ab-angle->ABCF C

Percentage Accurate: 80.7% → 80.7%

Time: 47.2s

Alternatives: 11

Speedup: 1.5×

• 37.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \cos \left(\frac{\frac{\pi}{{\left(\sqrt[3]{180} \cdot \frac{1}{\sqrt[3]{angle}}\right)}^{2}}}{\sqrt[3]{\frac{180}{angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Cos[N[(N[(Pi / N[Power[N[(N[Power[180.0, 1/3], $MachinePrecision] * N[(1.0 / N[Power[angle, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(180.0 / angle), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.5%

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

   1. add-sqr-sqrt 35.7%

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

   2. sqrt-unprod 69.9%

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

   3. associate-*r/ 69.8%

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

   4. associate-*r/ 69.8%

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

   5. frac-times 68.6%

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

   6. metadata-eval 68.6%

      \[\leadsto {\left(a \cdot \cos \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   7. metadata-eval 68.6%

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

   8. frac-times 69.8%

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

   9. associate-*l/ 69.8%

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

   10. associate-*l/ 69.8%

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

   11. sqrt-unprod 48.8%

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

   12. add-sqr-sqrt 84.5%

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

   13. *-un-lft-identity 84.5%

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

   14. *-commutative 84.5%

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

3. Applied egg-rr 84.5%

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

   1. metadata-eval 84.5%

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

   2. div-inv 84.5%

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

   3. clear-num 84.5%

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

   4. div-inv 84.4%

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

   5. add-cube-cbrt 84.5%

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

   6. associate-/r* 84.6%

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

   7. pow2 84.6%

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

5. Applied egg-rr 84.6%

   \[\leadsto {\left(a \cdot \left(\cos \color{blue}{\left(\frac{\frac{\pi}{{\left(\sqrt[3]{\frac{180}{angle}}\right)}^{2}}}{\sqrt[3]{\frac{180}{angle}}}\right)} \cdot 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
6. Step-by-step derivation

   1. cbrt-div 84.6%

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

   2. div-inv 84.7%

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

7. Applied egg-rr 84.7%

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

8. Final simplification 84.7%

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

Alternative 2: 80.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot \cos \left(\frac{\frac{\pi}{{\left(\sqrt[3]{180} \cdot \sqrt[3]{\frac{1}{angle}}\right)}^{2}}}{\sqrt[3]{\frac{180}{angle}}}\right)\right)}^{2} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(b * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(N[(Pi / N[Power[N[(N[Power[180.0, 1/3], $MachinePrecision] * N[Power[N[(1.0 / angle), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(180.0 / angle), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.5%

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

   1. add-sqr-sqrt 35.7%

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

   2. sqrt-unprod 69.9%

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

   3. associate-*r/ 69.8%

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

   4. associate-*r/ 69.8%

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

   5. frac-times 68.6%

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

   6. metadata-eval 68.6%

      \[\leadsto {\left(a \cdot \cos \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   7. metadata-eval 68.6%

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

   8. frac-times 69.8%

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

   9. associate-*l/ 69.8%

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

   10. associate-*l/ 69.8%

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

   11. sqrt-unprod 48.8%

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

   12. add-sqr-sqrt 84.5%

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

   13. *-un-lft-identity 84.5%

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

   14. *-commutative 84.5%

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

3. Applied egg-rr 84.5%

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

   1. metadata-eval 84.5%

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

   2. div-inv 84.5%

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

   3. clear-num 84.5%

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

   4. div-inv 84.4%

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

   5. add-cube-cbrt 84.5%

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

   6. associate-/r* 84.6%

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

   7. pow2 84.6%

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

5. Applied egg-rr 84.6%

   \[\leadsto {\left(a \cdot \left(\cos \color{blue}{\left(\frac{\frac{\pi}{{\left(\sqrt[3]{\frac{180}{angle}}\right)}^{2}}}{\sqrt[3]{\frac{180}{angle}}}\right)} \cdot 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
6. Step-by-step derivation

   1. pow1/3 36.1%

      \[\leadsto {\left(a \cdot \left(\cos \left(\frac{\frac{\pi}{{\color{blue}{\left({\left(\frac{180}{angle}\right)}^{0.3333333333333333}\right)}}^{2}}}{\sqrt[3]{\frac{180}{angle}}}\right) \cdot 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   2. div-inv 36.1%

      \[\leadsto {\left(a \cdot \left(\cos \left(\frac{\frac{\pi}{{\left({\color{blue}{\left(180 \cdot \frac{1}{angle}\right)}}^{0.3333333333333333}\right)}^{2}}}{\sqrt[3]{\frac{180}{angle}}}\right) \cdot 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   3. unpow-prod-down 35.7%

      \[\leadsto {\left(a \cdot \left(\cos \left(\frac{\frac{\pi}{{\color{blue}{\left({180}^{0.3333333333333333} \cdot {\left(\frac{1}{angle}\right)}^{0.3333333333333333}\right)}}^{2}}}{\sqrt[3]{\frac{180}{angle}}}\right) \cdot 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. pow1/3 35.7%

      \[\leadsto {\left(a \cdot \left(\cos \left(\frac{\frac{\pi}{{\left(\color{blue}{\sqrt[3]{180}} \cdot {\left(\frac{1}{angle}\right)}^{0.3333333333333333}\right)}^{2}}}{\sqrt[3]{\frac{180}{angle}}}\right) \cdot 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

7. Applied egg-rr 35.7%

   \[\leadsto {\left(a \cdot \left(\cos \left(\frac{\frac{\pi}{{\color{blue}{\left(\sqrt[3]{180} \cdot {\left(\frac{1}{angle}\right)}^{0.3333333333333333}\right)}}^{2}}}{\sqrt[3]{\frac{180}{angle}}}\right) \cdot 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
8. Step-by-step derivation

   1. unpow1/3 84.6%

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

9. Simplified 84.6%

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

10. Final simplification 84.6%

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

Alternative 3: 80.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(b * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(N[(Pi / N[Power[N[Power[N[(180.0 / angle), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[180.0, 1/3], $MachinePrecision] / N[Power[angle, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.5%

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

   1. add-sqr-sqrt 35.7%

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

   2. sqrt-unprod 69.9%

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

   3. associate-*r/ 69.8%

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

   4. associate-*r/ 69.8%

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

   5. frac-times 68.6%

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

   6. metadata-eval 68.6%

      \[\leadsto {\left(a \cdot \cos \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   7. metadata-eval 68.6%

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

   8. frac-times 69.8%

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

   9. associate-*l/ 69.8%

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

   10. associate-*l/ 69.8%

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

   11. sqrt-unprod 48.8%

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

   12. add-sqr-sqrt 84.5%

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

   13. *-un-lft-identity 84.5%

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

   14. *-commutative 84.5%

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

3. Applied egg-rr 84.5%

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

   1. metadata-eval 84.5%

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

   2. div-inv 84.5%

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

   3. clear-num 84.5%

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

   4. div-inv 84.4%

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

   5. add-cube-cbrt 84.5%

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

   6. associate-/r* 84.6%

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

   7. pow2 84.6%

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

5. Applied egg-rr 84.6%

   \[\leadsto {\left(a \cdot \left(\cos \color{blue}{\left(\frac{\frac{\pi}{{\left(\sqrt[3]{\frac{180}{angle}}\right)}^{2}}}{\sqrt[3]{\frac{180}{angle}}}\right)} \cdot 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
6. Step-by-step derivation

   1. cbrt-div 84.6%

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

   2. div-inv 84.7%

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

7. Applied egg-rr 84.7%

   \[\leadsto {\left(a \cdot \left(\cos \left(\frac{\frac{\pi}{{\left(\sqrt[3]{\frac{180}{angle}}\right)}^{2}}}{\color{blue}{\sqrt[3]{180} \cdot \frac{1}{\sqrt[3]{angle}}}}\right) \cdot 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
8. Step-by-step derivation

   1. associate-*r/ 84.6%

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

   2. *-rgt-identity 84.6%

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

9. Simplified 84.6%

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

10. Final simplification 84.6%

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

Alternative 4: 80.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.5%

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

   1. add-sqr-sqrt 35.7%

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

   2. sqrt-unprod 69.9%

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

   3. associate-*r/ 69.8%

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

   4. associate-*r/ 69.8%

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

   5. frac-times 68.6%

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

   6. metadata-eval 68.6%

      \[\leadsto {\left(a \cdot \cos \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   7. metadata-eval 68.6%

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

   8. frac-times 69.8%

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

   9. associate-*l/ 69.8%

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

   10. associate-*l/ 69.8%

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

   11. sqrt-unprod 48.8%

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

   12. add-sqr-sqrt 84.5%

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

   13. *-un-lft-identity 84.5%

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

   14. *-commutative 84.5%

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

3. Applied egg-rr 84.5%

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

   1. metadata-eval 84.5%

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

   2. div-inv 84.5%

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

   3. clear-num 84.5%

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

   4. div-inv 84.4%

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

   5. add-cube-cbrt 84.5%

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

   6. associate-/r* 84.6%

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

   7. pow2 84.6%

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

5. Applied egg-rr 84.6%

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

6. Final simplification 84.6%

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

Alternative 5: 80.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.5%

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

3. Simplified 84.6%

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

4. Taylor expanded in angle around inf 84.6%

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

5. Final simplification 84.6%

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

Alternative 6: 80.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.5%

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

   1. clear-num 84.5%

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

   2. un-div-inv 84.6%

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

3. Applied egg-rr 84.6%

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

4. Final simplification 84.6%

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

Alternative 7: 80.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.5%

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

2. Taylor expanded in angle around 0 84.5%

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

4. Applied egg-rr 83.5%

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

5. Taylor expanded in b around 0 73.1%

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

   1. unpow2 73.1%

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

   2. *-commutative 73.1%

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

   3. associate-*r* 73.1%

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

   4. unpow2 73.1%

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

   5. swap-sqr 84.5%

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

   6. unpow2 84.5%

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

7. Simplified 84.5%

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

8. Final simplification 84.5%

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

Alternative 8: 80.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.5%

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

2. Taylor expanded in angle around 0 84.5%

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

   1. add-sqr-sqrt 35.6%

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

   2. sqrt-unprod 64.5%

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

   3. associate-*r/ 64.5%

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

   4. associate-*r/ 64.5%

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

   5. frac-times 63.3%

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

   6. metadata-eval 63.3%

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

   7. metadata-eval 63.3%

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

   8. frac-times 64.5%

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

   9. associate-*l/ 64.5%

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

   10. associate-*l/ 64.5%

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

   11. sqrt-unprod 48.9%

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

   12. add-sqr-sqrt 84.5%

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

   13. *-commutative 84.5%

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

   14. clear-num 84.5%

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

   15. un-div-inv 84.6%

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

4. Applied egg-rr 84.6%

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

5. Final simplification 84.6%

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

Alternative 9: 67.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-105}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(angle \cdot \left(\pi \cdot b\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.1e-105)
   (+ (pow a 2.0) (pow (* b 0.0) 2.0))
   (+ (pow a 2.0) (* 3.08641975308642e-5 (pow (* angle (* PI b)) 2.0)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.1e-105) {
		tmp = pow(a, 2.0) + pow((b * 0.0), 2.0);
	} else {
		tmp = pow(a, 2.0) + (3.08641975308642e-5 * pow((angle * (((double) M_PI) * b)), 2.0));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.1e-105) {
		tmp = Math.pow(a, 2.0) + Math.pow((b * 0.0), 2.0);
	} else {
		tmp = Math.pow(a, 2.0) + (3.08641975308642e-5 * Math.pow((angle * (Math.PI * b)), 2.0));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.1e-105:
		tmp = math.pow(a, 2.0) + math.pow((b * 0.0), 2.0)
	else:
		tmp = math.pow(a, 2.0) + (3.08641975308642e-5 * math.pow((angle * (math.pi * b)), 2.0))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.1e-105)
		tmp = Float64((a ^ 2.0) + (Float64(b * 0.0) ^ 2.0));
	else
		tmp = Float64((a ^ 2.0) + Float64(3.08641975308642e-5 * (Float64(angle * Float64(pi * b)) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.1e-105)
		tmp = (a ^ 2.0) + ((b * 0.0) ^ 2.0);
	else
		tmp = (a ^ 2.0) + (3.08641975308642e-5 * ((angle * (pi * b)) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.1e-105], N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * 0.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(3.08641975308642e-5 * N[Power[N[(angle * N[(Pi * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.10000000000000002e-105

   1. Initial program 82.6%

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

   2. Taylor expanded in angle around 0 82.5%

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

      1. add-sqr-sqrt 35.6%

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

      2. sqrt-unprod 60.9%

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

      3. associate-*r/ 60.8%

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

      4. associate-*r/ 60.8%

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

      5. frac-times 59.6%

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

      6. metadata-eval 59.6%

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

      7. metadata-eval 59.6%

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

      8. frac-times 60.8%

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

      9. associate-*l/ 60.8%

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

      10. associate-*l/ 60.8%

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

      11. sqrt-unprod 47.0%

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

      12. add-sqr-sqrt 82.5%

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

      13. add-cube-cbrt 82.4%

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

      14. pow3 82.5%

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

   4. Applied egg-rr 82.4%

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

   5. Taylor expanded in angle around 0 62.7%

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

   if 1.10000000000000002e-105 < b

   1. Initial program 88.3%

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

   2. Taylor expanded in angle around 0 88.3%

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

   4. Applied egg-rr 86.9%

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

   5. Taylor expanded in angle around 0 71.1%

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

      1. *-commutative 71.1%

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

      2. *-commutative 71.1%

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

      3. unpow2 71.1%

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

      4. unpow2 71.1%

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

      5. swap-sqr 71.0%

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

      6. unpow2 71.0%

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

      7. swap-sqr 85.4%

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

      8. associate-*r* 85.3%

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

      9. associate-*r* 85.3%

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

      10. unpow2 85.3%

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

      11. associate-*r* 85.4%

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

      12. *-commutative 85.4%

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

      13. associate-*r* 85.4%

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

   7. Simplified 85.4%

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

   8. Taylor expanded in angle around 0 71.1%

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

      1. associate-*r* 71.1%

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

      2. unpow2 71.1%

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

      3. unpow2 71.1%

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

      4. swap-sqr 85.3%

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

      5. *-commutative 85.3%

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

      6. unpow2 85.3%

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

      7. swap-sqr 85.3%

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

      8. unpow2 85.3%

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

      9. *-commutative 85.3%

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

      10. associate-*r* 85.4%

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

      11. *-commutative 85.4%

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

   10. Simplified 85.4%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{{\left(angle \cdot \left(\pi \cdot b\right)\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-105}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(angle \cdot \left(\pi \cdot b\right)\right)}^{2}\\ \end{array} \]

Alternative 10: 67.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-103}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(b \cdot \left(\pi \cdot angle\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 2.4e-103)
   (+ (pow a 2.0) (pow (* b 0.0) 2.0))
   (+ (pow a 2.0) (* 3.08641975308642e-5 (pow (* b (* PI angle)) 2.0)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2.4e-103) {
		tmp = pow(a, 2.0) + pow((b * 0.0), 2.0);
	} else {
		tmp = pow(a, 2.0) + (3.08641975308642e-5 * pow((b * (((double) M_PI) * angle)), 2.0));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2.4e-103) {
		tmp = Math.pow(a, 2.0) + Math.pow((b * 0.0), 2.0);
	} else {
		tmp = Math.pow(a, 2.0) + (3.08641975308642e-5 * Math.pow((b * (Math.PI * angle)), 2.0));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 2.4e-103:
		tmp = math.pow(a, 2.0) + math.pow((b * 0.0), 2.0)
	else:
		tmp = math.pow(a, 2.0) + (3.08641975308642e-5 * math.pow((b * (math.pi * angle)), 2.0))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 2.4e-103)
		tmp = Float64((a ^ 2.0) + (Float64(b * 0.0) ^ 2.0));
	else
		tmp = Float64((a ^ 2.0) + Float64(3.08641975308642e-5 * (Float64(b * Float64(pi * angle)) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 2.4e-103)
		tmp = (a ^ 2.0) + ((b * 0.0) ^ 2.0);
	else
		tmp = (a ^ 2.0) + (3.08641975308642e-5 * ((b * (pi * angle)) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 2.4e-103], N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * 0.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(3.08641975308642e-5 * N[Power[N[(b * N[(Pi * angle), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.4000000000000002e-103

   1. Initial program 82.6%

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

   2. Taylor expanded in angle around 0 82.5%

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

      1. add-sqr-sqrt 35.6%

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

      2. sqrt-unprod 60.9%

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

      3. associate-*r/ 60.8%

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

      4. associate-*r/ 60.8%

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

      5. frac-times 59.6%

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

      6. metadata-eval 59.6%

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

      7. metadata-eval 59.6%

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

      8. frac-times 60.8%

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

      9. associate-*l/ 60.8%

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

      10. associate-*l/ 60.8%

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

      11. sqrt-unprod 47.0%

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

      12. add-sqr-sqrt 82.5%

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

      13. add-cube-cbrt 82.4%

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

      14. pow3 82.5%

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

   4. Applied egg-rr 82.4%

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

   5. Taylor expanded in angle around 0 62.7%

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

   if 2.4000000000000002e-103 < b

   1. Initial program 88.3%

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

   2. Taylor expanded in angle around 0 88.3%

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

   4. Applied egg-rr 86.9%

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

   5. Taylor expanded in angle around 0 71.1%

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

      1. *-commutative 71.1%

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

      2. *-commutative 71.1%

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

      3. unpow2 71.1%

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

      4. unpow2 71.1%

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

      5. swap-sqr 71.0%

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

      6. unpow2 71.0%

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

      7. swap-sqr 85.4%

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

      8. associate-*r* 85.3%

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

      9. associate-*r* 85.3%

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

      10. unpow2 85.3%

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

      11. associate-*r* 85.4%

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

      12. *-commutative 85.4%

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

      13. associate-*r* 85.4%

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

   7. Simplified 85.4%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot {\left(\left(angle \cdot \pi\right) \cdot b\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-103}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(b \cdot \left(\pi \cdot angle\right)\right)}^{2}\\ \end{array} \]

Alternative 11: 57.3% accurate, 3.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double angle) {
	return pow(a, 2.0) + pow((b * 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 = (a ** 2.0d0) + ((b * 0.0d0) ** 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.5%

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

2. Taylor expanded in angle around 0 84.5%

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

   1. add-sqr-sqrt 35.6%

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

   2. sqrt-unprod 64.5%

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

   3. associate-*r/ 64.5%

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

   4. associate-*r/ 64.5%

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

   5. frac-times 63.3%

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

   6. metadata-eval 63.3%

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

   7. metadata-eval 63.3%

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

   8. frac-times 64.5%

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

   9. associate-*l/ 64.5%

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

   10. associate-*l/ 64.5%

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

   11. sqrt-unprod 48.9%

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

   12. add-sqr-sqrt 84.5%

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

   13. add-cube-cbrt 84.3%

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

   14. pow3 84.3%

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

4. Applied egg-rr 84.3%

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

5. Taylor expanded in angle around 0 57.0%

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

6. Final simplification 57.0%

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

Reproduce

herbie shell --seed 2023310 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))