ab-angle->ABCF A

Percentage Accurate: 79.9% → 79.8%

Time: 25.2s

Alternatives: 14

Speedup: 1.5×

• 36.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{angle \cdot \pi}\\ {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left({t_0}^{2}\right)\right)}{\frac{180}{t_0}}\right)\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cbrt (* angle PI))))
   (+
    (pow (* a (sin (* (/ angle 180.0) PI))) 2.0)
    (pow (* b (cos (/ (expm1 (log1p (pow t_0 2.0))) (/ 180.0 t_0)))) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = cbrt((angle * ((double) M_PI)));
	return pow((a * sin(((angle / 180.0) * ((double) M_PI)))), 2.0) + pow((b * cos((expm1(log1p(pow(t_0, 2.0))) / (180.0 / t_0)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.cbrt((angle * Math.PI));
	return Math.pow((a * Math.sin(((angle / 180.0) * Math.PI))), 2.0) + Math.pow((b * Math.cos((Math.expm1(Math.log1p(Math.pow(t_0, 2.0))) / (180.0 / t_0)))), 2.0);
}

Julia

function code(a, b, angle)
	t_0 = cbrt(Float64(angle * pi))
	return Float64((Float64(a * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(b * cos(Float64(expm1(log1p((t_0 ^ 2.0))) / Float64(180.0 / t_0)))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Power[N[(angle * Pi), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[Power[N[(a * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(N[(Exp[N[Log[1 + N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] / N[(180.0 / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.1%

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

   1. associate-*l/ 79.1%

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

   2. add-cube-cbrt 79.2%

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

   3. associate-/l* 79.1%

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

   4. pow2 79.1%

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

3. Applied egg-rr 79.1%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{{\left(\sqrt[3]{angle \cdot \pi}\right)}^{2}}{\frac{180}{\sqrt[3]{angle \cdot \pi}}}\right)}\right)}^{2} \]
4. Step-by-step derivation

   1. expm1-log1p-u 79.2%

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

5. Applied egg-rr 79.2%

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

6. Final simplification 79.2%

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

Alternative 2: 79.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. associate-*l/ 79.1%

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

   2. add-cube-cbrt 79.2%

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

   3. associate-/l* 79.1%

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

   4. pow2 79.1%

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

3. Applied egg-rr 79.1%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{{\left(\sqrt[3]{angle \cdot \pi}\right)}^{2}}{\frac{180}{\sqrt[3]{angle \cdot \pi}}}\right)}\right)}^{2} \]
4. Step-by-step derivation

   1. expm1-log1p-u 79.2%

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

5. Applied egg-rr 79.2%

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

   1. add-log-exp 79.2%

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

   2. associate-/r/ 79.2%

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

   3. *-commutative 79.2%

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

   4. div-inv 79.2%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \log \left(e^{\cos \left(\sqrt[3]{angle \cdot \pi} \cdot \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{angle \cdot \pi}\right)}^{2}\right)\right) \cdot \frac{1}{180}\right)}\right)}\right)\right)}^{2} \]

   5. metadata-eval 79.2%

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

   6. associate-*l* 79.2%

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

   7. expm1-log1p-u 79.1%

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

   8. unpow2 79.1%

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

   9. rem-3cbrt-rft 79.0%

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

   10. associate-*l* 79.2%

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

7. Applied egg-rr 79.2%

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

8. Final simplification 79.2%

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

Alternative 3: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. associate-*l/ 79.1%

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

   2. clear-num 79.1%

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

   3. associate-/r* 79.1%

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

3. Applied egg-rr 79.1%

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

4. Final simplification 79.1%

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

Alternative 4: 79.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. associate-*l/ 79.1%

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

   2. associate-*r/ 79.1%

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

   3. associate-*l/ 79.0%

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

   4. associate-*r/ 79.1%

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

3. Simplified 79.1%

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

4. Final simplification 79.1%

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

Alternative 5: 79.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

2. Final simplification 79.1%

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

Alternative 6: 79.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. associate-*l/ 79.1%

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

   2. associate-*r/ 79.1%

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

   3. associate-*l/ 79.0%

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

   4. associate-*r/ 79.1%

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

3. Simplified 79.1%

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

4. Taylor expanded in angle around 0 79.0%

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

5. Taylor expanded in angle around inf 78.9%

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

6. Final simplification 78.9%

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

Alternative 7: 80.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. associate-*l/ 79.1%

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

   2. associate-*r/ 79.1%

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

   3. associate-*l/ 79.0%

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

   4. associate-*r/ 79.1%

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

3. Simplified 79.1%

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

4. Taylor expanded in angle around 0 79.0%

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

5. Final simplification 79.0%

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

Alternative 8: 80.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. associate-*l/ 79.1%

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

   2. associate-*r/ 79.1%

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

   3. associate-*l/ 79.0%

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

   4. associate-*r/ 79.1%

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

3. Simplified 79.1%

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

4. Taylor expanded in angle around 0 79.0%

   \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
5. Step-by-step derivation

   1. associate-*r/ 78.9%

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

   2. associate-*l/ 79.0%

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

   3. *-commutative 79.0%

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

   4. div-inv 79.0%

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

   5. metadata-eval 79.0%

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

   6. add-cube-cbrt 78.9%

      \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \color{blue}{\left(\left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   7. cbrt-prod 78.5%

      \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\color{blue}{\left(\sqrt[3]{angle} \cdot \sqrt[3]{0.005555555555555556}\right)} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   8. cbrt-prod 78.4%

      \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\left(\sqrt[3]{angle} \cdot \sqrt[3]{0.005555555555555556}\right) \cdot \color{blue}{\left(\sqrt[3]{angle} \cdot \sqrt[3]{0.005555555555555556}\right)}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   9. swap-sqr 78.4%

      \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\color{blue}{\left(\left(\sqrt[3]{angle} \cdot \sqrt[3]{angle}\right) \cdot \left(\sqrt[3]{0.005555555555555556} \cdot \sqrt[3]{0.005555555555555556}\right)\right)} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   10. cbrt-prod 56.5%

       \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\color{blue}{\sqrt[3]{angle \cdot angle}} \cdot \left(\sqrt[3]{0.005555555555555556} \cdot \sqrt[3]{0.005555555555555556}\right)\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   11. cbrt-unprod 56.7%

       \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\sqrt[3]{angle \cdot angle} \cdot \color{blue}{\sqrt[3]{0.005555555555555556 \cdot 0.005555555555555556}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   12. metadata-eval 56.7%

       \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\sqrt[3]{angle \cdot angle} \cdot \sqrt[3]{\color{blue}{3.08641975308642 \cdot 10^{-5}}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   13. cbrt-prod 56.7%

       \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\color{blue}{\sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   14. associate-*l* 56.7%

       \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   15. add-cube-cbrt 56.6%

       \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}} \cdot \sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}}\right) \cdot \sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   16. pow3 56.6%

       \[\leadsto {\left(a \cdot \sin \color{blue}{\left({\left(\sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}}\right)}^{3}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

6. Applied egg-rr 78.7%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left({\left(\sqrt[3]{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}\right)}^{3}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
7. Step-by-step derivation

   1. unpow3 78.7%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\sqrt[3]{\left(angle \cdot 0.005555555555555556\right) \cdot \pi} \cdot \sqrt[3]{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}\right) \cdot \sqrt[3]{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   2. add-cube-cbrt 79.0%

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

8. Applied egg-rr 79.0%

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

9. Final simplification 79.0%

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

Alternative 9: 77.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(a \cdot angle\right)\\ \mathbf{if}\;a \leq -1.1 \cdot 10^{-35}:\\ \;\;\;\;{b}^{2} + {t_0}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{-39}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + 0.005555555555555556 \cdot \left(t_0 \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* a angle))))
   (if (<= a -1.1e-35)
     (+ (pow b 2.0) (* (pow t_0 2.0) 3.08641975308642e-5))
     (if (<= a 1.7e-39)
       (+ (pow b 2.0) (pow (* a 0.0) 2.0))
       (+
        (pow b 2.0)
        (*
         0.005555555555555556
         (* t_0 (* (* angle PI) (* a 0.005555555555555556)))))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (a * angle);
	double tmp;
	if (a <= -1.1e-35) {
		tmp = pow(b, 2.0) + (pow(t_0, 2.0) * 3.08641975308642e-5);
	} else if (a <= 1.7e-39) {
		tmp = pow(b, 2.0) + pow((a * 0.0), 2.0);
	} else {
		tmp = pow(b, 2.0) + (0.005555555555555556 * (t_0 * ((angle * ((double) M_PI)) * (a * 0.005555555555555556))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (a * angle);
	double tmp;
	if (a <= -1.1e-35) {
		tmp = Math.pow(b, 2.0) + (Math.pow(t_0, 2.0) * 3.08641975308642e-5);
	} else if (a <= 1.7e-39) {
		tmp = Math.pow(b, 2.0) + Math.pow((a * 0.0), 2.0);
	} else {
		tmp = Math.pow(b, 2.0) + (0.005555555555555556 * (t_0 * ((angle * Math.PI) * (a * 0.005555555555555556))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pi * (a * angle)
	tmp = 0
	if a <= -1.1e-35:
		tmp = math.pow(b, 2.0) + (math.pow(t_0, 2.0) * 3.08641975308642e-5)
	elif a <= 1.7e-39:
		tmp = math.pow(b, 2.0) + math.pow((a * 0.0), 2.0)
	else:
		tmp = math.pow(b, 2.0) + (0.005555555555555556 * (t_0 * ((angle * math.pi) * (a * 0.005555555555555556))))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(a * angle))
	tmp = 0.0
	if (a <= -1.1e-35)
		tmp = Float64((b ^ 2.0) + Float64((t_0 ^ 2.0) * 3.08641975308642e-5));
	elseif (a <= 1.7e-39)
		tmp = Float64((b ^ 2.0) + (Float64(a * 0.0) ^ 2.0));
	else
		tmp = Float64((b ^ 2.0) + Float64(0.005555555555555556 * Float64(t_0 * Float64(Float64(angle * pi) * Float64(a * 0.005555555555555556)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = pi * (a * angle);
	tmp = 0.0;
	if (a <= -1.1e-35)
		tmp = (b ^ 2.0) + ((t_0 ^ 2.0) * 3.08641975308642e-5);
	elseif (a <= 1.7e-39)
		tmp = (b ^ 2.0) + ((a * 0.0) ^ 2.0);
	else
		tmp = (b ^ 2.0) + (0.005555555555555556 * (t_0 * ((angle * pi) * (a * 0.005555555555555556))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(a * angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.1e-35], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[Power[t$95$0, 2.0], $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.7e-39], N[(N[Power[b, 2.0], $MachinePrecision] + N[Power[N[(a * 0.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(0.005555555555555556 * N[(t$95$0 * N[(N[(angle * Pi), $MachinePrecision] * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.09999999999999997e-35

   1. Initial program 87.7%

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

      1. associate-*l/ 87.8%

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

      2. associate-*r/ 87.7%

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

      3. associate-*l/ 87.7%

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

      4. associate-*r/ 87.7%

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

   3. Simplified 87.7%

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

   4. Taylor expanded in angle around 0 87.7%

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

   5. Taylor expanded in angle around 0 85.7%

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

      1. *-commutative 85.7%

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

   7. Simplified 85.7%

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

      1. *-commutative 85.7%

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

      2. unpow-prod-down 85.8%

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

      3. associate-*r* 85.8%

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

      4. *-commutative 85.8%

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

      5. associate-*l* 85.9%

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

      6. metadata-eval 85.9%

         \[\leadsto {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2} \cdot \color{blue}{3.08641975308642 \cdot 10^{-5}} + {\left(b \cdot 1\right)}^{2} \]

   9. Applied egg-rr 85.9%

      \[\leadsto \color{blue}{{\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}} + {\left(b \cdot 1\right)}^{2} \]

   if -1.09999999999999997e-35 < a < 1.7e-39

   1. Initial program 70.8%

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

      1. associate-*l/ 70.8%

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

      2. associate-*r/ 70.8%

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

      3. associate-*l/ 70.7%

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

      4. associate-*r/ 70.9%

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

   3. Simplified 70.9%

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

   4. Taylor expanded in angle around 0 70.8%

      \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   5. Step-by-step derivation

      1. associate-*r/ 70.7%

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

      2. associate-*l/ 70.7%

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

      3. *-commutative 70.7%

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

      4. div-inv 70.7%

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

      5. metadata-eval 70.7%

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

      6. add-cube-cbrt 70.9%

         \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \color{blue}{\left(\left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      7. cbrt-prod 70.7%

         \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\color{blue}{\left(\sqrt[3]{angle} \cdot \sqrt[3]{0.005555555555555556}\right)} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      8. cbrt-prod 70.6%

         \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\left(\sqrt[3]{angle} \cdot \sqrt[3]{0.005555555555555556}\right) \cdot \color{blue}{\left(\sqrt[3]{angle} \cdot \sqrt[3]{0.005555555555555556}\right)}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      9. swap-sqr 70.6%

         \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\color{blue}{\left(\left(\sqrt[3]{angle} \cdot \sqrt[3]{angle}\right) \cdot \left(\sqrt[3]{0.005555555555555556} \cdot \sqrt[3]{0.005555555555555556}\right)\right)} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      10. cbrt-prod 52.4%

          \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\color{blue}{\sqrt[3]{angle \cdot angle}} \cdot \left(\sqrt[3]{0.005555555555555556} \cdot \sqrt[3]{0.005555555555555556}\right)\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      11. cbrt-unprod 52.5%

          \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\sqrt[3]{angle \cdot angle} \cdot \color{blue}{\sqrt[3]{0.005555555555555556 \cdot 0.005555555555555556}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      12. metadata-eval 52.5%

          \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\sqrt[3]{angle \cdot angle} \cdot \sqrt[3]{\color{blue}{3.08641975308642 \cdot 10^{-5}}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      13. cbrt-prod 52.5%

          \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\color{blue}{\sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      14. associate-*l* 52.5%

          \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      15. add-cube-cbrt 52.5%

          \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}} \cdot \sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}}\right) \cdot \sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      16. pow3 52.5%

          \[\leadsto {\left(a \cdot \sin \color{blue}{\left({\left(\sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}}\right)}^{3}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   6. Applied egg-rr 70.8%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left({\left(\sqrt[3]{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}\right)}^{3}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   7. Taylor expanded in angle around 0 70.6%

      \[\leadsto {\left(a \cdot \color{blue}{0}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   if 1.7e-39 < a

   1. Initial program 83.6%

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

      1. associate-*l/ 83.4%

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

      2. associate-*r/ 83.5%

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

      3. associate-*l/ 83.5%

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

      4. associate-*r/ 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in angle around 0 83.2%

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

   5. Taylor expanded in angle around 0 78.4%

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

      1. *-commutative 78.4%

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

   7. Simplified 78.4%

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

      1. unpow2 78.4%

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

      2. *-commutative 78.4%

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

      3. associate-*r* 78.4%

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

      4. *-commutative 78.4%

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

      5. associate-*r* 78.5%

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

      6. associate-*l* 78.5%

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

      7. associate-*r* 78.5%

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

      8. *-commutative 78.5%

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

      9. associate-*l* 78.5%

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

   9. Applied egg-rr 78.5%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-35}:\\ \;\;\;\;{b}^{2} + {\left(\pi \cdot \left(a \cdot angle\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{-39}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + 0.005555555555555556 \cdot \left(\left(\pi \cdot \left(a \cdot angle\right)\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]

Alternative 10: 77.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{-35} \lor \neg \left(a \leq 4 \cdot 10^{-39}\right):\\ \;\;\;\;{b}^{2} + {\left(\pi \cdot \left(a \cdot angle\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot 0\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (or (<= a -3.5e-35) (not (<= a 4e-39)))
   (+ (pow b 2.0) (* (pow (* PI (* a angle)) 2.0) 3.08641975308642e-5))
   (+ (pow b 2.0) (pow (* a 0.0) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((a <= -3.5e-35) || !(a <= 4e-39)) {
		tmp = pow(b, 2.0) + (pow((((double) M_PI) * (a * angle)), 2.0) * 3.08641975308642e-5);
	} else {
		tmp = pow(b, 2.0) + pow((a * 0.0), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((a <= -3.5e-35) || !(a <= 4e-39)) {
		tmp = Math.pow(b, 2.0) + (Math.pow((Math.PI * (a * angle)), 2.0) * 3.08641975308642e-5);
	} else {
		tmp = Math.pow(b, 2.0) + Math.pow((a * 0.0), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (a <= -3.5e-35) or not (a <= 4e-39):
		tmp = math.pow(b, 2.0) + (math.pow((math.pi * (a * angle)), 2.0) * 3.08641975308642e-5)
	else:
		tmp = math.pow(b, 2.0) + math.pow((a * 0.0), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if ((a <= -3.5e-35) || !(a <= 4e-39))
		tmp = Float64((b ^ 2.0) + Float64((Float64(pi * Float64(a * angle)) ^ 2.0) * 3.08641975308642e-5));
	else
		tmp = Float64((b ^ 2.0) + (Float64(a * 0.0) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((a <= -3.5e-35) || ~((a <= 4e-39)))
		tmp = (b ^ 2.0) + (((pi * (a * angle)) ^ 2.0) * 3.08641975308642e-5);
	else
		tmp = (b ^ 2.0) + ((a * 0.0) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[Or[LessEqual[a, -3.5e-35], N[Not[LessEqual[a, 4e-39]], $MachinePrecision]], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[Power[N[(Pi * N[(a * angle), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[Power[N[(a * 0.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.49999999999999996e-35 or 3.99999999999999972e-39 < a

   1. Initial program 85.3%

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

      1. associate-*l/ 85.2%

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

      2. associate-*r/ 85.2%

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

      3. associate-*l/ 85.2%

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

      4. associate-*r/ 85.2%

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

   3. Simplified 85.2%

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

   4. Taylor expanded in angle around 0 85.0%

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

   5. Taylor expanded in angle around 0 81.4%

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

      1. *-commutative 81.4%

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

   7. Simplified 81.4%

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

      1. *-commutative 81.4%

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

      2. unpow-prod-down 81.4%

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

      3. associate-*r* 81.5%

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

      4. *-commutative 81.5%

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

      5. associate-*l* 81.5%

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

      6. metadata-eval 81.5%

         \[\leadsto {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2} \cdot \color{blue}{3.08641975308642 \cdot 10^{-5}} + {\left(b \cdot 1\right)}^{2} \]

   9. Applied egg-rr 81.5%

      \[\leadsto \color{blue}{{\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}} + {\left(b \cdot 1\right)}^{2} \]

   if -3.49999999999999996e-35 < a < 3.99999999999999972e-39

   1. Initial program 70.8%

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

      1. associate-*l/ 70.8%

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

      2. associate-*r/ 70.8%

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

      3. associate-*l/ 70.7%

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

      4. associate-*r/ 70.9%

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

   3. Simplified 70.9%

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

   4. Taylor expanded in angle around 0 70.8%

      \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   5. Step-by-step derivation

      1. associate-*r/ 70.7%

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

      2. associate-*l/ 70.7%

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

      3. *-commutative 70.7%

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

      4. div-inv 70.7%

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

      5. metadata-eval 70.7%

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

      6. add-cube-cbrt 70.9%

         \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \color{blue}{\left(\left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      7. cbrt-prod 70.7%

         \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\color{blue}{\left(\sqrt[3]{angle} \cdot \sqrt[3]{0.005555555555555556}\right)} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      8. cbrt-prod 70.6%

         \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\left(\sqrt[3]{angle} \cdot \sqrt[3]{0.005555555555555556}\right) \cdot \color{blue}{\left(\sqrt[3]{angle} \cdot \sqrt[3]{0.005555555555555556}\right)}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      9. swap-sqr 70.6%

         \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\color{blue}{\left(\left(\sqrt[3]{angle} \cdot \sqrt[3]{angle}\right) \cdot \left(\sqrt[3]{0.005555555555555556} \cdot \sqrt[3]{0.005555555555555556}\right)\right)} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      10. cbrt-prod 52.4%

          \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\color{blue}{\sqrt[3]{angle \cdot angle}} \cdot \left(\sqrt[3]{0.005555555555555556} \cdot \sqrt[3]{0.005555555555555556}\right)\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      11. cbrt-unprod 52.5%

          \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\sqrt[3]{angle \cdot angle} \cdot \color{blue}{\sqrt[3]{0.005555555555555556 \cdot 0.005555555555555556}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      12. metadata-eval 52.5%

          \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\sqrt[3]{angle \cdot angle} \cdot \sqrt[3]{\color{blue}{3.08641975308642 \cdot 10^{-5}}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      13. cbrt-prod 52.5%

          \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\color{blue}{\sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      14. associate-*l* 52.5%

          \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      15. add-cube-cbrt 52.5%

          \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}} \cdot \sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}}\right) \cdot \sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      16. pow3 52.5%

          \[\leadsto {\left(a \cdot \sin \color{blue}{\left({\left(\sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}}\right)}^{3}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   6. Applied egg-rr 70.8%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left({\left(\sqrt[3]{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}\right)}^{3}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   7. Taylor expanded in angle around 0 70.6%

      \[\leadsto {\left(a \cdot \color{blue}{0}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{-35} \lor \neg \left(a \leq 4 \cdot 10^{-39}\right):\\ \;\;\;\;{b}^{2} + {\left(\pi \cdot \left(a \cdot angle\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot 0\right)}^{2}\\ \end{array} \]

Alternative 11: 77.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(a \cdot angle\right)\\ \mathbf{if}\;a \leq -2.2 \cdot 10^{-34}:\\ \;\;\;\;{b}^{2} + {t_0}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-38}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(0.005555555555555556 \cdot t_0\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* a angle))))
   (if (<= a -2.2e-34)
     (+ (pow b 2.0) (* (pow t_0 2.0) 3.08641975308642e-5))
     (if (<= a 1.25e-38)
       (+ (pow b 2.0) (pow (* a 0.0) 2.0))
       (+ (pow b 2.0) (pow (* 0.005555555555555556 t_0) 2.0))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (a * angle);
	double tmp;
	if (a <= -2.2e-34) {
		tmp = pow(b, 2.0) + (pow(t_0, 2.0) * 3.08641975308642e-5);
	} else if (a <= 1.25e-38) {
		tmp = pow(b, 2.0) + pow((a * 0.0), 2.0);
	} else {
		tmp = pow(b, 2.0) + pow((0.005555555555555556 * t_0), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (a * angle);
	double tmp;
	if (a <= -2.2e-34) {
		tmp = Math.pow(b, 2.0) + (Math.pow(t_0, 2.0) * 3.08641975308642e-5);
	} else if (a <= 1.25e-38) {
		tmp = Math.pow(b, 2.0) + Math.pow((a * 0.0), 2.0);
	} else {
		tmp = Math.pow(b, 2.0) + Math.pow((0.005555555555555556 * t_0), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pi * (a * angle)
	tmp = 0
	if a <= -2.2e-34:
		tmp = math.pow(b, 2.0) + (math.pow(t_0, 2.0) * 3.08641975308642e-5)
	elif a <= 1.25e-38:
		tmp = math.pow(b, 2.0) + math.pow((a * 0.0), 2.0)
	else:
		tmp = math.pow(b, 2.0) + math.pow((0.005555555555555556 * t_0), 2.0)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(a * angle))
	tmp = 0.0
	if (a <= -2.2e-34)
		tmp = Float64((b ^ 2.0) + Float64((t_0 ^ 2.0) * 3.08641975308642e-5));
	elseif (a <= 1.25e-38)
		tmp = Float64((b ^ 2.0) + (Float64(a * 0.0) ^ 2.0));
	else
		tmp = Float64((b ^ 2.0) + (Float64(0.005555555555555556 * t_0) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = pi * (a * angle);
	tmp = 0.0;
	if (a <= -2.2e-34)
		tmp = (b ^ 2.0) + ((t_0 ^ 2.0) * 3.08641975308642e-5);
	elseif (a <= 1.25e-38)
		tmp = (b ^ 2.0) + ((a * 0.0) ^ 2.0);
	else
		tmp = (b ^ 2.0) + ((0.005555555555555556 * t_0) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(a * angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.2e-34], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[Power[t$95$0, 2.0], $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.25e-38], N[(N[Power[b, 2.0], $MachinePrecision] + N[Power[N[(a * 0.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[Power[N[(0.005555555555555556 * t$95$0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.1999999999999999e-34

   1. Initial program 87.7%

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

      1. associate-*l/ 87.8%

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

      2. associate-*r/ 87.7%

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

      3. associate-*l/ 87.7%

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

      4. associate-*r/ 87.7%

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

   3. Simplified 87.7%

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

   4. Taylor expanded in angle around 0 87.7%

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

   5. Taylor expanded in angle around 0 85.7%

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

      1. *-commutative 85.7%

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

   7. Simplified 85.7%

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

      1. *-commutative 85.7%

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

      2. unpow-prod-down 85.8%

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

      3. associate-*r* 85.8%

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

      4. *-commutative 85.8%

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

      5. associate-*l* 85.9%

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

      6. metadata-eval 85.9%

         \[\leadsto {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2} \cdot \color{blue}{3.08641975308642 \cdot 10^{-5}} + {\left(b \cdot 1\right)}^{2} \]

   9. Applied egg-rr 85.9%

      \[\leadsto \color{blue}{{\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}} + {\left(b \cdot 1\right)}^{2} \]

   if -2.1999999999999999e-34 < a < 1.25000000000000008e-38

   1. Initial program 70.8%

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

      1. associate-*l/ 70.8%

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

      2. associate-*r/ 70.8%

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

      3. associate-*l/ 70.7%

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

      4. associate-*r/ 70.9%

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

   3. Simplified 70.9%

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

   4. Taylor expanded in angle around 0 70.8%

      \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   5. Step-by-step derivation

      1. associate-*r/ 70.7%

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

      2. associate-*l/ 70.7%

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

      3. *-commutative 70.7%

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

      4. div-inv 70.7%

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

      5. metadata-eval 70.7%

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

      6. add-cube-cbrt 70.9%

         \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \color{blue}{\left(\left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      7. cbrt-prod 70.7%

         \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\color{blue}{\left(\sqrt[3]{angle} \cdot \sqrt[3]{0.005555555555555556}\right)} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      8. cbrt-prod 70.6%

         \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\left(\sqrt[3]{angle} \cdot \sqrt[3]{0.005555555555555556}\right) \cdot \color{blue}{\left(\sqrt[3]{angle} \cdot \sqrt[3]{0.005555555555555556}\right)}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      9. swap-sqr 70.6%

         \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\color{blue}{\left(\left(\sqrt[3]{angle} \cdot \sqrt[3]{angle}\right) \cdot \left(\sqrt[3]{0.005555555555555556} \cdot \sqrt[3]{0.005555555555555556}\right)\right)} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      10. cbrt-prod 52.4%

          \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\color{blue}{\sqrt[3]{angle \cdot angle}} \cdot \left(\sqrt[3]{0.005555555555555556} \cdot \sqrt[3]{0.005555555555555556}\right)\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      11. cbrt-unprod 52.5%

          \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\sqrt[3]{angle \cdot angle} \cdot \color{blue}{\sqrt[3]{0.005555555555555556 \cdot 0.005555555555555556}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      12. metadata-eval 52.5%

          \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\sqrt[3]{angle \cdot angle} \cdot \sqrt[3]{\color{blue}{3.08641975308642 \cdot 10^{-5}}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      13. cbrt-prod 52.5%

          \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\color{blue}{\sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      14. associate-*l* 52.5%

          \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      15. add-cube-cbrt 52.5%

          \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}} \cdot \sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}}\right) \cdot \sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      16. pow3 52.5%

          \[\leadsto {\left(a \cdot \sin \color{blue}{\left({\left(\sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}}\right)}^{3}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   6. Applied egg-rr 70.8%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left({\left(\sqrt[3]{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}\right)}^{3}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   7. Taylor expanded in angle around 0 70.6%

      \[\leadsto {\left(a \cdot \color{blue}{0}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   if 1.25000000000000008e-38 < a

   1. Initial program 83.6%

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

      1. associate-*l/ 83.4%

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

      2. associate-*r/ 83.5%

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

      3. associate-*l/ 83.5%

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

      4. associate-*r/ 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in angle around 0 83.2%

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

   5. Taylor expanded in angle around 0 78.4%

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

      1. associate-*r* 78.4%

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

      2. *-commutative 78.4%

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

   7. Simplified 78.4%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.2 \cdot 10^{-34}:\\ \;\;\;\;{b}^{2} + {\left(\pi \cdot \left(a \cdot angle\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-38}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(0.005555555555555556 \cdot \left(\pi \cdot \left(a \cdot angle\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 12: 77.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.15 \cdot 10^{-34}:\\ \;\;\;\;{b}^{2} + {\left(\pi \cdot \left(a \cdot angle\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{-39}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a -1.15e-34)
   (+ (pow b 2.0) (* (pow (* PI (* a angle)) 2.0) 3.08641975308642e-5))
   (if (<= a 1.3e-39)
     (+ (pow b 2.0) (pow (* a 0.0) 2.0))
     (+ (pow b 2.0) (pow (* a (* (* angle PI) 0.005555555555555556)) 2.0)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= -1.15e-34) {
		tmp = pow(b, 2.0) + (pow((((double) M_PI) * (a * angle)), 2.0) * 3.08641975308642e-5);
	} else if (a <= 1.3e-39) {
		tmp = pow(b, 2.0) + pow((a * 0.0), 2.0);
	} else {
		tmp = pow(b, 2.0) + pow((a * ((angle * ((double) M_PI)) * 0.005555555555555556)), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= -1.15e-34) {
		tmp = Math.pow(b, 2.0) + (Math.pow((Math.PI * (a * angle)), 2.0) * 3.08641975308642e-5);
	} else if (a <= 1.3e-39) {
		tmp = Math.pow(b, 2.0) + Math.pow((a * 0.0), 2.0);
	} else {
		tmp = Math.pow(b, 2.0) + Math.pow((a * ((angle * Math.PI) * 0.005555555555555556)), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= -1.15e-34:
		tmp = math.pow(b, 2.0) + (math.pow((math.pi * (a * angle)), 2.0) * 3.08641975308642e-5)
	elif a <= 1.3e-39:
		tmp = math.pow(b, 2.0) + math.pow((a * 0.0), 2.0)
	else:
		tmp = math.pow(b, 2.0) + math.pow((a * ((angle * math.pi) * 0.005555555555555556)), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= -1.15e-34)
		tmp = Float64((b ^ 2.0) + Float64((Float64(pi * Float64(a * angle)) ^ 2.0) * 3.08641975308642e-5));
	elseif (a <= 1.3e-39)
		tmp = Float64((b ^ 2.0) + (Float64(a * 0.0) ^ 2.0));
	else
		tmp = Float64((b ^ 2.0) + (Float64(a * Float64(Float64(angle * pi) * 0.005555555555555556)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= -1.15e-34)
		tmp = (b ^ 2.0) + (((pi * (a * angle)) ^ 2.0) * 3.08641975308642e-5);
	elseif (a <= 1.3e-39)
		tmp = (b ^ 2.0) + ((a * 0.0) ^ 2.0);
	else
		tmp = (b ^ 2.0) + ((a * ((angle * pi) * 0.005555555555555556)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, -1.15e-34], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[Power[N[(Pi * N[(a * angle), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.3e-39], N[(N[Power[b, 2.0], $MachinePrecision] + N[Power[N[(a * 0.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[Power[N[(a * N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.15000000000000006e-34

   1. Initial program 87.7%

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

      1. associate-*l/ 87.8%

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

      2. associate-*r/ 87.7%

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

      3. associate-*l/ 87.7%

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

      4. associate-*r/ 87.7%

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

   3. Simplified 87.7%

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

   4. Taylor expanded in angle around 0 87.7%

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

   5. Taylor expanded in angle around 0 85.7%

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

      1. *-commutative 85.7%

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

   7. Simplified 85.7%

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

      1. *-commutative 85.7%

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

      2. unpow-prod-down 85.8%

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

      3. associate-*r* 85.8%

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

      4. *-commutative 85.8%

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

      5. associate-*l* 85.9%

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

      6. metadata-eval 85.9%

         \[\leadsto {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2} \cdot \color{blue}{3.08641975308642 \cdot 10^{-5}} + {\left(b \cdot 1\right)}^{2} \]

   9. Applied egg-rr 85.9%

      \[\leadsto \color{blue}{{\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}} + {\left(b \cdot 1\right)}^{2} \]

   if -1.15000000000000006e-34 < a < 1.3e-39

   1. Initial program 70.8%

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

      1. associate-*l/ 70.8%

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

      2. associate-*r/ 70.8%

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

      3. associate-*l/ 70.7%

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

      4. associate-*r/ 70.9%

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

   3. Simplified 70.9%

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

   4. Taylor expanded in angle around 0 70.8%

      \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   5. Step-by-step derivation

      1. associate-*r/ 70.7%

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

      2. associate-*l/ 70.7%

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

      3. *-commutative 70.7%

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

      4. div-inv 70.7%

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

      5. metadata-eval 70.7%

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

      6. add-cube-cbrt 70.9%

         \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \color{blue}{\left(\left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      7. cbrt-prod 70.7%

         \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\color{blue}{\left(\sqrt[3]{angle} \cdot \sqrt[3]{0.005555555555555556}\right)} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      8. cbrt-prod 70.6%

         \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\left(\sqrt[3]{angle} \cdot \sqrt[3]{0.005555555555555556}\right) \cdot \color{blue}{\left(\sqrt[3]{angle} \cdot \sqrt[3]{0.005555555555555556}\right)}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      9. swap-sqr 70.6%

         \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\color{blue}{\left(\left(\sqrt[3]{angle} \cdot \sqrt[3]{angle}\right) \cdot \left(\sqrt[3]{0.005555555555555556} \cdot \sqrt[3]{0.005555555555555556}\right)\right)} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      10. cbrt-prod 52.4%

          \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\color{blue}{\sqrt[3]{angle \cdot angle}} \cdot \left(\sqrt[3]{0.005555555555555556} \cdot \sqrt[3]{0.005555555555555556}\right)\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      11. cbrt-unprod 52.5%

          \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\sqrt[3]{angle \cdot angle} \cdot \color{blue}{\sqrt[3]{0.005555555555555556 \cdot 0.005555555555555556}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      12. metadata-eval 52.5%

          \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\sqrt[3]{angle \cdot angle} \cdot \sqrt[3]{\color{blue}{3.08641975308642 \cdot 10^{-5}}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      13. cbrt-prod 52.5%

          \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\color{blue}{\sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      14. associate-*l* 52.5%

          \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      15. add-cube-cbrt 52.5%

          \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}} \cdot \sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}}\right) \cdot \sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      16. pow3 52.5%

          \[\leadsto {\left(a \cdot \sin \color{blue}{\left({\left(\sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}}\right)}^{3}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   6. Applied egg-rr 70.8%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left({\left(\sqrt[3]{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}\right)}^{3}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   7. Taylor expanded in angle around 0 70.6%

      \[\leadsto {\left(a \cdot \color{blue}{0}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   if 1.3e-39 < a

   1. Initial program 83.6%

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

      1. associate-*l/ 83.4%

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

      2. associate-*r/ 83.5%

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

      3. associate-*l/ 83.5%

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

      4. associate-*r/ 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in angle around 0 83.2%

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

   5. Taylor expanded in angle around 0 78.4%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.15 \cdot 10^{-34}:\\ \;\;\;\;{b}^{2} + {\left(\pi \cdot \left(a \cdot angle\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{-39}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)}^{2}\\ \end{array} \]

Alternative 13: 77.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{-36}:\\ \;\;\;\;{b}^{2} + {\left(\pi \cdot \left(a \cdot angle\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-39}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a -5.4e-36)
   (+ (pow b 2.0) (* (pow (* PI (* a angle)) 2.0) 3.08641975308642e-5))
   (if (<= a 1.8e-39)
     (+ (pow b 2.0) (pow (* a 0.0) 2.0))
     (+ (pow b 2.0) (pow (* a (* angle (/ PI 180.0))) 2.0)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= -5.4e-36) {
		tmp = pow(b, 2.0) + (pow((((double) M_PI) * (a * angle)), 2.0) * 3.08641975308642e-5);
	} else if (a <= 1.8e-39) {
		tmp = pow(b, 2.0) + pow((a * 0.0), 2.0);
	} else {
		tmp = pow(b, 2.0) + pow((a * (angle * (((double) M_PI) / 180.0))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= -5.4e-36) {
		tmp = Math.pow(b, 2.0) + (Math.pow((Math.PI * (a * angle)), 2.0) * 3.08641975308642e-5);
	} else if (a <= 1.8e-39) {
		tmp = Math.pow(b, 2.0) + Math.pow((a * 0.0), 2.0);
	} else {
		tmp = Math.pow(b, 2.0) + Math.pow((a * (angle * (Math.PI / 180.0))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= -5.4e-36:
		tmp = math.pow(b, 2.0) + (math.pow((math.pi * (a * angle)), 2.0) * 3.08641975308642e-5)
	elif a <= 1.8e-39:
		tmp = math.pow(b, 2.0) + math.pow((a * 0.0), 2.0)
	else:
		tmp = math.pow(b, 2.0) + math.pow((a * (angle * (math.pi / 180.0))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= -5.4e-36)
		tmp = Float64((b ^ 2.0) + Float64((Float64(pi * Float64(a * angle)) ^ 2.0) * 3.08641975308642e-5));
	elseif (a <= 1.8e-39)
		tmp = Float64((b ^ 2.0) + (Float64(a * 0.0) ^ 2.0));
	else
		tmp = Float64((b ^ 2.0) + (Float64(a * Float64(angle * Float64(pi / 180.0))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= -5.4e-36)
		tmp = (b ^ 2.0) + (((pi * (a * angle)) ^ 2.0) * 3.08641975308642e-5);
	elseif (a <= 1.8e-39)
		tmp = (b ^ 2.0) + ((a * 0.0) ^ 2.0);
	else
		tmp = (b ^ 2.0) + ((a * (angle * (pi / 180.0))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, -5.4e-36], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[Power[N[(Pi * N[(a * angle), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.8e-39], N[(N[Power[b, 2.0], $MachinePrecision] + N[Power[N[(a * 0.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[Power[N[(a * N[(angle * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.40000000000000015e-36

   1. Initial program 87.7%

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

      1. associate-*l/ 87.8%

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

      2. associate-*r/ 87.7%

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

      3. associate-*l/ 87.7%

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

      4. associate-*r/ 87.7%

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

   3. Simplified 87.7%

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

   4. Taylor expanded in angle around 0 87.7%

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

   5. Taylor expanded in angle around 0 85.7%

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

      1. *-commutative 85.7%

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

   7. Simplified 85.7%

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

      1. *-commutative 85.7%

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

      2. unpow-prod-down 85.8%

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

      3. associate-*r* 85.8%

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

      4. *-commutative 85.8%

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

      5. associate-*l* 85.9%

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

      6. metadata-eval 85.9%

         \[\leadsto {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2} \cdot \color{blue}{3.08641975308642 \cdot 10^{-5}} + {\left(b \cdot 1\right)}^{2} \]

   9. Applied egg-rr 85.9%

      \[\leadsto \color{blue}{{\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}} + {\left(b \cdot 1\right)}^{2} \]

   if -5.40000000000000015e-36 < a < 1.8e-39

   1. Initial program 70.8%

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

      1. associate-*l/ 70.8%

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

      2. associate-*r/ 70.8%

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

      3. associate-*l/ 70.7%

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

      4. associate-*r/ 70.9%

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

   3. Simplified 70.9%

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

   4. Taylor expanded in angle around 0 70.8%

      \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   5. Step-by-step derivation

      1. associate-*r/ 70.7%

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

      2. associate-*l/ 70.7%

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

      3. *-commutative 70.7%

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

      4. div-inv 70.7%

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

      5. metadata-eval 70.7%

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

      6. add-cube-cbrt 70.9%

         \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \color{blue}{\left(\left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      7. cbrt-prod 70.7%

         \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\color{blue}{\left(\sqrt[3]{angle} \cdot \sqrt[3]{0.005555555555555556}\right)} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      8. cbrt-prod 70.6%

         \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\left(\sqrt[3]{angle} \cdot \sqrt[3]{0.005555555555555556}\right) \cdot \color{blue}{\left(\sqrt[3]{angle} \cdot \sqrt[3]{0.005555555555555556}\right)}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      9. swap-sqr 70.6%

         \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\color{blue}{\left(\left(\sqrt[3]{angle} \cdot \sqrt[3]{angle}\right) \cdot \left(\sqrt[3]{0.005555555555555556} \cdot \sqrt[3]{0.005555555555555556}\right)\right)} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      10. cbrt-prod 52.4%

          \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\color{blue}{\sqrt[3]{angle \cdot angle}} \cdot \left(\sqrt[3]{0.005555555555555556} \cdot \sqrt[3]{0.005555555555555556}\right)\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      11. cbrt-unprod 52.5%

          \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\sqrt[3]{angle \cdot angle} \cdot \color{blue}{\sqrt[3]{0.005555555555555556 \cdot 0.005555555555555556}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      12. metadata-eval 52.5%

          \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\sqrt[3]{angle \cdot angle} \cdot \sqrt[3]{\color{blue}{3.08641975308642 \cdot 10^{-5}}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      13. cbrt-prod 52.5%

          \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\color{blue}{\sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      14. associate-*l* 52.5%

          \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      15. add-cube-cbrt 52.5%

          \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}} \cdot \sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}}\right) \cdot \sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      16. pow3 52.5%

          \[\leadsto {\left(a \cdot \sin \color{blue}{\left({\left(\sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}}\right)}^{3}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   6. Applied egg-rr 70.8%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left({\left(\sqrt[3]{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}\right)}^{3}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   7. Taylor expanded in angle around 0 70.6%

      \[\leadsto {\left(a \cdot \color{blue}{0}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   if 1.8e-39 < a

   1. Initial program 83.6%

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

      1. associate-*l/ 83.4%

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

      2. associate-*r/ 83.5%

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

      3. associate-*l/ 83.5%

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

      4. associate-*r/ 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in angle around 0 83.2%

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

   5. Taylor expanded in angle around 0 78.4%

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

      1. associate-*r* 78.4%

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

      2. metadata-eval 78.4%

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

      3. associate-/r/ 78.3%

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

      4. associate-*l/ 78.4%

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

      5. *-lft-identity 78.4%

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

      6. associate-/r/ 78.5%

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

      7. *-commutative 78.5%

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

   7. Simplified 78.5%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{-36}:\\ \;\;\;\;{b}^{2} + {\left(\pi \cdot \left(a \cdot angle\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-39}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}\\ \end{array} \]

Alternative 14: 57.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ {b}^{2} + {\left(a \cdot 0\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle) :precision binary64 (+ (pow b 2.0) (pow (* a 0.0) 2.0)))

C

double code(double a, double b, double angle) {
	return pow(b, 2.0) + pow((a * 0.0), 2.0);
}

Fortran

real(8) function code(a, b, angle)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    code = (b ** 2.0d0) + ((a * 0.0d0) ** 2.0d0)
end function

Java

public static double code(double a, double b, double angle) {
	return Math.pow(b, 2.0) + Math.pow((a * 0.0), 2.0);
}

Python

def code(a, b, angle):
	return math.pow(b, 2.0) + math.pow((a * 0.0), 2.0)

Julia

function code(a, b, angle)
	return Float64((b ^ 2.0) + (Float64(a * 0.0) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (b ^ 2.0) + ((a * 0.0) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[b, 2.0], $MachinePrecision] + N[Power[N[(a * 0.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.1%

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

   1. associate-*l/ 79.1%

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

   2. associate-*r/ 79.1%

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

   3. associate-*l/ 79.0%

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

   4. associate-*r/ 79.1%

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

3. Simplified 79.1%

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

4. Taylor expanded in angle around 0 79.0%

   \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
5. Step-by-step derivation

   1. associate-*r/ 78.9%

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

   2. associate-*l/ 79.0%

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

   3. *-commutative 79.0%

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

   4. div-inv 79.0%

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

   5. metadata-eval 79.0%

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

   6. add-cube-cbrt 78.9%

      \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \color{blue}{\left(\left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   7. cbrt-prod 78.5%

      \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\color{blue}{\left(\sqrt[3]{angle} \cdot \sqrt[3]{0.005555555555555556}\right)} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   8. cbrt-prod 78.4%

      \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\left(\sqrt[3]{angle} \cdot \sqrt[3]{0.005555555555555556}\right) \cdot \color{blue}{\left(\sqrt[3]{angle} \cdot \sqrt[3]{0.005555555555555556}\right)}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   9. swap-sqr 78.4%

      \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\color{blue}{\left(\left(\sqrt[3]{angle} \cdot \sqrt[3]{angle}\right) \cdot \left(\sqrt[3]{0.005555555555555556} \cdot \sqrt[3]{0.005555555555555556}\right)\right)} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   10. cbrt-prod 56.5%

       \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\color{blue}{\sqrt[3]{angle \cdot angle}} \cdot \left(\sqrt[3]{0.005555555555555556} \cdot \sqrt[3]{0.005555555555555556}\right)\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   11. cbrt-unprod 56.7%

       \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\sqrt[3]{angle \cdot angle} \cdot \color{blue}{\sqrt[3]{0.005555555555555556 \cdot 0.005555555555555556}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   12. metadata-eval 56.7%

       \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\left(\sqrt[3]{angle \cdot angle} \cdot \sqrt[3]{\color{blue}{3.08641975308642 \cdot 10^{-5}}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   13. cbrt-prod 56.7%

       \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \left(\color{blue}{\sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}} \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   14. associate-*l* 56.7%

       \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   15. add-cube-cbrt 56.6%

       \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}} \cdot \sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}}\right) \cdot \sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   16. pow3 56.6%

       \[\leadsto {\left(a \cdot \sin \color{blue}{\left({\left(\sqrt[3]{\left(\pi \cdot \sqrt[3]{\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}}\right) \cdot \sqrt[3]{angle \cdot 0.005555555555555556}}\right)}^{3}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

6. Applied egg-rr 78.7%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left({\left(\sqrt[3]{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}\right)}^{3}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

7. Taylor expanded in angle around 0 50.3%

   \[\leadsto {\left(a \cdot \color{blue}{0}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

8. Final simplification 50.3%

   \[\leadsto {b}^{2} + {\left(a \cdot 0\right)}^{2} \]

Reproduce

herbie shell --seed 2023187 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))