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

3. Simplified 78.4%

   \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \]
4. Add Preprocessing

5. Taylor expanded in angle around 0 78.7%

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

   1. add-sqr-sqrt 78.7%

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

   2. pow2 78.7%

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

7. Applied egg-rr 78.8%

   \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(a, b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}^{2}} \]
8. Add Preprocessing

Alternative 2: 58.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 7.6 \cdot 10^{-149}:\\ \;\;\;\;{\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(\left(angle \cdot 0.005555555555555556\right) \cdot \left(b \cdot \pi\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 7.6e-149)
   (pow (* b (sin (* angle (* PI 0.005555555555555556)))) 2.0)
   (+ (pow a 2.0) (pow (* (* angle 0.005555555555555556) (* b PI)) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 7.6e-149) {
		tmp = pow((b * sin((angle * (((double) M_PI) * 0.005555555555555556)))), 2.0);
	} else {
		tmp = pow(a, 2.0) + pow(((angle * 0.005555555555555556) * (b * ((double) M_PI))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 7.6e-149) {
		tmp = Math.pow((b * Math.sin((angle * (Math.PI * 0.005555555555555556)))), 2.0);
	} else {
		tmp = Math.pow(a, 2.0) + Math.pow(((angle * 0.005555555555555556) * (b * Math.PI)), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 7.6e-149:
		tmp = math.pow((b * math.sin((angle * (math.pi * 0.005555555555555556)))), 2.0)
	else:
		tmp = math.pow(a, 2.0) + math.pow(((angle * 0.005555555555555556) * (b * math.pi)), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 7.6e-149)
		tmp = Float64(b * sin(Float64(angle * Float64(pi * 0.005555555555555556)))) ^ 2.0;
	else
		tmp = Float64((a ^ 2.0) + (Float64(Float64(angle * 0.005555555555555556) * Float64(b * pi)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 7.6e-149)
		tmp = (b * sin((angle * (pi * 0.005555555555555556)))) ^ 2.0;
	else
		tmp = (a ^ 2.0) + (((angle * 0.005555555555555556) * (b * pi)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 7.6e-149], N[Power[N[(b * N[Sin[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(N[(angle * 0.005555555555555556), $MachinePrecision] * N[(b * Pi), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7.6000000000000001e-149

   1. Initial program 76.0%

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

      1. associate-*r/ 76.0%

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

      2. metadata-eval 76.0%

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

      3. metadata-eval 76.0%

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

      4. distribute-neg-frac2 76.0%

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

      5. distribute-frac-neg 76.0%

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

      6. distribute-rgt-neg-out 76.0%

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

      7. associate-/l* 76.0%

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

      8. neg-mul-1 76.0%

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

      9. *-commutative 76.0%

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

      10. associate-/l* 76.0%

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

      11. metadata-eval 76.0%

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

      12. metadata-eval 76.0%

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

   3. Simplified 75.9%

      \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 39.8%

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

      1. unpow2 39.8%

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

      2. *-commutative 39.8%

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

      3. unpow2 39.8%

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

      4. swap-sqr 47.9%

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

      5. unpow2 47.9%

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

      6. *-commutative 47.9%

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

      7. *-commutative 47.9%

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

      8. associate-*r* 48.0%

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

   7. Simplified 48.0%

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

   if 7.6000000000000001e-149 < a

   1. Initial program 82.0%

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

      1. associate-*r/ 81.9%

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

      2. metadata-eval 81.9%

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

      3. metadata-eval 81.9%

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

      4. distribute-neg-frac2 81.9%

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

      5. distribute-frac-neg 81.9%

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

      6. distribute-rgt-neg-out 81.9%

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

      7. associate-/l* 82.0%

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

      8. neg-mul-1 82.0%

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

      9. *-commutative 82.0%

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

      10. associate-/l* 82.0%

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

      11. metadata-eval 82.0%

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

      12. metadata-eval 82.0%

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

   3. Simplified 82.1%

      \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0 81.7%

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

   6. Taylor expanded in angle around 0 79.8%

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

      1. associate-*r* 79.8%

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

      2. *-commutative 79.8%

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

   8. Simplified 79.8%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7.6 \cdot 10^{-149}:\\ \;\;\;\;{\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(\left(angle \cdot 0.005555555555555556\right) \cdot \left(b \cdot \pi\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 7.6 \cdot 10^{-149}:\\ \;\;\;\;{\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, b \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 7.6e-149)
   (pow (* b (sin (* angle (* PI 0.005555555555555556)))) 2.0)
   (pow (hypot a (* b (* 0.005555555555555556 (* angle PI)))) 2.0)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 7.6e-149) {
		tmp = pow((b * sin((angle * (((double) M_PI) * 0.005555555555555556)))), 2.0);
	} else {
		tmp = pow(hypot(a, (b * (0.005555555555555556 * (angle * ((double) M_PI))))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 7.6e-149) {
		tmp = Math.pow((b * Math.sin((angle * (Math.PI * 0.005555555555555556)))), 2.0);
	} else {
		tmp = Math.pow(Math.hypot(a, (b * (0.005555555555555556 * (angle * Math.PI)))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 7.6e-149:
		tmp = math.pow((b * math.sin((angle * (math.pi * 0.005555555555555556)))), 2.0)
	else:
		tmp = math.pow(math.hypot(a, (b * (0.005555555555555556 * (angle * math.pi)))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 7.6e-149)
		tmp = Float64(b * sin(Float64(angle * Float64(pi * 0.005555555555555556)))) ^ 2.0;
	else
		tmp = hypot(a, Float64(b * Float64(0.005555555555555556 * Float64(angle * pi)))) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 7.6e-149)
		tmp = (b * sin((angle * (pi * 0.005555555555555556)))) ^ 2.0;
	else
		tmp = hypot(a, (b * (0.005555555555555556 * (angle * pi)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 7.6e-149], N[Power[N[(b * N[Sin[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[Power[N[Sqrt[a ^ 2 + N[(b * N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7.6000000000000001e-149

   1. Initial program 76.0%

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

      1. associate-*r/ 76.0%

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

      2. metadata-eval 76.0%

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

      3. metadata-eval 76.0%

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

      4. distribute-neg-frac2 76.0%

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

      5. distribute-frac-neg 76.0%

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

      6. distribute-rgt-neg-out 76.0%

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

      7. associate-/l* 76.0%

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

      8. neg-mul-1 76.0%

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

      9. *-commutative 76.0%

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

      10. associate-/l* 76.0%

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

      11. metadata-eval 76.0%

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

      12. metadata-eval 76.0%

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

   3. Simplified 75.9%

      \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 39.8%

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

      1. unpow2 39.8%

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

      2. *-commutative 39.8%

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

      3. unpow2 39.8%

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

      4. swap-sqr 47.9%

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

      5. unpow2 47.9%

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

      6. *-commutative 47.9%

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

      7. *-commutative 47.9%

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

      8. associate-*r* 48.0%

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

   7. Simplified 48.0%

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

   if 7.6000000000000001e-149 < a

   1. Initial program 82.0%

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

      1. associate-*r/ 81.9%

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

      2. metadata-eval 81.9%

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

      3. metadata-eval 81.9%

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

      4. distribute-neg-frac2 81.9%

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

      5. distribute-frac-neg 81.9%

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

      6. distribute-rgt-neg-out 81.9%

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

      7. associate-/l* 82.0%

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

      8. neg-mul-1 82.0%

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

      9. *-commutative 82.0%

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

      10. associate-/l* 82.0%

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

      11. metadata-eval 82.0%

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

      12. metadata-eval 82.0%

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

   3. Simplified 82.1%

      \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0 81.7%

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

      1. add-sqr-sqrt 81.7%

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

      2. pow2 81.7%

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

   7. Applied egg-rr 81.7%

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

   8. Taylor expanded in angle around 0 79.8%

      \[\leadsto {\left(\mathsf{hypot}\left(a, b \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7.6 \cdot 10^{-149}:\\ \;\;\;\;{\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, b \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5.6 \cdot 10^{-22}:\\ \;\;\;\;{\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 5.6e-22)
   (pow (* b (sin (* angle (* PI 0.005555555555555556)))) 2.0)
   (* a a)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5.6e-22) {
		tmp = pow((b * sin((angle * (((double) M_PI) * 0.005555555555555556)))), 2.0);
	} else {
		tmp = a * a;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5.6e-22) {
		tmp = Math.pow((b * Math.sin((angle * (Math.PI * 0.005555555555555556)))), 2.0);
	} else {
		tmp = a * a;
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 5.6e-22:
		tmp = math.pow((b * math.sin((angle * (math.pi * 0.005555555555555556)))), 2.0)
	else:
		tmp = a * a
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 5.6e-22)
		tmp = Float64(b * sin(Float64(angle * Float64(pi * 0.005555555555555556)))) ^ 2.0;
	else
		tmp = Float64(a * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 5.6e-22)
		tmp = (b * sin((angle * (pi * 0.005555555555555556)))) ^ 2.0;
	else
		tmp = a * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 5.6e-22], N[Power[N[(b * N[Sin[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(a * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.5999999999999999e-22

   1. Initial program 76.1%

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

      1. associate-*r/ 76.0%

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

      2. metadata-eval 76.0%

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

      3. metadata-eval 76.0%

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

      4. distribute-neg-frac2 76.0%

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

      5. distribute-frac-neg 76.0%

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

      6. distribute-rgt-neg-out 76.0%

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

      7. associate-/l* 76.1%

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

      8. neg-mul-1 76.1%

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

      9. *-commutative 76.1%

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

      10. associate-/l* 76.0%

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

      11. metadata-eval 76.0%

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

      12. metadata-eval 76.0%

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

   3. Simplified 76.0%

      \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 39.5%

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

      1. unpow2 39.5%

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

      2. *-commutative 39.5%

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

      3. unpow2 39.5%

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

      4. swap-sqr 47.2%

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

      5. unpow2 47.2%

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

      6. *-commutative 47.2%

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

      7. *-commutative 47.2%

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

      8. associate-*r* 47.2%

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

   7. Simplified 47.2%

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

   if 5.5999999999999999e-22 < a

   1. Initial program 84.6%

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

      1. associate-*r/ 84.7%

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

      2. metadata-eval 84.7%

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

      3. metadata-eval 84.7%

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

      4. distribute-neg-frac2 84.7%

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

      5. distribute-frac-neg 84.7%

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

      6. distribute-rgt-neg-out 84.7%

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

      7. associate-/l* 84.6%

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

      8. neg-mul-1 84.6%

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

      9. *-commutative 84.6%

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

      10. associate-/l* 84.7%

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

      11. metadata-eval 84.7%

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

      12. metadata-eval 84.7%

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

   3. Simplified 84.7%

      \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0 77.3%

      \[\leadsto \color{blue}{{a}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 85.5%

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

   7. Applied egg-rr 77.3%

      \[\leadsto \color{blue}{a \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.6 \cdot 10^{-22}:\\ \;\;\;\;{\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.8 \cdot 10^{-22}:\\ \;\;\;\;{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.8e-22)
   (pow (* b (sin (* 0.005555555555555556 (* angle PI)))) 2.0)
   (* a a)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.8e-22) {
		tmp = pow((b * sin((0.005555555555555556 * (angle * ((double) M_PI))))), 2.0);
	} else {
		tmp = a * a;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.8e-22) {
		tmp = Math.pow((b * Math.sin((0.005555555555555556 * (angle * Math.PI)))), 2.0);
	} else {
		tmp = a * a;
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 1.8e-22:
		tmp = math.pow((b * math.sin((0.005555555555555556 * (angle * math.pi)))), 2.0)
	else:
		tmp = a * a
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.8e-22)
		tmp = Float64(b * sin(Float64(0.005555555555555556 * Float64(angle * pi)))) ^ 2.0;
	else
		tmp = Float64(a * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1.8e-22)
		tmp = (b * sin((0.005555555555555556 * (angle * pi)))) ^ 2.0;
	else
		tmp = a * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.8e-22], N[Power[N[(b * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(a * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.7999999999999999e-22

   1. Initial program 76.1%

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

      1. associate-*r/ 76.0%

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

      2. metadata-eval 76.0%

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

      3. metadata-eval 76.0%

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

      4. distribute-neg-frac2 76.0%

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

      5. distribute-frac-neg 76.0%

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

      6. distribute-rgt-neg-out 76.0%

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

      7. associate-/l* 76.1%

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

      8. neg-mul-1 76.1%

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

      9. *-commutative 76.1%

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

      10. associate-/l* 76.0%

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

      11. metadata-eval 76.0%

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

      12. metadata-eval 76.0%

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

   3. Simplified 76.0%

      \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 39.5%

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

      1. unpow2 39.5%

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

      2. *-commutative 39.5%

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

      3. unpow2 39.5%

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

      4. swap-sqr 47.2%

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

      5. unpow2 47.2%

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

      6. *-commutative 47.2%

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

   7. Simplified 47.2%

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

   if 1.7999999999999999e-22 < a

   1. Initial program 84.6%

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

      1. associate-*r/ 84.7%

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

      2. metadata-eval 84.7%

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

      3. metadata-eval 84.7%

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

      4. distribute-neg-frac2 84.7%

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

      5. distribute-frac-neg 84.7%

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

      6. distribute-rgt-neg-out 84.7%

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

      7. associate-/l* 84.6%

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

      8. neg-mul-1 84.6%

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

      9. *-commutative 84.6%

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

      10. associate-/l* 84.7%

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

      11. metadata-eval 84.7%

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

      12. metadata-eval 84.7%

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

   3. Simplified 84.7%

      \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0 77.3%

      \[\leadsto \color{blue}{{a}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 85.5%

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

   7. Applied egg-rr 77.3%

      \[\leadsto \color{blue}{a \cdot a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 79.2% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+ (* a a) (pow (* b (sin (* PI (* angle 0.005555555555555556)))) 2.0)))

C

double code(double a, double b, double angle) {
	return (a * a) + pow((b * sin((((double) M_PI) * (angle * 0.005555555555555556)))), 2.0);
}

Java

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

Python

def code(a, b, angle):
	return (a * a) + math.pow((b * math.sin((math.pi * (angle * 0.005555555555555556)))), 2.0)

Julia

function code(a, b, angle)
	return Float64(Float64(a * a) + (Float64(b * sin(Float64(pi * Float64(angle * 0.005555555555555556)))) ^ 2.0))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. associate-*r/ 78.4%

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

   2. metadata-eval 78.4%

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

   3. metadata-eval 78.4%

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

   4. distribute-neg-frac2 78.4%

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

   5. distribute-frac-neg 78.4%

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

   6. distribute-rgt-neg-out 78.4%

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

   7. associate-/l* 78.4%

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

   8. neg-mul-1 78.4%

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

   9. *-commutative 78.4%

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

   10. associate-/l* 78.4%

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

   11. metadata-eval 78.4%

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

   12. metadata-eval 78.4%

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

3. Simplified 78.4%

   \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \]
4. Add Preprocessing

5. Taylor expanded in angle around 0 78.7%

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

   1. unpow2 78.7%

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

7. Applied egg-rr 78.7%

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

Alternative 7: 56.9% accurate, 139.0× speedup

Math

\[\begin{array}{l} \\ a \cdot a \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b, angle)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    code = a * a
end function

Java

public static double code(double a, double b, double angle) {
	return a * a;
}

Python

def code(a, b, angle):
	return a * a

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[(a * a), $MachinePrecision]

Derivation

1. Initial program 78.4%

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

   1. associate-*r/ 78.4%

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

   2. metadata-eval 78.4%

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

   3. metadata-eval 78.4%

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

   4. distribute-neg-frac2 78.4%

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

   5. distribute-frac-neg 78.4%

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

   6. distribute-rgt-neg-out 78.4%

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

   7. associate-/l* 78.4%

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

   8. neg-mul-1 78.4%

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

   9. *-commutative 78.4%

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

   10. associate-/l* 78.4%

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

   11. metadata-eval 78.4%

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

   12. metadata-eval 78.4%

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

3. Simplified 78.4%

   \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \]
4. Add Preprocessing

5. Taylor expanded in angle around 0 53.9%

   \[\leadsto \color{blue}{{a}^{2}} \]
6. Step-by-step derivation

   1. unpow2 78.7%

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

7. Applied egg-rr 53.9%

   \[\leadsto \color{blue}{a \cdot a} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024185 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))