ab-angle->ABCF A

Percentage Accurate: 79.6% → 79.6%

Time: 44.7s

Alternatives: 12

Speedup: 1.0×

• 35.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 79.6% 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: 79.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* angle (/ PI 180.0)))) 2.0)
  (pow
   (* b (log1p (cbrt (pow (expm1 (cos (/ PI (/ -180.0 angle)))) 3.0))))
   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 * log1p(cbrt(pow(expm1(cos((((double) M_PI) / (-180.0 / angle)))), 3.0)))), 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 * Math.log1p(Math.cbrt(Math.pow(Math.expm1(Math.cos((Math.PI / (-180.0 / angle)))), 3.0)))), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(angle * Float64(pi / 180.0)))) ^ 2.0) + (Float64(b * log1p(cbrt((expm1(cos(Float64(pi / Float64(-180.0 / angle)))) ^ 3.0)))) ^ 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[N[(b * N[Log[1 + N[Power[N[Power[N[(Exp[N[Cos[N[(Pi / N[(-180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.7%

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

   1. associate-*l/ 81.7%

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

   2. associate-/l* 81.7%

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

   3. cos-neg 81.7%

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

   4. distribute-lft-neg-out 81.7%

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

   5. distribute-frac-neg 81.7%

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

   6. distribute-frac-neg 81.7%

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

   7. distribute-lft-neg-out 81.7%

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

   8. cos-neg 81.7%

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

   9. associate-*l/ 81.7%

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

   10. associate-/l* 81.7%

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

3. Simplified 81.7%

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

   1. associate-*r/ 81.7%

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

   2. associate-*l/ 81.7%

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

   3. log1p-expm1-u 81.7%

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

   4. associate-*l/ 81.7%

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

   5. associate-*r/ 81.7%

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

   6. div-inv 81.7%

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

   7. metadata-eval 81.7%

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

6. Applied egg-rr 81.7%

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

   1. add-cbrt-cube 81.7%

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

   2. pow3 81.7%

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

   3. metadata-eval 81.7%

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

   4. div-inv 81.7%

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

   5. *-commutative 81.7%

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

   6. associate-/r/ 81.8%

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

   7. frac-2neg 81.8%

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

   8. distribute-frac-neg 81.8%

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

   9. cos-neg 81.8%

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

   10. distribute-neg-frac 81.8%

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

   11. metadata-eval 81.8%

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

8. Applied egg-rr 81.8%

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

9. Final simplification 81.8%

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

Alternative 2: 65.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (expm1 (log1p (* angle (* PI 0.005555555555555556)))))) 2.0)
  (pow (* b (cos (/ PI (/ 180.0 angle)))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * sin(expm1(log1p((angle * (((double) M_PI) * 0.005555555555555556)))))), 2.0) + pow((b * cos((((double) M_PI) / (180.0 / angle)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin(Math.expm1(Math.log1p((angle * (Math.PI * 0.005555555555555556)))))), 2.0) + Math.pow((b * Math.cos((Math.PI / (180.0 / angle)))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((a * math.sin(math.expm1(math.log1p((angle * (math.pi * 0.005555555555555556)))))), 2.0) + math.pow((b * math.cos((math.pi / (180.0 / angle)))), 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(expm1(log1p(Float64(angle * Float64(pi * 0.005555555555555556)))))) ^ 2.0) + (Float64(b * cos(Float64(pi / Float64(180.0 / angle)))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(Exp[N[Log[1 + N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.7%

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

   1. *-commutative 81.7%

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

   2. clear-num 81.7%

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

   3. un-div-inv 81.8%

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

4. Applied egg-rr 81.8%

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

   1. expm1-log1p-u 67.8%

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

   2. associate-*l/ 67.8%

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

   3. associate-*r/ 67.8%

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

   4. div-inv 67.8%

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

   5. metadata-eval 67.8%

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

6. Applied egg-rr 67.8%

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

7. Final simplification 67.8%

   \[\leadsto {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]
8. Add Preprocessing

Alternative 3: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \frac{\pi}{\frac{1}{angle}}\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* PI (/ angle 180.0)))) 2.0)
  (pow (* b (cos (* 0.005555555555555556 (/ PI (/ 1.0 angle))))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * sin((((double) M_PI) * (angle / 180.0)))), 2.0) + pow((b * cos((0.005555555555555556 * (((double) M_PI) / (1.0 / angle))))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin((Math.PI * (angle / 180.0)))), 2.0) + Math.pow((b * Math.cos((0.005555555555555556 * (Math.PI / (1.0 / angle))))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((a * math.sin((math.pi * (angle / 180.0)))), 2.0) + math.pow((b * math.cos((0.005555555555555556 * (math.pi / (1.0 / angle))))), 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0) + (Float64(b * cos(Float64(0.005555555555555556 * Float64(pi / Float64(1.0 / angle))))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * sin((pi * (angle / 180.0)))) ^ 2.0) + ((b * cos((0.005555555555555556 * (pi / (1.0 / angle))))) ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. *-commutative 81.7%

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

   2. clear-num 81.7%

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

   3. un-div-inv 81.8%

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

4. Applied egg-rr 81.8%

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

   1. *-un-lft-identity 81.8%

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

   2. div-inv 81.8%

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

   3. times-frac 81.8%

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

   4. metadata-eval 81.8%

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

6. Applied egg-rr 81.8%

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

7. Final simplification 81.8%

   \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \frac{\pi}{\frac{1}{angle}}\right)\right)}^{2} \]
8. Add Preprocessing

Alternative 4: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. associate-*l/ 81.7%

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

   2. associate-/l* 81.7%

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

   3. cos-neg 81.7%

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

   4. distribute-lft-neg-out 81.7%

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

   5. distribute-frac-neg 81.7%

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

   6. distribute-frac-neg 81.7%

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

   7. distribute-lft-neg-out 81.7%

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

   8. cos-neg 81.7%

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

   9. associate-*l/ 81.7%

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

   10. associate-/l* 81.7%

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

3. Simplified 81.7%

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

5. Taylor expanded in angle around inf 81.7%

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

6. Final simplification 81.7%

   \[\leadsto {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} \]
7. Add Preprocessing

Alternative 5: 79.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* angle (/ PI 180.0)))) 2.0)
  (pow (* b (cos (* 0.005555555555555556 (* angle PI)))) 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 * cos((0.005555555555555556 * (angle * ((double) M_PI))))), 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 * Math.cos((0.005555555555555556 * (angle * Math.PI)))), 2.0);
}

Python

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

Julia

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

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * sin((angle * (pi / 180.0)))) ^ 2.0) + ((b * cos((0.005555555555555556 * (angle * pi)))) ^ 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[N[(b * N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.7%

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

   1. associate-*l/ 81.7%

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

   2. associate-/l* 81.7%

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

   3. cos-neg 81.7%

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

   4. distribute-lft-neg-out 81.7%

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

   5. distribute-frac-neg 81.7%

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

   6. distribute-frac-neg 81.7%

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

   7. distribute-lft-neg-out 81.7%

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

   8. cos-neg 81.7%

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

   9. associate-*l/ 81.7%

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

   10. associate-/l* 81.7%

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

3. Simplified 81.7%

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

5. Taylor expanded in angle around inf 81.8%

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

6. Final simplification 81.8%

   \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} \]
7. Add Preprocessing

Alternative 6: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. *-commutative 81.7%

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

   2. clear-num 81.7%

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

   3. un-div-inv 81.8%

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

4. Applied egg-rr 81.8%

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

5. Final simplification 81.8%

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

Alternative 7: 79.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. associate-*l/ 81.7%

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

   2. associate-/l* 81.7%

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

   3. cos-neg 81.7%

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

   4. distribute-lft-neg-out 81.7%

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

   5. distribute-frac-neg 81.7%

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

   6. distribute-frac-neg 81.7%

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

   7. distribute-lft-neg-out 81.7%

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

   8. cos-neg 81.7%

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

   9. associate-*l/ 81.7%

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

   10. associate-/l* 81.7%

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

3. Simplified 81.7%

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

5. Taylor expanded in angle around 0 81.4%

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

6. Taylor expanded in angle around inf 81.4%

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

7. Final simplification 81.4%

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

Alternative 8: 79.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. associate-*l/ 81.7%

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

   2. associate-/l* 81.7%

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

   3. cos-neg 81.7%

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

   4. distribute-lft-neg-out 81.7%

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

   5. distribute-frac-neg 81.7%

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

   6. distribute-frac-neg 81.7%

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

   7. distribute-lft-neg-out 81.7%

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

   8. cos-neg 81.7%

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

   9. associate-*l/ 81.7%

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

   10. associate-/l* 81.7%

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

3. Simplified 81.7%

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

5. Taylor expanded in angle around 0 81.4%

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

   1. *-commutative 81.4%

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

   2. associate-/r/ 81.4%

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

7. Applied egg-rr 81.4%

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

8. Final simplification 81.4%

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

Alternative 9: 74.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. associate-*l/ 81.7%

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

   2. associate-/l* 81.7%

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

   3. cos-neg 81.7%

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

   4. distribute-lft-neg-out 81.7%

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

   5. distribute-frac-neg 81.7%

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

   6. distribute-frac-neg 81.7%

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

   7. distribute-lft-neg-out 81.7%

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

   8. cos-neg 81.7%

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

   9. associate-*l/ 81.7%

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

   10. associate-/l* 81.7%

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

3. Simplified 81.7%

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

5. Taylor expanded in angle around 0 81.4%

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

6. Taylor expanded in angle around 0 76.6%

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

   1. *-commutative 76.6%

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

   2. *-commutative 76.6%

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

   3. associate-*l* 76.5%

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

8. Simplified 76.5%

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

9. Taylor expanded in angle around 0 76.6%

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

    1. associate-*r* 76.6%

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

11. Simplified 76.6%

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

12. Final simplification 76.6%

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

Alternative 10: 74.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(a \cdot angle\right)\\ {b}^{2} + 0.005555555555555556 \cdot \left(t\_0 \cdot \left(0.005555555555555556 \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. associate-*l/ 81.7%

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

   2. associate-/l* 81.7%

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

   3. cos-neg 81.7%

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

   4. distribute-lft-neg-out 81.7%

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

   5. distribute-frac-neg 81.7%

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

   6. distribute-frac-neg 81.7%

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

   7. distribute-lft-neg-out 81.7%

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

   8. cos-neg 81.7%

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

   9. associate-*l/ 81.7%

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

   10. associate-/l* 81.7%

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

3. Simplified 81.7%

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

5. Taylor expanded in angle around 0 81.4%

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

6. Taylor expanded in angle around 0 76.6%

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

   1. *-commutative 76.6%

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

   2. *-commutative 76.6%

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

   3. associate-*l* 76.5%

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

8. Simplified 76.5%

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

   1. unpow2 76.5%

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

   2. associate-*l* 76.5%

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

   3. *-commutative 76.5%

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

   4. associate-*r* 76.5%

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

   5. associate-*r* 76.6%

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

   6. *-commutative 76.6%

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

   7. associate-*r* 76.5%

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

10. Applied egg-rr 76.5%

    \[\leadsto \color{blue}{0.005555555555555556 \cdot \left(\left(\left(\pi \cdot a\right) \cdot angle\right) \cdot \left(\left(\frac{\pi}{180} \cdot angle\right) \cdot a\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
11. Step-by-step derivation

    1. *-commutative 76.5%

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

12. Simplified 76.5%

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

13. Taylor expanded in a around 0 76.6%

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

    1. associate-*r* 76.5%

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

    2. *-commutative 76.5%

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

    3. *-commutative 76.5%

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

15. Simplified 76.5%

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

16. Final simplification 76.5%

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

Alternative 11: 74.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. associate-*l/ 81.7%

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

   2. associate-/l* 81.7%

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

   3. cos-neg 81.7%

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

   4. distribute-lft-neg-out 81.7%

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

   5. distribute-frac-neg 81.7%

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

   6. distribute-frac-neg 81.7%

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

   7. distribute-lft-neg-out 81.7%

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

   8. cos-neg 81.7%

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

   9. associate-*l/ 81.7%

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

   10. associate-/l* 81.7%

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

3. Simplified 81.7%

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

5. Taylor expanded in angle around 0 81.4%

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

6. Taylor expanded in angle around 0 76.6%

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

   1. *-commutative 76.6%

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

   2. *-commutative 76.6%

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

   3. associate-*l* 76.5%

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

8. Simplified 76.5%

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

   1. unpow2 76.5%

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

   2. associate-*l* 76.5%

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

   3. *-commutative 76.5%

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

   4. associate-*r* 76.5%

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

   5. associate-*r* 76.6%

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

   6. *-commutative 76.6%

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

   7. associate-*r* 76.5%

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

10. Applied egg-rr 76.5%

    \[\leadsto \color{blue}{0.005555555555555556 \cdot \left(\left(\left(\pi \cdot a\right) \cdot angle\right) \cdot \left(\left(\frac{\pi}{180} \cdot angle\right) \cdot a\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
11. Step-by-step derivation

    1. *-commutative 76.5%

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

12. Simplified 76.5%

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

13. Final simplification 76.5%

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

Alternative 12: 74.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. associate-*l/ 81.7%

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

   2. associate-/l* 81.7%

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

   3. cos-neg 81.7%

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

   4. distribute-lft-neg-out 81.7%

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

   5. distribute-frac-neg 81.7%

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

   6. distribute-frac-neg 81.7%

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

   7. distribute-lft-neg-out 81.7%

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

   8. cos-neg 81.7%

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

   9. associate-*l/ 81.7%

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

   10. associate-/l* 81.7%

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

3. Simplified 81.7%

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

5. Taylor expanded in angle around 0 81.4%

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

6. Taylor expanded in angle around 0 76.6%

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

   1. *-commutative 76.6%

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

   2. *-commutative 76.6%

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

   3. associate-*l* 76.5%

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

8. Simplified 76.5%

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

   1. unpow2 76.5%

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

   2. associate-*l* 76.5%

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

   3. *-commutative 76.5%

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

   4. associate-*r* 76.5%

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

   5. associate-*r* 76.6%

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

   6. *-commutative 76.6%

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

   7. associate-*r* 76.5%

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

10. Applied egg-rr 76.5%

    \[\leadsto \color{blue}{0.005555555555555556 \cdot \left(\left(\left(\pi \cdot a\right) \cdot angle\right) \cdot \left(\left(\frac{\pi}{180} \cdot angle\right) \cdot a\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
11. Step-by-step derivation

    1. *-commutative 76.5%

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

12. Simplified 76.5%

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

13. Taylor expanded in a around 0 76.6%

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

    1. associate-*r* 76.6%

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

15. Simplified 76.6%

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

16. Final simplification 76.6%

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

Reproduce

herbie shell --seed 2024053 
(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)))