ab-angle->ABCF C

Percentage Accurate: 79.6% → 79.4%

Time: 16.4s

Alternatives: 15

Speedup: N/A×

• 36.9% 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: 79.6% 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: 79.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ a \cdot a + {\left(b \cdot \sin \left(\frac{\frac{\pi}{\frac{-1}{angle}}}{-180}\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+ (* a a) (pow (* b (sin (/ (/ PI (/ -1.0 angle)) -180.0))) 2.0)))

C

double code(double a, double b, double angle) {
	return (a * a) + pow((b * sin(((((double) M_PI) / (-1.0 / angle)) / -180.0))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return (a * a) + Math.pow((b * Math.sin(((Math.PI / (-1.0 / angle)) / -180.0))), 2.0);
}

Python

def code(a, b, angle):
	return (a * a) + math.pow((b * math.sin(((math.pi / (-1.0 / angle)) / -180.0))), 2.0)

Julia

function code(a, b, angle)
	return Float64(Float64(a * a) + (Float64(b * sin(Float64(Float64(pi / Float64(-1.0 / angle)) / -180.0))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (a * a) + ((b * sin(((pi / (-1.0 / angle)) / -180.0))) ^ 2.0);
end

Wolfram

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

Derivation

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. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right), \color{blue}{\left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\color{blue}{\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\left(\color{blue}{b} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{2}\right)\right) \]

3. Simplified 82.1%

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

5. Taylor expanded in angle around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right)\right) \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

   2. *-lowering-*.f64 82.5%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

7. Simplified 82.5%

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

   1. associate-*l/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{180} \cdot angle\right)\right)\right), 2\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right), 2\right)\right) \]

   3. *-un-lft-identity N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{1 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right), 2\right)\right) \]

   4. div-inv N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{1 \cdot \mathsf{PI}\left(\right)}{180 \cdot \frac{1}{angle}}\right)\right)\right), 2\right)\right) \]

   5. times-frac N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{angle}}\right)\right)\right), 2\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{angle}}\right)\right)\right), 2\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \left(\frac{\mathsf{PI}\left(\right)}{\frac{1}{angle}}\right)\right)\right)\right), 2\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(\frac{1}{angle}\right)\right)\right)\right)\right), 2\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{1}{angle}\right)\right)\right)\right)\right), 2\right)\right) \]

   10. /-lowering-/.f64 82.5%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(1, angle\right)\right)\right)\right)\right), 2\right)\right) \]

9. Applied egg-rr 82.5%

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

    1. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{angle}}\right)\right)\right), 2\right)\right) \]

    2. times-frac N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{1 \cdot \mathsf{PI}\left(\right)}{180 \cdot \frac{1}{angle}}\right)\right)\right), 2\right)\right) \]

    3. div-inv N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{1 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right), 2\right)\right) \]

    4. *-un-lft-identity N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right), 2\right)\right) \]

    5. associate-/r/ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{180} \cdot angle\right)\right)\right), 2\right)\right) \]

    6. associate-*l/ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), 2\right)\right) \]

    7. add-sqr-sqrt N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot angle}{180}\right)\right)\right), 2\right)\right) \]

    8. associate-*r* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot angle\right)}{180}\right)\right)\right), 2\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot \left(angle \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}{180}\right)\right)\right), 2\right)\right) \]

    10. rem-exp-log N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\log \left(angle \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}}{180}\right)\right)\right), 2\right)\right) \]

    11. frac-2neg N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\log \left(angle \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)}{\mathsf{neg}\left(180\right)}\right)\right)\right), 2\right)\right) \]

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

        \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\log \left(angle \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right), \left(\mathsf{neg}\left(180\right)\right)\right)\right)\right), 2\right)\right) \]

11. Applied egg-rr 82.6%

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

Alternative 2: 79.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right), \color{blue}{\left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\color{blue}{\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\left(\color{blue}{b} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{2}\right)\right) \]

3. Simplified 82.1%

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

5. Taylor expanded in angle around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right)\right) \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

   2. *-lowering-*.f64 82.5%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

7. Simplified 82.5%

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

Alternative 3: 79.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   2. *-lowering-*.f64 82.4%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

5. Simplified 82.4%

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

Alternative 4: 77.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 5 \cdot 10^{-5}:\\ \;\;\;\;a \cdot a + {\left(angle \cdot \left(b \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + b \cdot \left(b \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\pi}{\frac{180}{angle}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 5e-5)
   (+ (* a a) (pow (* angle (* b (* PI 0.005555555555555556))) 2.0))
   (+
    (* a a)
    (* b (* b (- 0.5 (* 0.5 (cos (* 2.0 (/ PI (/ 180.0 angle)))))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 5e-5) {
		tmp = (a * a) + pow((angle * (b * (((double) M_PI) * 0.005555555555555556))), 2.0);
	} else {
		tmp = (a * a) + (b * (b * (0.5 - (0.5 * cos((2.0 * (((double) M_PI) / (180.0 / angle))))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 5e-5) {
		tmp = (a * a) + Math.pow((angle * (b * (Math.PI * 0.005555555555555556))), 2.0);
	} else {
		tmp = (a * a) + (b * (b * (0.5 - (0.5 * Math.cos((2.0 * (Math.PI / (180.0 / angle))))))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 5e-5:
		tmp = (a * a) + math.pow((angle * (b * (math.pi * 0.005555555555555556))), 2.0)
	else:
		tmp = (a * a) + (b * (b * (0.5 - (0.5 * math.cos((2.0 * (math.pi / (180.0 / angle))))))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 5e-5)
		tmp = Float64(Float64(a * a) + (Float64(angle * Float64(b * Float64(pi * 0.005555555555555556))) ^ 2.0));
	else
		tmp = Float64(Float64(a * a) + Float64(b * Float64(b * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(pi / Float64(180.0 / angle)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 5e-5)
		tmp = (a * a) + ((angle * (b * (pi * 0.005555555555555556))) ^ 2.0);
	else
		tmp = (a * a) + (b * (b * (0.5 - (0.5 * cos((2.0 * (pi / (180.0 / angle))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 5e-5], N[(N[(a * a), $MachinePrecision] + N[Power[N[(angle * N[(b * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(b * N[(b * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 5.00000000000000024e-5

   1. Initial program 87.4%

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

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

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right), \color{blue}{\left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\color{blue}{\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\left(\color{blue}{b} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{2}\right)\right) \]

   3. Simplified 87.5%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 87.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

   7. Simplified 87.9%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, 2\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(\left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{180}\right), 2\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(angle \cdot \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right), 2\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(angle \cdot \left(\frac{1}{180} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{180} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right), 2\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{180}\right)\right)\right), 2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(b \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

      10. PI-lowering-PI.f64 83.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right)\right) \]

   10. Simplified 83.9%

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

   if 5.00000000000000024e-5 < angle

   1. Initial program 61.2%

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

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

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right), \color{blue}{\left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\color{blue}{\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\left(\color{blue}{b} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{2}\right)\right) \]

   3. Simplified 61.3%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 61.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

   7. Simplified 61.7%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-/r/ N/A

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

      7. associate-*r* N/A

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

      8. associate-*l/ N/A

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

      9. associate-/r/ N/A

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

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

          \[\leadsto \mathsf{+.f64}\left(\left(\left(\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b \cdot b\right)\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), \color{blue}{\left(a \cdot a\right)}\right) \]

   9. Applied egg-rr 61.6%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\pi}{\frac{180}{angle}}\right)\right)\right) + a \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 5 \cdot 10^{-5}:\\ \;\;\;\;a \cdot a + {\left(angle \cdot \left(b \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + b \cdot \left(b \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\pi}{\frac{180}{angle}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 0.0038:\\ \;\;\;\;a \cdot a + {\left(angle \cdot \left(b \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 0.0038)
   (+ (* a a) (pow (* angle (* b (* PI 0.005555555555555556))) 2.0))
   (+
    (* a a)
    (* (- 0.5 (* 0.5 (cos (* 2.0 (/ PI (/ 180.0 angle)))))) (* b b)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 0.0038) {
		tmp = (a * a) + pow((angle * (b * (((double) M_PI) * 0.005555555555555556))), 2.0);
	} else {
		tmp = (a * a) + ((0.5 - (0.5 * cos((2.0 * (((double) M_PI) / (180.0 / angle)))))) * (b * b));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 0.0038) {
		tmp = (a * a) + Math.pow((angle * (b * (Math.PI * 0.005555555555555556))), 2.0);
	} else {
		tmp = (a * a) + ((0.5 - (0.5 * Math.cos((2.0 * (Math.PI / (180.0 / angle)))))) * (b * b));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 0.0038:
		tmp = (a * a) + math.pow((angle * (b * (math.pi * 0.005555555555555556))), 2.0)
	else:
		tmp = (a * a) + ((0.5 - (0.5 * math.cos((2.0 * (math.pi / (180.0 / angle)))))) * (b * b))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 0.0038)
		tmp = Float64(Float64(a * a) + (Float64(angle * Float64(b * Float64(pi * 0.005555555555555556))) ^ 2.0));
	else
		tmp = Float64(Float64(a * a) + Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(pi / Float64(180.0 / angle)))))) * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 0.0038)
		tmp = (a * a) + ((angle * (b * (pi * 0.005555555555555556))) ^ 2.0);
	else
		tmp = (a * a) + ((0.5 - (0.5 * cos((2.0 * (pi / (180.0 / angle)))))) * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 0.0038], N[(N[(a * a), $MachinePrecision] + N[Power[N[(angle * N[(b * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 0.00379999999999999999

   1. Initial program 87.4%

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

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

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right), \color{blue}{\left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\color{blue}{\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\left(\color{blue}{b} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{2}\right)\right) \]

   3. Simplified 87.5%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 87.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

   7. Simplified 87.9%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, 2\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(\left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{180}\right), 2\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(angle \cdot \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right), 2\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(angle \cdot \left(\frac{1}{180} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{180} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right), 2\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{180}\right)\right)\right), 2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(b \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

      10. PI-lowering-PI.f64 83.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right)\right) \]

   10. Simplified 83.9%

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

   if 0.00379999999999999999 < angle

   1. Initial program 61.2%

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

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

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right), \color{blue}{\left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\color{blue}{\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\left(\color{blue}{b} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{2}\right)\right) \]

   3. Simplified 61.3%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 61.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

   7. Simplified 61.7%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot b\right)}^{2}\right)\right) \]

      2. unpow-prod-down N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}^{2} \cdot \color{blue}{{b}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}^{2}\right), \color{blue}{\left({b}^{2}\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left({\color{blue}{b}}^{2}\right)\right)\right) \]

      5. sqr-sin-a N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left({\color{blue}{b}}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \cos \left(2 \cdot \frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), \left({\color{blue}{b}}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \cos \left(2 \cdot \frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), \left({b}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\left(2 \cdot \frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right)\right), \left({b}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right)\right)\right), \left({b}^{2}\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{\mathsf{PI}\left(\right)}{180} \cdot angle\right)\right)\right)\right)\right), \left({b}^{2}\right)\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)\right)\right), \left({b}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(\frac{180}{angle}\right)\right)\right)\right)\right)\right), \left({b}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{180}{angle}\right)\right)\right)\right)\right)\right), \left({b}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right), \left({b}^{2}\right)\right)\right) \]

      15. pow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right), \left(b \cdot \color{blue}{b}\right)\right)\right) \]

      16. *-lowering-*.f64 61.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   9. Applied egg-rr 61.6%

      \[\leadsto a \cdot a + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 0.0038:\\ \;\;\;\;a \cdot a + {\left(angle \cdot \left(b \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 0.0038:\\ \;\;\;\;a \cdot a + {\left(angle \cdot \left(b \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(b \cdot b\right) \cdot \left(0.5 + \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 0.0038)
   (+ (* a a) (pow (* angle (* b (* PI 0.005555555555555556))) 2.0))
   (+
    (* a a)
    (* (* b b) (+ 0.5 (* (cos (* (* PI angle) 0.011111111111111112)) -0.5))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 0.0038) {
		tmp = (a * a) + pow((angle * (b * (((double) M_PI) * 0.005555555555555556))), 2.0);
	} else {
		tmp = (a * a) + ((b * b) * (0.5 + (cos(((((double) M_PI) * angle) * 0.011111111111111112)) * -0.5)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 0.0038) {
		tmp = (a * a) + Math.pow((angle * (b * (Math.PI * 0.005555555555555556))), 2.0);
	} else {
		tmp = (a * a) + ((b * b) * (0.5 + (Math.cos(((Math.PI * angle) * 0.011111111111111112)) * -0.5)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 0.0038:
		tmp = (a * a) + math.pow((angle * (b * (math.pi * 0.005555555555555556))), 2.0)
	else:
		tmp = (a * a) + ((b * b) * (0.5 + (math.cos(((math.pi * angle) * 0.011111111111111112)) * -0.5)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 0.0038)
		tmp = Float64(Float64(a * a) + (Float64(angle * Float64(b * Float64(pi * 0.005555555555555556))) ^ 2.0));
	else
		tmp = Float64(Float64(a * a) + Float64(Float64(b * b) * Float64(0.5 + Float64(cos(Float64(Float64(pi * angle) * 0.011111111111111112)) * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 0.0038)
		tmp = (a * a) + ((angle * (b * (pi * 0.005555555555555556))) ^ 2.0);
	else
		tmp = (a * a) + ((b * b) * (0.5 + (cos(((pi * angle) * 0.011111111111111112)) * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 0.0038], N[(N[(a * a), $MachinePrecision] + N[Power[N[(angle * N[(b * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(0.5 + N[(N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 0.00379999999999999999

   1. Initial program 87.4%

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

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

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right), \color{blue}{\left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\color{blue}{\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\left(\color{blue}{b} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{2}\right)\right) \]

   3. Simplified 87.5%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 87.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

   7. Simplified 87.9%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, 2\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(\left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{180}\right), 2\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(angle \cdot \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right), 2\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(angle \cdot \left(\frac{1}{180} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{180} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right), 2\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{180}\right)\right)\right), 2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(b \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

      10. PI-lowering-PI.f64 83.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right)\right) \]

   10. Simplified 83.9%

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

   if 0.00379999999999999999 < angle

   1. Initial program 61.2%

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

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

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right), \color{blue}{\left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\color{blue}{\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\left(\color{blue}{b} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{2}\right)\right) \]

   3. Simplified 61.3%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 61.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

   7. Simplified 61.7%

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

   9. Applied egg-rr 4.1%

      \[\leadsto \color{blue}{\frac{\left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) - {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{8}\right) \cdot 1}{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{4}\right) \cdot \left(a \cdot a - b \cdot \left(b \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\pi}{\frac{180}{angle}}\right)\right)\right)\right)}} \]

   10. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

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

          \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left({b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{{b}^{2}} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{{b}^{2}} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

       8. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

       11. distribute-rgt-neg-in N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right), \frac{1}{90}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right), \frac{1}{90}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), \frac{1}{90}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right) \]

       18. metadata-eval 61.8%

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), \frac{1}{90}\right)\right), \frac{-1}{2}\right)\right)\right)\right) \]

   12. Simplified 61.8%

       \[\leadsto \color{blue}{a \cdot a + \left(b \cdot b\right) \cdot \left(0.5 + \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot -0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 0.0038:\\ \;\;\;\;a \cdot a + {\left(angle \cdot \left(b \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(b \cdot b\right) \cdot \left(0.5 + \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.28 \cdot 10^{-23}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(angle \cdot \left(b \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.28e-23)
   (* a a)
   (+ (* a a) (pow (* angle (* b (* PI 0.005555555555555556))) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.28e-23) {
		tmp = a * a;
	} else {
		tmp = (a * a) + pow((angle * (b * (((double) M_PI) * 0.005555555555555556))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.28e-23) {
		tmp = a * a;
	} else {
		tmp = (a * a) + Math.pow((angle * (b * (Math.PI * 0.005555555555555556))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.28e-23:
		tmp = a * a
	else:
		tmp = (a * a) + math.pow((angle * (b * (math.pi * 0.005555555555555556))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.28e-23)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + (Float64(angle * Float64(b * Float64(pi * 0.005555555555555556))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.28e-23)
		tmp = a * a;
	else
		tmp = (a * a) + ((angle * (b * (pi * 0.005555555555555556))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.28e-23], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[Power[N[(angle * N[(b * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.28000000000000005e-23

   1. Initial program 83.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

         \[\leadsto a \cdot \color{blue}{a} \]

      2. *-lowering-*.f64 69.1%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

   5. Simplified 69.1%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 1.28000000000000005e-23 < b

   1. Initial program 76.8%

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

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

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right), \color{blue}{\left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\color{blue}{\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\left(\color{blue}{b} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{2}\right)\right) \]

   3. Simplified 76.9%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 78.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

   7. Simplified 78.3%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, 2\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(\left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{180}\right), 2\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(angle \cdot \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right), 2\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(angle \cdot \left(\frac{1}{180} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{180} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right), 2\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{180}\right)\right)\right), 2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \left(b \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI}\left(\right)\right)\right)\right), 2\right)\right) \]

      10. PI-lowering-PI.f64 72.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right)\right) \]

   10. Simplified 72.3%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.28 \cdot 10^{-23}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(angle \cdot \left(b \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.6% accurate, 18.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-23}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(b \cdot angle\right)\right) \cdot \left(b \cdot angle\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.15e-23)
   (* a a)
   (+
    (* a a)
    (* (* (* PI (* PI 3.08641975308642e-5)) (* b angle)) (* b angle)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.15e-23) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (((((double) M_PI) * (((double) M_PI) * 3.08641975308642e-5)) * (b * angle)) * (b * angle));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.15e-23) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (((Math.PI * (Math.PI * 3.08641975308642e-5)) * (b * angle)) * (b * angle));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.15e-23:
		tmp = a * a
	else:
		tmp = (a * a) + (((math.pi * (math.pi * 3.08641975308642e-5)) * (b * angle)) * (b * angle))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.15e-23)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + Float64(Float64(Float64(pi * Float64(pi * 3.08641975308642e-5)) * Float64(b * angle)) * Float64(b * angle)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.15e-23)
		tmp = a * a;
	else
		tmp = (a * a) + (((pi * (pi * 3.08641975308642e-5)) * (b * angle)) * (b * angle));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.15e-23], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(N[(N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] * N[(b * angle), $MachinePrecision]), $MachinePrecision] * N[(b * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.15000000000000005e-23

   1. Initial program 83.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

         \[\leadsto a \cdot \color{blue}{a} \]

      2. *-lowering-*.f64 69.1%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

   5. Simplified 69.1%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 1.15000000000000005e-23 < b

   1. Initial program 76.8%

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

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

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right), \color{blue}{\left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\color{blue}{\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\left(\color{blue}{b} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{2}\right)\right) \]

   3. Simplified 76.9%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 78.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

   7. Simplified 78.3%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400} + {\color{blue}{a}}^{2} \]

      2. associate-*r* N/A

         \[\leadsto {angle}^{2} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right) + {\color{blue}{a}}^{2} \]

      3. *-commutative N/A

         \[\leadsto {angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2} \]

      4. +-commutative N/A

         \[\leadsto {a}^{2} + \color{blue}{{angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({angle}^{2} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({angle}^{2} \cdot {b}^{2}\right), \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({b}^{2} \cdot {angle}^{2}\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left({angle}^{2}\right)\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

   10. Simplified 64.7%

       \[\leadsto \color{blue}{a \cdot a + \left(\left(b \cdot b\right) \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right) \cdot \color{blue}{\left(\left(b \cdot b\right) \cdot \left(angle \cdot angle\right)\right)}\right)\right) \]

       2. unswap-sqr N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right) \cdot \left(\left(b \cdot angle\right) \cdot \color{blue}{\left(b \cdot angle\right)}\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right) \cdot \left(b \cdot angle\right)\right) \cdot \color{blue}{\left(b \cdot angle\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right) \cdot \left(b \cdot angle\right)\right), \color{blue}{\left(b \cdot angle\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right), \left(b \cdot angle\right)\right), \left(\color{blue}{b} \cdot angle\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right), \left(b \cdot angle\right)\right), \left(b \cdot angle\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right), \left(b \cdot angle\right)\right), \left(b \cdot angle\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right), \left(b \cdot angle\right)\right), \left(b \cdot angle\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \frac{1}{32400}\right)\right), \left(b \cdot angle\right)\right), \left(b \cdot angle\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right)\right), \left(b \cdot angle\right)\right), \left(b \cdot angle\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right)\right), \mathsf{*.f64}\left(b, angle\right)\right), \left(b \cdot angle\right)\right)\right) \]

       12. *-lowering-*.f64 72.3%

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right)\right), \mathsf{*.f64}\left(b, angle\right)\right), \mathsf{*.f64}\left(b, \color{blue}{angle}\right)\right)\right) \]

   12. Applied egg-rr 72.3%

       \[\leadsto a \cdot a + \color{blue}{\left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(b \cdot angle\right)\right) \cdot \left(b \cdot angle\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 64.8% accurate, 18.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.8 \cdot 10^{-23}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(b \cdot \left(b \cdot \left(angle \cdot angle\right)\right)\right) \cdot \left(\pi \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 9.8e-23)
   (* a a)
   (+
    (* a a)
    (* 3.08641975308642e-5 (* (* b (* b (* angle angle))) (* PI PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 9.8e-23) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (3.08641975308642e-5 * ((b * (b * (angle * angle))) * (((double) M_PI) * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 9.8e-23) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (3.08641975308642e-5 * ((b * (b * (angle * angle))) * (Math.PI * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 9.8e-23:
		tmp = a * a
	else:
		tmp = (a * a) + (3.08641975308642e-5 * ((b * (b * (angle * angle))) * (math.pi * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 9.8e-23)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + Float64(3.08641975308642e-5 * Float64(Float64(b * Float64(b * Float64(angle * angle))) * Float64(pi * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 9.8e-23)
		tmp = a * a;
	else
		tmp = (a * a) + (3.08641975308642e-5 * ((b * (b * (angle * angle))) * (pi * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 9.8e-23], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(3.08641975308642e-5 * N[(N[(b * N[(b * N[(angle * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.7999999999999996e-23

   1. Initial program 83.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

         \[\leadsto a \cdot \color{blue}{a} \]

      2. *-lowering-*.f64 69.1%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

   5. Simplified 69.1%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 9.7999999999999996e-23 < b

   1. Initial program 76.8%

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

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

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right), \color{blue}{\left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\color{blue}{\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\left(\color{blue}{b} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{2}\right)\right) \]

   3. Simplified 76.9%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 78.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

   7. Simplified 78.3%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400} + {\color{blue}{a}}^{2} \]

      2. associate-*r* N/A

         \[\leadsto {angle}^{2} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right) + {\color{blue}{a}}^{2} \]

      3. *-commutative N/A

         \[\leadsto {angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2} \]

      4. +-commutative N/A

         \[\leadsto {a}^{2} + \color{blue}{{angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({angle}^{2} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({angle}^{2} \cdot {b}^{2}\right), \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({b}^{2} \cdot {angle}^{2}\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left({angle}^{2}\right)\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

   10. Simplified 64.7%

       \[\leadsto \color{blue}{a \cdot a + \left(\left(b \cdot b\right) \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left(\left(b \cdot b\right) \cdot \left(angle \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left(\left(\left(b \cdot b\right) \cdot \left(angle \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\frac{1}{32400}}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(b \cdot b\right) \cdot \left(angle \cdot angle\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{32400}\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot \left(b \cdot \left(angle \cdot angle\right)\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{32400}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left(angle \cdot angle\right)\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{32400}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(angle \cdot angle\right)\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{32400}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, angle\right)\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{32400}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, angle\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right), \frac{1}{32400}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, angle\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right), \frac{1}{32400}\right)\right) \]

       10. PI-lowering-PI.f64 70.6%

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, angle\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right), \frac{1}{32400}\right)\right) \]

   12. Applied egg-rr 70.6%

       \[\leadsto a \cdot a + \color{blue}{\left(\left(b \cdot \left(b \cdot \left(angle \cdot angle\right)\right)\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.8 \cdot 10^{-23}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(b \cdot \left(b \cdot \left(angle \cdot angle\right)\right)\right) \cdot \left(\pi \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.8% accurate, 18.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.4 \cdot 10^{-23}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(\pi \cdot \pi\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot \left(b \cdot \left(angle \cdot angle\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 8.4e-23)
   (* a a)
   (+
    (* a a)
    (* (* PI PI) (* 3.08641975308642e-5 (* b (* b (* angle angle))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 8.4e-23) {
		tmp = a * a;
	} else {
		tmp = (a * a) + ((((double) M_PI) * ((double) M_PI)) * (3.08641975308642e-5 * (b * (b * (angle * angle)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 8.4e-23) {
		tmp = a * a;
	} else {
		tmp = (a * a) + ((Math.PI * Math.PI) * (3.08641975308642e-5 * (b * (b * (angle * angle)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 8.4e-23:
		tmp = a * a
	else:
		tmp = (a * a) + ((math.pi * math.pi) * (3.08641975308642e-5 * (b * (b * (angle * angle)))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 8.4e-23)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + Float64(Float64(pi * pi) * Float64(3.08641975308642e-5 * Float64(b * Float64(b * Float64(angle * angle))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 8.4e-23)
		tmp = a * a;
	else
		tmp = (a * a) + ((pi * pi) * (3.08641975308642e-5 * (b * (b * (angle * angle)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 8.4e-23], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(N[(Pi * Pi), $MachinePrecision] * N[(3.08641975308642e-5 * N[(b * N[(b * N[(angle * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.4000000000000003e-23

   1. Initial program 83.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

         \[\leadsto a \cdot \color{blue}{a} \]

      2. *-lowering-*.f64 69.1%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

   5. Simplified 69.1%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 8.4000000000000003e-23 < b

   1. Initial program 76.8%

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

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

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right), \color{blue}{\left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\color{blue}{\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\left(\color{blue}{b} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{2}\right)\right) \]

   3. Simplified 76.9%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 78.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

   7. Simplified 78.3%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400} + {\color{blue}{a}}^{2} \]

      2. associate-*r* N/A

         \[\leadsto {angle}^{2} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right) + {\color{blue}{a}}^{2} \]

      3. *-commutative N/A

         \[\leadsto {angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2} \]

      4. +-commutative N/A

         \[\leadsto {a}^{2} + \color{blue}{{angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({angle}^{2} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({angle}^{2} \cdot {b}^{2}\right), \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({b}^{2} \cdot {angle}^{2}\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left({angle}^{2}\right)\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

   10. Simplified 64.7%

       \[\leadsto \color{blue}{a \cdot a + \left(\left(b \cdot b\right) \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left(b \cdot b\right) \cdot \left(angle \cdot angle\right)\right) \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left(\left(b \cdot b\right) \cdot \left(angle \cdot angle\right)\right) \cdot \frac{1}{32400}\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left(\left(\left(b \cdot b\right) \cdot \left(angle \cdot angle\right)\right) \cdot \frac{1}{32400}\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(b \cdot b\right) \cdot \left(angle \cdot angle\right)\right), \frac{1}{32400}\right), \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot \left(b \cdot \left(angle \cdot angle\right)\right)\right), \frac{1}{32400}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left(angle \cdot angle\right)\right)\right), \frac{1}{32400}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(angle \cdot angle\right)\right)\right), \frac{1}{32400}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, angle\right)\right)\right), \frac{1}{32400}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, angle\right)\right)\right), \frac{1}{32400}\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, angle\right)\right)\right), \frac{1}{32400}\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right) \]

       11. PI-lowering-PI.f64 70.6%

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, angle\right)\right)\right), \frac{1}{32400}\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

   12. Applied egg-rr 70.6%

       \[\leadsto a \cdot a + \color{blue}{\left(\left(b \cdot \left(b \cdot \left(angle \cdot angle\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\pi \cdot \pi\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.4 \cdot 10^{-23}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(\pi \cdot \pi\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot \left(b \cdot \left(angle \cdot angle\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.8% accurate, 18.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.35 \cdot 10^{-23}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \pi \cdot \left(\left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(b \cdot \left(b \cdot \left(angle \cdot angle\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 2.35e-23)
   (* a a)
   (+
    (* a a)
    (* PI (* (* PI 3.08641975308642e-5) (* b (* b (* angle angle))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2.35e-23) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (((double) M_PI) * ((((double) M_PI) * 3.08641975308642e-5) * (b * (b * (angle * angle)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2.35e-23) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (Math.PI * ((Math.PI * 3.08641975308642e-5) * (b * (b * (angle * angle)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 2.35e-23:
		tmp = a * a
	else:
		tmp = (a * a) + (math.pi * ((math.pi * 3.08641975308642e-5) * (b * (b * (angle * angle)))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 2.35e-23)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + Float64(pi * Float64(Float64(pi * 3.08641975308642e-5) * Float64(b * Float64(b * Float64(angle * angle))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 2.35e-23)
		tmp = a * a;
	else
		tmp = (a * a) + (pi * ((pi * 3.08641975308642e-5) * (b * (b * (angle * angle)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 2.35e-23], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(Pi * N[(N[(Pi * 3.08641975308642e-5), $MachinePrecision] * N[(b * N[(b * N[(angle * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.35e-23

   1. Initial program 83.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

         \[\leadsto a \cdot \color{blue}{a} \]

      2. *-lowering-*.f64 69.1%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

   5. Simplified 69.1%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 2.35e-23 < b

   1. Initial program 76.8%

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

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

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right), \color{blue}{\left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\color{blue}{\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\left(\color{blue}{b} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{2}\right)\right) \]

   3. Simplified 76.9%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 78.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

   7. Simplified 78.3%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400} + {\color{blue}{a}}^{2} \]

      2. associate-*r* N/A

         \[\leadsto {angle}^{2} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right) + {\color{blue}{a}}^{2} \]

      3. *-commutative N/A

         \[\leadsto {angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2} \]

      4. +-commutative N/A

         \[\leadsto {a}^{2} + \color{blue}{{angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({angle}^{2} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({angle}^{2} \cdot {b}^{2}\right), \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({b}^{2} \cdot {angle}^{2}\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left({angle}^{2}\right)\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

   10. Simplified 64.7%

       \[\leadsto \color{blue}{a \cdot a + \left(\left(b \cdot b\right) \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right) \cdot \color{blue}{\left(\left(b \cdot b\right) \cdot \left(angle \cdot angle\right)\right)}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \left(angle \cdot angle\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right) \cdot \left(\left(b \cdot b\right) \cdot \left(angle \cdot angle\right)\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right) \cdot \left(\left(b \cdot b\right) \cdot \left(angle \cdot angle\right)\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)} \cdot \left(\left(b \cdot b\right) \cdot \left(angle \cdot angle\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right), \color{blue}{\left(\left(b \cdot b\right) \cdot \left(angle \cdot angle\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \frac{1}{32400}\right), \left(\color{blue}{\left(b \cdot b\right)} \cdot \left(angle \cdot angle\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right), \left(\left(\color{blue}{b} \cdot b\right) \cdot \left(angle \cdot angle\right)\right)\right)\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right), \left(b \cdot \color{blue}{\left(b \cdot \left(angle \cdot angle\right)\right)}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right), \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot \left(angle \cdot angle\right)\right)}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left(angle \cdot angle\right)}\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 70.5%

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, \color{blue}{angle}\right)\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 70.5%

       \[\leadsto a \cdot a + \color{blue}{\pi \cdot \left(\left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(b \cdot \left(b \cdot \left(angle \cdot angle\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 62.6% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.55 \cdot 10^{+147}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot angle\right) \cdot \left(angle \cdot \left(b \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.55e+147)
   (* a a)
   (* (* b angle) (* angle (* b (* PI (* PI 3.08641975308642e-5)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.55e+147) {
		tmp = a * a;
	} else {
		tmp = (b * angle) * (angle * (b * (((double) M_PI) * (((double) M_PI) * 3.08641975308642e-5))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.55e+147) {
		tmp = a * a;
	} else {
		tmp = (b * angle) * (angle * (b * (Math.PI * (Math.PI * 3.08641975308642e-5))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.55e+147:
		tmp = a * a
	else:
		tmp = (b * angle) * (angle * (b * (math.pi * (math.pi * 3.08641975308642e-5))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.55e+147)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(b * angle) * Float64(angle * Float64(b * Float64(pi * Float64(pi * 3.08641975308642e-5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.55e+147)
		tmp = a * a;
	else
		tmp = (b * angle) * (angle * (b * (pi * (pi * 3.08641975308642e-5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.55e+147], N[(a * a), $MachinePrecision], N[(N[(b * angle), $MachinePrecision] * N[(angle * N[(b * N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.55e147

   1. Initial program 80.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

         \[\leadsto a \cdot \color{blue}{a} \]

      2. *-lowering-*.f64 65.2%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

   5. Simplified 65.2%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 1.55e147 < b

   1. Initial program 96.6%

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

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

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right), \color{blue}{\left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\color{blue}{\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\left(\color{blue}{b} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{2}\right)\right) \]

   3. Simplified 96.6%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 96.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

   7. Simplified 96.6%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400} + {\color{blue}{a}}^{2} \]

      2. associate-*r* N/A

         \[\leadsto {angle}^{2} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right) + {\color{blue}{a}}^{2} \]

      3. *-commutative N/A

         \[\leadsto {angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2} \]

      4. +-commutative N/A

         \[\leadsto {a}^{2} + \color{blue}{{angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({angle}^{2} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({angle}^{2} \cdot {b}^{2}\right), \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({b}^{2} \cdot {angle}^{2}\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left({angle}^{2}\right)\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

   10. Simplified 76.8%

       \[\leadsto \color{blue}{a \cdot a + \left(\left(b \cdot b\right) \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)} \]

   11. Taylor expanded in a around 0

       \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}} \]

       2. associate-*r* N/A

          \[\leadsto {angle}^{2} \cdot \color{blue}{\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \]

       3. *-commutative N/A

          \[\leadsto {angle}^{2} \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({angle}^{2}\right), \color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot angle\right), \left(\color{blue}{\frac{1}{32400}} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\color{blue}{\frac{1}{32400}} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left({b}^{2} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left({b}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right)\right)\right)\right) \]

       10. distribute-rgt-neg-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left({b}^{2} \cdot \left(\mathsf{neg}\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{32400}\right)\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left({b}^{2} \cdot \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}}\right)\right)\right)\right) \]

       15. distribute-lft-neg-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right)\right) \]

       16. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{32400} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \color{blue}{\left({\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

       21. PI-lowering-PI.f64 76.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

   13. Simplified 76.8%

       \[\leadsto \color{blue}{\left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)} \]
   14. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto angle \cdot \color{blue}{\left(angle \cdot \left(\left(b \cdot b\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(angle \cdot \left(\left(b \cdot b\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \color{blue}{angle} \]

       3. associate-*l* N/A

          \[\leadsto \left(angle \cdot \left(b \cdot \left(b \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \cdot angle \]

       4. associate-*r* N/A

          \[\leadsto \left(\left(angle \cdot b\right) \cdot \left(b \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot angle \]

       5. associate-*l* N/A

          \[\leadsto \left(angle \cdot b\right) \cdot \color{blue}{\left(\left(b \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot angle\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot b\right), \color{blue}{\left(\left(b \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot angle\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \left(\color{blue}{\left(b \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot angle\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(\left(b \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \color{blue}{angle}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), angle\right)\right) \]

       10. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \left(\left(\frac{1}{32400} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right), angle\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \mathsf{PI}\left(\right)\right)\right)\right), angle\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\frac{1}{32400} \cdot \mathsf{PI}\left(\right)\right)\right)\right), angle\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{1}{32400} \cdot \mathsf{PI}\left(\right)\right)\right)\right), angle\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right), angle\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \frac{1}{32400}\right)\right)\right), angle\right)\right) \]

       16. PI-lowering-PI.f64 84.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right)\right)\right), angle\right)\right) \]

   15. Applied egg-rr 84.4%

       \[\leadsto \color{blue}{\left(angle \cdot b\right) \cdot \left(\left(b \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right) \cdot angle\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.55 \cdot 10^{+147}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot angle\right) \cdot \left(angle \cdot \left(b \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.6% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.48 \cdot 10^{+147}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(b \cdot angle\right)\right) \cdot \left(b \cdot angle\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.48e+147)
   (* a a)
   (* (* (* PI (* PI 3.08641975308642e-5)) (* b angle)) (* b angle))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.48e+147) {
		tmp = a * a;
	} else {
		tmp = ((((double) M_PI) * (((double) M_PI) * 3.08641975308642e-5)) * (b * angle)) * (b * angle);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.48e+147) {
		tmp = a * a;
	} else {
		tmp = ((Math.PI * (Math.PI * 3.08641975308642e-5)) * (b * angle)) * (b * angle);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.48e+147:
		tmp = a * a
	else:
		tmp = ((math.pi * (math.pi * 3.08641975308642e-5)) * (b * angle)) * (b * angle)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.48e+147)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(Float64(pi * Float64(pi * 3.08641975308642e-5)) * Float64(b * angle)) * Float64(b * angle));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.48e+147)
		tmp = a * a;
	else
		tmp = ((pi * (pi * 3.08641975308642e-5)) * (b * angle)) * (b * angle);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.48e+147], N[(a * a), $MachinePrecision], N[(N[(N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] * N[(b * angle), $MachinePrecision]), $MachinePrecision] * N[(b * angle), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.48000000000000002e147

   1. Initial program 80.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

         \[\leadsto a \cdot \color{blue}{a} \]

      2. *-lowering-*.f64 65.2%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

   5. Simplified 65.2%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 1.48000000000000002e147 < b

   1. Initial program 96.6%

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

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

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right), \color{blue}{\left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\color{blue}{\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\left(\color{blue}{b} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{2}\right)\right) \]

   3. Simplified 96.6%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 96.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

   7. Simplified 96.6%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400} + {\color{blue}{a}}^{2} \]

      2. associate-*r* N/A

         \[\leadsto {angle}^{2} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right) + {\color{blue}{a}}^{2} \]

      3. *-commutative N/A

         \[\leadsto {angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2} \]

      4. +-commutative N/A

         \[\leadsto {a}^{2} + \color{blue}{{angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({angle}^{2} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({angle}^{2} \cdot {b}^{2}\right), \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({b}^{2} \cdot {angle}^{2}\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left({angle}^{2}\right)\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

   10. Simplified 76.8%

       \[\leadsto \color{blue}{a \cdot a + \left(\left(b \cdot b\right) \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)} \]

   11. Taylor expanded in a around 0

       \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}} \]

       2. associate-*r* N/A

          \[\leadsto {angle}^{2} \cdot \color{blue}{\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \]

       3. *-commutative N/A

          \[\leadsto {angle}^{2} \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({angle}^{2}\right), \color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot angle\right), \left(\color{blue}{\frac{1}{32400}} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\color{blue}{\frac{1}{32400}} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left({b}^{2} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left({b}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right)\right)\right)\right) \]

       10. distribute-rgt-neg-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left({b}^{2} \cdot \left(\mathsf{neg}\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{32400}\right)\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left({b}^{2} \cdot \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}}\right)\right)\right)\right) \]

       15. distribute-lft-neg-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right)\right) \]

       16. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{32400} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \color{blue}{\left({\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

       21. PI-lowering-PI.f64 76.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

   13. Simplified 76.8%

       \[\leadsto \color{blue}{\left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)} \]
   14. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left({b}^{2} \cdot \left(\color{blue}{\frac{1}{32400}} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\left(angle \cdot angle\right) \cdot {b}^{2}\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)} \]

       3. pow2 N/A

          \[\leadsto \left(\left(angle \cdot angle\right) \cdot \left(b \cdot b\right)\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \]

       4. unswap-sqr N/A

          \[\leadsto \left(\left(angle \cdot b\right) \cdot \left(angle \cdot b\right)\right) \cdot \left(\color{blue}{\frac{1}{32400}} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \left(angle \cdot b\right) \cdot \color{blue}{\left(\left(angle \cdot b\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot b\right), \color{blue}{\left(\left(angle \cdot b\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \left(\color{blue}{\left(angle \cdot b\right)} \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(\left(angle \cdot b\right), \color{blue}{\left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \left(\color{blue}{\frac{1}{32400}} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

       10. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \left(\left(\frac{1}{32400} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(\frac{1}{32400} \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{\frac{1}{32400}} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\frac{1}{32400}}\right)\right)\right)\right) \]

       16. PI-lowering-PI.f64 84.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right)\right)\right)\right) \]

   15. Applied egg-rr 84.4%

       \[\leadsto \color{blue}{\left(angle \cdot b\right) \cdot \left(\left(angle \cdot b\right) \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.48 \cdot 10^{+147}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(b \cdot angle\right)\right) \cdot \left(b \cdot angle\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.1% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{+147}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(\left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(b \cdot \left(b \cdot \left(angle \cdot angle\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 2.1e+147)
   (* a a)
   (* PI (* (* PI 3.08641975308642e-5) (* b (* b (* angle angle)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2.1e+147) {
		tmp = a * a;
	} else {
		tmp = ((double) M_PI) * ((((double) M_PI) * 3.08641975308642e-5) * (b * (b * (angle * angle))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2.1e+147) {
		tmp = a * a;
	} else {
		tmp = Math.PI * ((Math.PI * 3.08641975308642e-5) * (b * (b * (angle * angle))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 2.1e+147:
		tmp = a * a
	else:
		tmp = math.pi * ((math.pi * 3.08641975308642e-5) * (b * (b * (angle * angle))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 2.1e+147)
		tmp = Float64(a * a);
	else
		tmp = Float64(pi * Float64(Float64(pi * 3.08641975308642e-5) * Float64(b * Float64(b * Float64(angle * angle)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 2.1e+147)
		tmp = a * a;
	else
		tmp = pi * ((pi * 3.08641975308642e-5) * (b * (b * (angle * angle))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 2.1e+147], N[(a * a), $MachinePrecision], N[(Pi * N[(N[(Pi * 3.08641975308642e-5), $MachinePrecision] * N[(b * N[(b * N[(angle * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.10000000000000006e147

   1. Initial program 80.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

         \[\leadsto a \cdot \color{blue}{a} \]

      2. *-lowering-*.f64 65.2%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

   5. Simplified 65.2%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 2.10000000000000006e147 < b

   1. Initial program 96.6%

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

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

         \[\leadsto \mathsf{+.f64}\left(\left({\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right), \color{blue}{\left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\color{blue}{\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), 2\right), \left({\left(\color{blue}{b} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \left({\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{2}\right)\right) \]

   3. Simplified 96.6%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right), 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 96.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right)\right)}, 2\right)\right) \]

   7. Simplified 96.6%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400} + {\color{blue}{a}}^{2} \]

      2. associate-*r* N/A

         \[\leadsto {angle}^{2} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right) + {\color{blue}{a}}^{2} \]

      3. *-commutative N/A

         \[\leadsto {angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2} \]

      4. +-commutative N/A

         \[\leadsto {a}^{2} + \color{blue}{{angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({angle}^{2} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({angle}^{2} \cdot {b}^{2}\right), \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({b}^{2} \cdot {angle}^{2}\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left({angle}^{2}\right)\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

   10. Simplified 76.8%

       \[\leadsto \color{blue}{a \cdot a + \left(\left(b \cdot b\right) \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)} \]

   11. Taylor expanded in a around 0

       \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}} \]

       2. associate-*r* N/A

          \[\leadsto {angle}^{2} \cdot \color{blue}{\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \]

       3. *-commutative N/A

          \[\leadsto {angle}^{2} \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({angle}^{2}\right), \color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot angle\right), \left(\color{blue}{\frac{1}{32400}} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\color{blue}{\frac{1}{32400}} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left({b}^{2} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left({b}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right)\right)\right)\right) \]

       10. distribute-rgt-neg-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left({b}^{2} \cdot \left(\mathsf{neg}\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{32400}\right)\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left({b}^{2} \cdot \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}}\right)\right)\right)\right) \]

       15. distribute-lft-neg-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right)\right) \]

       16. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{32400} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \color{blue}{\left({\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

       21. PI-lowering-PI.f64 76.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

   13. Simplified 76.8%

       \[\leadsto \color{blue}{\left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)} \]
   14. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left({b}^{2} \cdot \left(\color{blue}{\frac{1}{32400}} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\left(angle \cdot angle\right) \cdot {b}^{2}\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(angle \cdot angle\right) \cdot {b}^{2}\right)} \]

       4. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{32400} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot {b}^{2}\right) \]

       5. *-commutative N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot {b}^{2}\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot {b}^{2}\right)\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(\left(\frac{1}{32400} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot {b}^{2}\right)\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{\left(\frac{1}{32400} \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\left(angle \cdot angle\right) \cdot {b}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\left(\frac{1}{32400} \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\left(\left(angle \cdot angle\right) \cdot {b}^{2}\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right), \left(\color{blue}{\left(angle \cdot angle\right)} \cdot {b}^{2}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \frac{1}{32400}\right), \left(\color{blue}{\left(angle \cdot angle\right)} \cdot {b}^{2}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right), \left(\left(\color{blue}{angle} \cdot angle\right) \cdot {b}^{2}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right), \left({b}^{2} \cdot \color{blue}{\left(angle \cdot angle\right)}\right)\right)\right) \]

       14. pow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right), \left(\left(b \cdot b\right) \cdot \left(\color{blue}{angle} \cdot angle\right)\right)\right)\right) \]

       15. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right), \left(b \cdot \color{blue}{\left(b \cdot \left(angle \cdot angle\right)\right)}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right), \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot \left(angle \cdot angle\right)\right)}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\left(angle \cdot angle\right)}\right)\right)\right)\right) \]

       18. *-lowering-*.f64 80.3%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, \color{blue}{angle}\right)\right)\right)\right)\right) \]

   15. Applied egg-rr 80.3%

       \[\leadsto \color{blue}{\pi \cdot \left(\left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(b \cdot \left(b \cdot \left(angle \cdot angle\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 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 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. Add Preprocessing

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

      \[\leadsto a \cdot \color{blue}{a} \]

   2. *-lowering-*.f64 61.9%

      \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

5. Simplified 61.9%

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

Reproduce

herbie shell --seed 2024138 
(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)))