ab-angle->ABCF C

Percentage Accurate: 79.1% → 78.9%

Time: 1.3min

Alternatives: 8

Speedup: N/A×

• 36.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.1% 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: 78.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

   1. unpow2 79.7%

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

   2. swap-sqr 73.0%

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

   3. sqr-neg 73.0%

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

   4. swap-sqr 79.7%

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

   5. unpow2 79.7%

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

   6. distribute-lft-neg-out 79.7%

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

   7. distribute-rgt-neg-in 79.7%

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

   8. sin-neg 79.7%

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

   9. distribute-rgt-neg-out 79.7%

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

   10. distribute-frac-neg 79.7%

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

   11. unpow2 79.7%

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

   12. associate-*l* 79.1%

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

3. Simplified 79.8%

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

4. Taylor expanded in angle around 0 81.0%

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

   1. add-sqr-sqrt 44.8%

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

   2. sqrt-unprod 64.9%

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

   3. swap-sqr 64.4%

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

   4. metadata-eval 64.4%

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

   5. metadata-eval 64.4%

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

   6. metadata-eval 64.4%

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

   7. metadata-eval 64.4%

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

   8. swap-sqr 64.9%

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

   9. div-inv 64.8%

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

   10. div-inv 64.8%

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

   11. sqrt-unprod 36.2%

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

   12. add-sqr-sqrt 81.0%

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

   13. associate-*r/ 81.0%

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

   14. clear-num 81.1%

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

6. Applied egg-rr 81.1%

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

7. Final simplification 81.1%

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

Alternative 2: 79.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

   1. unpow2 79.7%

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

   2. swap-sqr 73.0%

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

   3. sqr-neg 73.0%

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

   4. swap-sqr 79.7%

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

   5. unpow2 79.7%

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

   6. distribute-lft-neg-out 79.7%

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

   7. distribute-rgt-neg-in 79.7%

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

   8. sin-neg 79.7%

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

   9. distribute-rgt-neg-out 79.7%

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

   10. distribute-frac-neg 79.7%

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

   11. unpow2 79.7%

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

   12. associate-*l* 79.1%

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

3. Simplified 79.8%

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

4. Taylor expanded in angle around 0 81.0%

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

5. Final simplification 81.0%

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

Alternative 3: 79.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

   1. unpow2 79.7%

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

   2. swap-sqr 73.0%

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

   3. sqr-neg 73.0%

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

   4. swap-sqr 79.7%

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

   5. unpow2 79.7%

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

   6. distribute-lft-neg-out 79.7%

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

   7. distribute-rgt-neg-in 79.7%

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

   8. sin-neg 79.7%

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

   9. distribute-rgt-neg-out 79.7%

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

   10. distribute-frac-neg 79.7%

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

   11. unpow2 79.7%

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

   12. associate-*l* 79.1%

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

3. Simplified 79.8%

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

4. Taylor expanded in angle around 0 81.0%

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

   1. add-sqr-sqrt 44.8%

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

   2. sqrt-unprod 64.9%

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

   3. swap-sqr 64.4%

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

   4. metadata-eval 64.4%

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

   5. metadata-eval 64.4%

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

   6. metadata-eval 64.4%

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

   7. metadata-eval 64.4%

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

   8. swap-sqr 64.9%

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

   9. div-inv 64.8%

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

   10. div-inv 64.8%

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

   11. sqrt-unprod 36.2%

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

   12. add-sqr-sqrt 81.0%

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

   13. associate-*r/ 81.0%

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

6. Applied egg-rr 81.0%

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

7. Final simplification 81.0%

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

Alternative 4: 66.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{+25}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(0.005555555555555556 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 9e+25)
   (+ (pow a 2.0) (pow (* b 0.0) 2.0))
   (+ (pow a 2.0) (pow (* 0.005555555555555556 (* angle (* b PI))) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 9e+25) {
		tmp = pow(a, 2.0) + pow((b * 0.0), 2.0);
	} else {
		tmp = pow(a, 2.0) + pow((0.005555555555555556 * (angle * (b * ((double) M_PI)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 9e+25) {
		tmp = Math.pow(a, 2.0) + Math.pow((b * 0.0), 2.0);
	} else {
		tmp = Math.pow(a, 2.0) + Math.pow((0.005555555555555556 * (angle * (b * Math.PI))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 9e+25:
		tmp = math.pow(a, 2.0) + math.pow((b * 0.0), 2.0)
	else:
		tmp = math.pow(a, 2.0) + math.pow((0.005555555555555556 * (angle * (b * math.pi))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 9e+25)
		tmp = Float64((a ^ 2.0) + (Float64(b * 0.0) ^ 2.0));
	else
		tmp = Float64((a ^ 2.0) + (Float64(0.005555555555555556 * Float64(angle * Float64(b * pi))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 9e+25)
		tmp = (a ^ 2.0) + ((b * 0.0) ^ 2.0);
	else
		tmp = (a ^ 2.0) + ((0.005555555555555556 * (angle * (b * pi))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 9e+25], 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[Power[N[(0.005555555555555556 * N[(angle * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.0000000000000006e25

   1. Initial program 76.2%

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

      1. unpow2 76.2%

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

      2. swap-sqr 70.9%

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

      3. sqr-neg 70.9%

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

      4. swap-sqr 76.2%

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

      5. unpow2 76.2%

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

      6. distribute-lft-neg-out 76.2%

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

      7. distribute-rgt-neg-in 76.2%

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

      8. sin-neg 76.2%

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

      9. distribute-rgt-neg-out 76.2%

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

      10. distribute-frac-neg 76.2%

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

      11. unpow2 76.2%

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

      12. associate-*l* 75.7%

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

   3. Simplified 76.2%

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

   4. Taylor expanded in angle around 0 77.7%

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

      1. add-cube-cbrt 77.5%

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

      2. pow3 77.4%

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

   6. Applied egg-rr 77.4%

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

   7. Taylor expanded in angle around 0 64.5%

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

   if 9.0000000000000006e25 < b

   1. Initial program 94.9%

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

   2. Taylor expanded in angle around 0 94.9%

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

   3. Taylor expanded in angle around 0 94.3%

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

      1. *-commutative 94.3%

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

   5. Simplified 94.3%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{+25}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(0.005555555555555556 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 5: 66.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.2 \cdot 10^{+25}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 8.2e+25)
   (+ (pow a 2.0) (pow (* b 0.0) 2.0))
   (+ (pow a 2.0) (pow (* b (* -0.005555555555555556 (* PI angle))) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 8.2e+25) {
		tmp = pow(a, 2.0) + pow((b * 0.0), 2.0);
	} else {
		tmp = pow(a, 2.0) + pow((b * (-0.005555555555555556 * (((double) M_PI) * angle))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 8.2e+25) {
		tmp = Math.pow(a, 2.0) + Math.pow((b * 0.0), 2.0);
	} else {
		tmp = Math.pow(a, 2.0) + Math.pow((b * (-0.005555555555555556 * (Math.PI * angle))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 8.2e+25:
		tmp = math.pow(a, 2.0) + math.pow((b * 0.0), 2.0)
	else:
		tmp = math.pow(a, 2.0) + math.pow((b * (-0.005555555555555556 * (math.pi * angle))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 8.2e+25)
		tmp = Float64((a ^ 2.0) + (Float64(b * 0.0) ^ 2.0));
	else
		tmp = Float64((a ^ 2.0) + (Float64(b * Float64(-0.005555555555555556 * Float64(pi * angle))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 8.2e+25)
		tmp = (a ^ 2.0) + ((b * 0.0) ^ 2.0);
	else
		tmp = (a ^ 2.0) + ((b * (-0.005555555555555556 * (pi * angle))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 8.2e+25], 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[Power[N[(b * N[(-0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.19999999999999933e25

   1. Initial program 76.2%

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

      1. unpow2 76.2%

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

      2. swap-sqr 70.9%

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

      3. sqr-neg 70.9%

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

      4. swap-sqr 76.2%

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

      5. unpow2 76.2%

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

      6. distribute-lft-neg-out 76.2%

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

      7. distribute-rgt-neg-in 76.2%

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

      8. sin-neg 76.2%

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

      9. distribute-rgt-neg-out 76.2%

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

      10. distribute-frac-neg 76.2%

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

      11. unpow2 76.2%

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

      12. associate-*l* 75.7%

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

   3. Simplified 76.2%

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

   4. Taylor expanded in angle around 0 77.7%

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

      1. add-cube-cbrt 77.5%

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

      2. pow3 77.4%

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

   6. Applied egg-rr 77.4%

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

   7. Taylor expanded in angle around 0 64.5%

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

   if 8.19999999999999933e25 < b

   1. Initial program 94.9%

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

      1. unpow2 94.9%

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

      2. swap-sqr 81.7%

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

      3. sqr-neg 81.7%

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

      4. swap-sqr 94.9%

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

      5. unpow2 94.9%

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

      6. distribute-lft-neg-out 94.9%

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

      7. distribute-rgt-neg-in 94.9%

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

      8. sin-neg 94.9%

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

      9. distribute-rgt-neg-out 94.9%

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

      10. distribute-frac-neg 94.9%

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

      11. unpow2 94.9%

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

      12. associate-*l* 93.2%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in angle around 0 94.8%

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

   5. Taylor expanded in angle around 0 94.3%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.2 \cdot 10^{+25}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 6: 66.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.2 \cdot 10^{+25}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 8.2e+25)
   (+ (pow a 2.0) (pow (* b 0.0) 2.0))
   (+ (pow a 2.0) (pow (* b (* angle (* PI -0.005555555555555556))) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 8.2e+25) {
		tmp = pow(a, 2.0) + pow((b * 0.0), 2.0);
	} else {
		tmp = pow(a, 2.0) + pow((b * (angle * (((double) M_PI) * -0.005555555555555556))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 8.2e+25) {
		tmp = Math.pow(a, 2.0) + Math.pow((b * 0.0), 2.0);
	} else {
		tmp = Math.pow(a, 2.0) + Math.pow((b * (angle * (Math.PI * -0.005555555555555556))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 8.2e+25:
		tmp = math.pow(a, 2.0) + math.pow((b * 0.0), 2.0)
	else:
		tmp = math.pow(a, 2.0) + math.pow((b * (angle * (math.pi * -0.005555555555555556))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 8.2e+25)
		tmp = Float64((a ^ 2.0) + (Float64(b * 0.0) ^ 2.0));
	else
		tmp = Float64((a ^ 2.0) + (Float64(b * Float64(angle * Float64(pi * -0.005555555555555556))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 8.2e+25)
		tmp = (a ^ 2.0) + ((b * 0.0) ^ 2.0);
	else
		tmp = (a ^ 2.0) + ((b * (angle * (pi * -0.005555555555555556))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 8.2e+25], 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[Power[N[(b * N[(angle * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.19999999999999933e25

   1. Initial program 76.2%

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

      1. unpow2 76.2%

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

      2. swap-sqr 70.9%

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

      3. sqr-neg 70.9%

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

      4. swap-sqr 76.2%

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

      5. unpow2 76.2%

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

      6. distribute-lft-neg-out 76.2%

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

      7. distribute-rgt-neg-in 76.2%

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

      8. sin-neg 76.2%

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

      9. distribute-rgt-neg-out 76.2%

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

      10. distribute-frac-neg 76.2%

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

      11. unpow2 76.2%

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

      12. associate-*l* 75.7%

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

   3. Simplified 76.2%

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

   4. Taylor expanded in angle around 0 77.7%

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

      1. add-cube-cbrt 77.5%

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

      2. pow3 77.4%

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

   6. Applied egg-rr 77.4%

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

   7. Taylor expanded in angle around 0 64.5%

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

   if 8.19999999999999933e25 < b

   1. Initial program 94.9%

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

      1. unpow2 94.9%

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

      2. swap-sqr 81.7%

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

      3. sqr-neg 81.7%

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

      4. swap-sqr 94.9%

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

      5. unpow2 94.9%

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

      6. distribute-lft-neg-out 94.9%

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

      7. distribute-rgt-neg-in 94.9%

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

      8. sin-neg 94.9%

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

      9. distribute-rgt-neg-out 94.9%

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

      10. distribute-frac-neg 94.9%

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

      11. unpow2 94.9%

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

      12. associate-*l* 93.2%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in angle around 0 94.8%

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

   5. Taylor expanded in angle around 0 94.3%

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

      1. associate-*r* 94.3%

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

      2. *-commutative 94.3%

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

      3. associate-*l* 94.3%

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

   7. Simplified 94.3%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.2 \cdot 10^{+25}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 7: 66.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.2 \cdot 10^{+25}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(\left(b \cdot angle\right) \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 8.2e+25)
   (+ (pow a 2.0) (pow (* b 0.0) 2.0))
   (+ (pow a 2.0) (pow (* (* b angle) (* PI 0.005555555555555556)) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 8.2e+25) {
		tmp = pow(a, 2.0) + pow((b * 0.0), 2.0);
	} else {
		tmp = pow(a, 2.0) + pow(((b * angle) * (((double) M_PI) * 0.005555555555555556)), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 8.2e+25) {
		tmp = Math.pow(a, 2.0) + Math.pow((b * 0.0), 2.0);
	} else {
		tmp = Math.pow(a, 2.0) + Math.pow(((b * angle) * (Math.PI * 0.005555555555555556)), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 8.2e+25:
		tmp = math.pow(a, 2.0) + math.pow((b * 0.0), 2.0)
	else:
		tmp = math.pow(a, 2.0) + math.pow(((b * angle) * (math.pi * 0.005555555555555556)), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 8.2e+25)
		tmp = Float64((a ^ 2.0) + (Float64(b * 0.0) ^ 2.0));
	else
		tmp = Float64((a ^ 2.0) + (Float64(Float64(b * angle) * Float64(pi * 0.005555555555555556)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 8.2e+25)
		tmp = (a ^ 2.0) + ((b * 0.0) ^ 2.0);
	else
		tmp = (a ^ 2.0) + (((b * angle) * (pi * 0.005555555555555556)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 8.2e+25], 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[Power[N[(N[(b * angle), $MachinePrecision] * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.19999999999999933e25

   1. Initial program 76.2%

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

      1. unpow2 76.2%

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

      2. swap-sqr 70.9%

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

      3. sqr-neg 70.9%

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

      4. swap-sqr 76.2%

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

      5. unpow2 76.2%

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

      6. distribute-lft-neg-out 76.2%

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

      7. distribute-rgt-neg-in 76.2%

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

      8. sin-neg 76.2%

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

      9. distribute-rgt-neg-out 76.2%

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

      10. distribute-frac-neg 76.2%

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

      11. unpow2 76.2%

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

      12. associate-*l* 75.7%

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

   3. Simplified 76.2%

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

   4. Taylor expanded in angle around 0 77.7%

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

      1. add-cube-cbrt 77.5%

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

      2. pow3 77.4%

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

   6. Applied egg-rr 77.4%

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

   7. Taylor expanded in angle around 0 64.5%

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

   if 8.19999999999999933e25 < b

   1. Initial program 94.9%

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

   2. Taylor expanded in angle around 0 94.9%

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

   3. Taylor expanded in angle around 0 94.3%

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

      1. *-commutative 94.3%

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

      2. associate-*r* 94.3%

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

      3. associate-*l* 94.4%

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

   5. Simplified 94.4%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.2 \cdot 10^{+25}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(\left(b \cdot angle\right) \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{2}\\ \end{array} \]

Alternative 8: 57.1% 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 79.7%

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

   1. unpow2 79.7%

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

   2. swap-sqr 73.0%

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

   3. sqr-neg 73.0%

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

   4. swap-sqr 79.7%

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

   5. unpow2 79.7%

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

   6. distribute-lft-neg-out 79.7%

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

   7. distribute-rgt-neg-in 79.7%

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

   8. sin-neg 79.7%

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

   9. distribute-rgt-neg-out 79.7%

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

   10. distribute-frac-neg 79.7%

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

   11. unpow2 79.7%

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

   12. associate-*l* 79.1%

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

3. Simplified 79.8%

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

4. Taylor expanded in angle around 0 81.0%

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

   1. add-cube-cbrt 80.7%

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

   2. pow3 80.7%

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

6. Applied egg-rr 80.7%

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

7. Taylor expanded in angle around 0 61.4%

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

8. Final simplification 61.4%

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

Reproduce

herbie shell --seed 2023279 
(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)))