ab-angle->ABCF A

Percentage Accurate: 79.6% → 79.5%

Time: 25.6s

Alternatives: 10

Speedup: N/A×

• 37.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. *-commutative 79.4%

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

   2. associate-*r/ 79.5%

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

   3. associate-*l/ 79.6%

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

   4. *-commutative 79.6%

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

   5. *-commutative 79.6%

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

   6. associate-*r/ 79.4%

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

   7. associate-*l/ 79.5%

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

   8. *-commutative 79.5%

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

3. Simplified 79.5%

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

5. Taylor expanded in angle around 0 79.8%

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

   1. associate-*r/ 79.8%

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

   2. associate-*l/ 79.7%

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

   3. add-cbrt-cube 50.9%

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

   4. pow1/3 36.6%

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

   5. pow3 36.6%

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

   6. associate-*l/ 36.6%

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

   7. associate-*r/ 36.6%

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

   8. div-inv 36.6%

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

   9. metadata-eval 36.6%

      \[\leadsto {\left(a \cdot \sin \left({\left({\left(angle \cdot \left(\pi \cdot \color{blue}{0.005555555555555556}\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

7. Applied egg-rr 36.6%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left({\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{3}\right)}^{0.3333333333333333}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
8. Step-by-step derivation

   1. unpow1/3 50.9%

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

   2. rem-cbrt-cube 79.8%

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

   3. *-commutative 79.8%

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

   4. metadata-eval 79.8%

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

   5. div-inv 79.8%

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

   6. associate-/r/ 79.8%

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

   7. div-inv 79.8%

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

   8. associate-/r* 79.8%

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

   9. div-inv 79.8%

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

   10. metadata-eval 79.8%

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

9. Applied egg-rr 79.8%

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

10. Final simplification 79.8%

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

Alternative 2: 79.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. *-commutative 79.4%

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

   2. associate-*r/ 79.5%

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

   3. associate-*l/ 79.6%

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

   4. *-commutative 79.6%

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

   5. *-commutative 79.6%

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

   6. associate-*r/ 79.4%

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

   7. associate-*l/ 79.5%

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

   8. *-commutative 79.5%

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

3. Simplified 79.5%

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

5. Taylor expanded in angle around 0 79.8%

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

6. Taylor expanded in angle around 0 79.8%

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

7. Final simplification 79.8%

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

Alternative 3: 79.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. *-commutative 79.4%

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

   2. associate-*r/ 79.5%

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

   3. associate-*l/ 79.6%

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

   4. *-commutative 79.6%

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

   5. *-commutative 79.6%

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

   6. associate-*r/ 79.4%

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

   7. associate-*l/ 79.5%

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

   8. *-commutative 79.5%

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

3. Simplified 79.5%

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

5. Taylor expanded in angle around 0 79.8%

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

6. Final simplification 79.8%

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

Alternative 4: 79.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. *-commutative 79.4%

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

   2. associate-*r/ 79.5%

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

   3. associate-*l/ 79.6%

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

   4. *-commutative 79.6%

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

   5. *-commutative 79.6%

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

   6. associate-*r/ 79.4%

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

   7. associate-*l/ 79.5%

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

   8. *-commutative 79.5%

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

3. Simplified 79.5%

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

5. Taylor expanded in angle around 0 79.8%

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

   1. associate-*r/ 79.8%

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

   2. associate-*l/ 79.7%

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

   3. *-commutative 79.7%

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

   4. clear-num 79.8%

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

   5. un-div-inv 79.8%

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

7. Applied egg-rr 79.8%

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

8. Final simplification 79.8%

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

Alternative 5: 67.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.2 \cdot 10^{-75}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(0.005555555555555556 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \left(a \cdot \pi\right)\right)\right)\right) \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.2e-75)
   (pow b 2.0)
   (+
    (pow b 2.0)
    (*
     (* 0.005555555555555556 (* angle (* 0.005555555555555556 (* a PI))))
     (* angle (* a PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.2e-75) {
		tmp = pow(b, 2.0);
	} else {
		tmp = pow(b, 2.0) + ((0.005555555555555556 * (angle * (0.005555555555555556 * (a * ((double) M_PI))))) * (angle * (a * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.2e-75) {
		tmp = Math.pow(b, 2.0);
	} else {
		tmp = Math.pow(b, 2.0) + ((0.005555555555555556 * (angle * (0.005555555555555556 * (a * Math.PI)))) * (angle * (a * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 1.2e-75:
		tmp = math.pow(b, 2.0)
	else:
		tmp = math.pow(b, 2.0) + ((0.005555555555555556 * (angle * (0.005555555555555556 * (a * math.pi)))) * (angle * (a * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.2e-75)
		tmp = b ^ 2.0;
	else
		tmp = Float64((b ^ 2.0) + Float64(Float64(0.005555555555555556 * Float64(angle * Float64(0.005555555555555556 * Float64(a * pi)))) * Float64(angle * Float64(a * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1.2e-75)
		tmp = b ^ 2.0;
	else
		tmp = (b ^ 2.0) + ((0.005555555555555556 * (angle * (0.005555555555555556 * (a * pi)))) * (angle * (a * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.2e-75], N[Power[b, 2.0], $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(0.005555555555555556 * N[(angle * N[(0.005555555555555556 * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.2000000000000001e-75

   1. Initial program 81.4%

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

      1. *-commutative 81.4%

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

      2. associate-*r/ 81.5%

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

      3. associate-*l/ 81.6%

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

      4. *-commutative 81.6%

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

      5. *-commutative 81.6%

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

      6. associate-*r/ 81.5%

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

      7. associate-*l/ 81.6%

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

      8. *-commutative 81.6%

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

   3. Simplified 81.6%

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

   5. Taylor expanded in angle around 0 82.0%

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

   6. Taylor expanded in angle around 0 78.0%

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

      1. *-commutative 78.0%

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

      2. associate-*l* 77.9%

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

   8. Simplified 77.9%

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

   9. Taylor expanded in angle around 0 64.9%

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

       1. *-commutative 64.9%

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

       2. associate-*l* 64.9%

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

       3. unpow2 64.9%

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

       4. unpow2 64.9%

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

       5. unpow2 64.9%

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

       6. swap-sqr 64.9%

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

       7. swap-sqr 77.9%

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

       8. unpow2 77.9%

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

       9. associate-*r* 77.9%

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

       10. *-commutative 77.9%

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

       11. associate-*l* 77.9%

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

   11. Simplified 77.9%

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

   12. Taylor expanded in angle around 0 60.0%

       \[\leadsto \color{blue}{{b}^{2}} \]

   if 1.2000000000000001e-75 < a

   1. Initial program 75.3%

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

      1. *-commutative 75.3%

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

      2. associate-*r/ 75.4%

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

      3. associate-*l/ 75.4%

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

      4. *-commutative 75.4%

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

      5. *-commutative 75.4%

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

      6. associate-*r/ 75.3%

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

      7. associate-*l/ 75.4%

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

      8. *-commutative 75.4%

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

   3. Simplified 75.4%

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

   5. Taylor expanded in angle around 0 75.3%

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

   6. Taylor expanded in angle around 0 70.7%

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

      1. *-commutative 70.7%

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

      2. associate-*l* 70.7%

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

   8. Simplified 70.7%

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

      1. unpow2 70.7%

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

      2. associate-*r* 70.8%

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

      3. *-commutative 70.8%

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

      4. associate-*l* 70.8%

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

   10. Applied egg-rr 70.8%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.2 \cdot 10^{-75}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(0.005555555555555556 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \left(a \cdot \pi\right)\right)\right)\right) \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{-68}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(\left(\pi \cdot angle\right) \cdot \left(a \cdot 0.005555555555555556\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.5e-68)
   (pow b 2.0)
   (+ (pow b 2.0) (pow (* (* PI angle) (* a 0.005555555555555556)) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.5e-68) {
		tmp = pow(b, 2.0);
	} else {
		tmp = pow(b, 2.0) + pow(((((double) M_PI) * angle) * (a * 0.005555555555555556)), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.5e-68) {
		tmp = Math.pow(b, 2.0);
	} else {
		tmp = Math.pow(b, 2.0) + Math.pow(((Math.PI * angle) * (a * 0.005555555555555556)), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 3.5e-68:
		tmp = math.pow(b, 2.0)
	else:
		tmp = math.pow(b, 2.0) + math.pow(((math.pi * angle) * (a * 0.005555555555555556)), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 3.5e-68)
		tmp = b ^ 2.0;
	else
		tmp = Float64((b ^ 2.0) + (Float64(Float64(pi * angle) * Float64(a * 0.005555555555555556)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 3.5e-68)
		tmp = b ^ 2.0;
	else
		tmp = (b ^ 2.0) + (((pi * angle) * (a * 0.005555555555555556)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 3.5e-68], N[Power[b, 2.0], $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[Power[N[(N[(Pi * angle), $MachinePrecision] * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.50000000000000013e-68

   1. Initial program 81.1%

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

      1. *-commutative 81.1%

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

      2. associate-*r/ 81.2%

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

      3. associate-*l/ 81.2%

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

      4. *-commutative 81.2%

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

      5. *-commutative 81.2%

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

      6. associate-*r/ 81.1%

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

      7. associate-*l/ 81.2%

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

      8. *-commutative 81.2%

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

   3. Simplified 81.2%

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

   5. Taylor expanded in angle around 0 81.6%

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

   6. Taylor expanded in angle around 0 77.6%

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

      1. *-commutative 77.6%

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

      2. associate-*l* 77.6%

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

   8. Simplified 77.6%

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

   9. Taylor expanded in angle around 0 64.6%

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

       1. *-commutative 64.6%

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

       2. associate-*l* 64.6%

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

       3. unpow2 64.6%

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

       4. unpow2 64.6%

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

       5. unpow2 64.6%

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

       6. swap-sqr 64.6%

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

       7. swap-sqr 77.6%

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

       8. unpow2 77.6%

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

       9. associate-*r* 77.6%

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

       10. *-commutative 77.6%

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

       11. associate-*l* 77.6%

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

   11. Simplified 77.6%

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

   12. Taylor expanded in angle around 0 59.7%

       \[\leadsto \color{blue}{{b}^{2}} \]

   if 3.50000000000000013e-68 < a

   1. Initial program 76.0%

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

      1. *-commutative 76.0%

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

      2. associate-*r/ 76.1%

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

      3. associate-*l/ 76.1%

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

      4. *-commutative 76.1%

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

      5. *-commutative 76.1%

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

      6. associate-*r/ 76.0%

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

      7. associate-*l/ 76.1%

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

      8. *-commutative 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in angle around 0 76.0%

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

   6. Taylor expanded in angle around 0 71.4%

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

      1. associate-*r* 71.3%

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

   8. Simplified 71.3%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{-68}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(\left(\pi \cdot angle\right) \cdot \left(a \cdot 0.005555555555555556\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.9 \cdot 10^{-75}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b, {\left(a \cdot \left(\pi \cdot angle\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 2.9e-75)
   (pow b 2.0)
   (fma b b (* (pow (* a (* PI angle)) 2.0) 3.08641975308642e-5))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.9e-75) {
		tmp = pow(b, 2.0);
	} else {
		tmp = fma(b, b, (pow((a * (((double) M_PI) * angle)), 2.0) * 3.08641975308642e-5));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 2.9e-75)
		tmp = b ^ 2.0;
	else
		tmp = fma(b, b, Float64((Float64(a * Float64(pi * angle)) ^ 2.0) * 3.08641975308642e-5));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 2.9e-75], N[Power[b, 2.0], $MachinePrecision], N[(b * b + N[(N[Power[N[(a * N[(Pi * angle), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.9000000000000002e-75

   1. Initial program 81.4%

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

      1. *-commutative 81.4%

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

      2. associate-*r/ 81.5%

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

      3. associate-*l/ 81.6%

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

      4. *-commutative 81.6%

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

      5. *-commutative 81.6%

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

      6. associate-*r/ 81.5%

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

      7. associate-*l/ 81.6%

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

      8. *-commutative 81.6%

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

   3. Simplified 81.6%

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

   5. Taylor expanded in angle around 0 82.0%

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

   6. Taylor expanded in angle around 0 78.0%

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

      1. *-commutative 78.0%

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

      2. associate-*l* 77.9%

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

   8. Simplified 77.9%

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

   9. Taylor expanded in angle around 0 64.9%

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

       1. *-commutative 64.9%

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

       2. associate-*l* 64.9%

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

       3. unpow2 64.9%

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

       4. unpow2 64.9%

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

       5. unpow2 64.9%

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

       6. swap-sqr 64.9%

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

       7. swap-sqr 77.9%

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

       8. unpow2 77.9%

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

       9. associate-*r* 77.9%

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

       10. *-commutative 77.9%

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

       11. associate-*l* 77.9%

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

   11. Simplified 77.9%

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

   12. Taylor expanded in angle around 0 60.0%

       \[\leadsto \color{blue}{{b}^{2}} \]

   if 2.9000000000000002e-75 < a

   1. Initial program 75.3%

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

      1. *-commutative 75.3%

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

      2. associate-*r/ 75.4%

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

      3. associate-*l/ 75.4%

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

      4. *-commutative 75.4%

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

      5. *-commutative 75.4%

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

      6. associate-*r/ 75.3%

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

      7. associate-*l/ 75.4%

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

      8. *-commutative 75.4%

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

   3. Simplified 75.4%

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

   5. Taylor expanded in angle around 0 75.3%

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

   6. Taylor expanded in angle around 0 70.7%

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

      1. *-commutative 70.7%

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

      2. associate-*l* 70.7%

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

   8. Simplified 70.7%

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

   9. Taylor expanded in angle around 0 56.6%

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

       1. *-commutative 56.6%

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

       2. associate-*l* 56.6%

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

       3. unpow2 56.6%

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

       4. unpow2 56.6%

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

       5. unpow2 56.6%

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

       6. swap-sqr 56.6%

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

       7. swap-sqr 70.7%

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

       8. unpow2 70.7%

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

       9. associate-*r* 70.8%

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

       10. *-commutative 70.8%

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

       11. associate-*l* 70.8%

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

   11. Simplified 70.8%

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

       1. +-commutative 70.8%

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

       2. *-rgt-identity 70.8%

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

       3. pow2 70.8%

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

       4. fma-def 70.8%

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

       5. *-commutative 70.8%

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

       6. associate-*r* 70.8%

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

   13. Applied egg-rr 70.8%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.9 \cdot 10^{-75}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b, {\left(a \cdot \left(\pi \cdot angle\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 6.5 \cdot 10^{-69}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(b, 0.005555555555555556 \cdot \left(\pi \cdot \left(a \cdot angle\right)\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 6.5e-69)
   (pow b 2.0)
   (pow (hypot b (* 0.005555555555555556 (* PI (* a angle)))) 2.0)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 6.5e-69) {
		tmp = pow(b, 2.0);
	} else {
		tmp = pow(hypot(b, (0.005555555555555556 * (((double) M_PI) * (a * angle)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 6.5e-69) {
		tmp = Math.pow(b, 2.0);
	} else {
		tmp = Math.pow(Math.hypot(b, (0.005555555555555556 * (Math.PI * (a * angle)))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 6.5e-69:
		tmp = math.pow(b, 2.0)
	else:
		tmp = math.pow(math.hypot(b, (0.005555555555555556 * (math.pi * (a * angle)))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 6.5e-69)
		tmp = b ^ 2.0;
	else
		tmp = hypot(b, Float64(0.005555555555555556 * Float64(pi * Float64(a * angle)))) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 6.5e-69)
		tmp = b ^ 2.0;
	else
		tmp = hypot(b, (0.005555555555555556 * (pi * (a * angle)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 6.5e-69], N[Power[b, 2.0], $MachinePrecision], N[Power[N[Sqrt[b ^ 2 + N[(0.005555555555555556 * N[(Pi * N[(a * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 6.49999999999999951e-69

   1. Initial program 81.1%

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

      1. *-commutative 81.1%

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

      2. associate-*r/ 81.2%

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

      3. associate-*l/ 81.2%

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

      4. *-commutative 81.2%

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

      5. *-commutative 81.2%

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

      6. associate-*r/ 81.1%

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

      7. associate-*l/ 81.2%

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

      8. *-commutative 81.2%

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

   3. Simplified 81.2%

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

   5. Taylor expanded in angle around 0 81.6%

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

   6. Taylor expanded in angle around 0 77.6%

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

      1. *-commutative 77.6%

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

      2. associate-*l* 77.6%

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

   8. Simplified 77.6%

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

   9. Taylor expanded in angle around 0 64.6%

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

       1. *-commutative 64.6%

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

       2. associate-*l* 64.6%

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

       3. unpow2 64.6%

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

       4. unpow2 64.6%

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

       5. unpow2 64.6%

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

       6. swap-sqr 64.6%

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

       7. swap-sqr 77.6%

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

       8. unpow2 77.6%

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

       9. associate-*r* 77.6%

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

       10. *-commutative 77.6%

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

       11. associate-*l* 77.6%

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

   11. Simplified 77.6%

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

   12. Taylor expanded in angle around 0 59.7%

       \[\leadsto \color{blue}{{b}^{2}} \]

   if 6.49999999999999951e-69 < a

   1. Initial program 76.0%

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

      1. *-commutative 76.0%

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

      2. associate-*r/ 76.1%

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

      3. associate-*l/ 76.1%

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

      4. *-commutative 76.1%

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

      5. *-commutative 76.1%

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

      6. associate-*r/ 76.0%

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

      7. associate-*l/ 76.1%

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

      8. *-commutative 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in angle around 0 76.0%

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

   6. Taylor expanded in angle around 0 71.4%

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

      1. *-commutative 71.4%

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

      2. associate-*l* 71.4%

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

   8. Simplified 71.4%

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

   9. Taylor expanded in angle around 0 57.1%

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

       1. *-commutative 57.1%

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

       2. associate-*l* 57.1%

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

       3. unpow2 57.1%

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

       4. unpow2 57.1%

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

       5. unpow2 57.1%

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

       6. swap-sqr 57.1%

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

       7. swap-sqr 71.4%

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

       8. unpow2 71.4%

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

       9. associate-*r* 71.5%

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

       10. *-commutative 71.5%

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

       11. associate-*l* 71.4%

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

   11. Simplified 71.4%

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

       1. expm1-log1p-u 70.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)} \]

       2. expm1-udef 57.3%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)} - 1} \]

   13. Applied egg-rr 57.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(b, 0.005555555555555556 \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right)\right)}^{2}\right)} - 1} \]
   14. Step-by-step derivation

       1. expm1-def 70.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(b, 0.005555555555555556 \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right)\right)}^{2}\right)\right)} \]

       2. expm1-log1p 71.4%

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

       3. associate-*l* 71.4%

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

   15. Simplified 71.4%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6.5 \cdot 10^{-69}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(b, 0.005555555555555556 \cdot \left(\pi \cdot \left(a \cdot angle\right)\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4.4 \cdot 10^{-66}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(b, 0.005555555555555556 \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 4.4e-66)
   (pow b 2.0)
   (pow (hypot b (* 0.005555555555555556 (* a (* PI angle)))) 2.0)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 4.4e-66) {
		tmp = pow(b, 2.0);
	} else {
		tmp = pow(hypot(b, (0.005555555555555556 * (a * (((double) M_PI) * angle)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 4.4e-66) {
		tmp = Math.pow(b, 2.0);
	} else {
		tmp = Math.pow(Math.hypot(b, (0.005555555555555556 * (a * (Math.PI * angle)))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 4.4e-66:
		tmp = math.pow(b, 2.0)
	else:
		tmp = math.pow(math.hypot(b, (0.005555555555555556 * (a * (math.pi * angle)))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 4.4e-66)
		tmp = b ^ 2.0;
	else
		tmp = hypot(b, Float64(0.005555555555555556 * Float64(a * Float64(pi * angle)))) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 4.4e-66)
		tmp = b ^ 2.0;
	else
		tmp = hypot(b, (0.005555555555555556 * (a * (pi * angle)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 4.4e-66], N[Power[b, 2.0], $MachinePrecision], N[Power[N[Sqrt[b ^ 2 + N[(0.005555555555555556 * N[(a * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.4000000000000002e-66

   1. Initial program 81.1%

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

      1. *-commutative 81.1%

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

      2. associate-*r/ 81.2%

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

      3. associate-*l/ 81.2%

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

      4. *-commutative 81.2%

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

      5. *-commutative 81.2%

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

      6. associate-*r/ 81.1%

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

      7. associate-*l/ 81.2%

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

      8. *-commutative 81.2%

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

   3. Simplified 81.2%

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

   5. Taylor expanded in angle around 0 81.6%

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

   6. Taylor expanded in angle around 0 77.6%

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

      1. *-commutative 77.6%

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

      2. associate-*l* 77.6%

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

   8. Simplified 77.6%

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

   9. Taylor expanded in angle around 0 64.6%

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

       1. *-commutative 64.6%

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

       2. associate-*l* 64.6%

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

       3. unpow2 64.6%

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

       4. unpow2 64.6%

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

       5. unpow2 64.6%

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

       6. swap-sqr 64.6%

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

       7. swap-sqr 77.6%

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

       8. unpow2 77.6%

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

       9. associate-*r* 77.6%

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

       10. *-commutative 77.6%

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

       11. associate-*l* 77.6%

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

   11. Simplified 77.6%

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

   12. Taylor expanded in angle around 0 59.7%

       \[\leadsto \color{blue}{{b}^{2}} \]

   if 4.4000000000000002e-66 < a

   1. Initial program 76.0%

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

      1. *-commutative 76.0%

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

      2. associate-*r/ 76.1%

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

      3. associate-*l/ 76.1%

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

      4. *-commutative 76.1%

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

      5. *-commutative 76.1%

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

      6. associate-*r/ 76.0%

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

      7. associate-*l/ 76.1%

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

      8. *-commutative 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in angle around 0 76.0%

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

   6. Taylor expanded in angle around 0 71.4%

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

      1. *-commutative 71.4%

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

      2. associate-*l* 71.4%

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

   8. Simplified 71.4%

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

   9. Taylor expanded in angle around 0 57.1%

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

       1. *-commutative 57.1%

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

       2. associate-*l* 57.1%

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

       3. unpow2 57.1%

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

       4. unpow2 57.1%

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

       5. unpow2 57.1%

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

       6. swap-sqr 57.1%

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

       7. swap-sqr 71.4%

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

       8. unpow2 71.4%

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

       9. associate-*r* 71.5%

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

       10. *-commutative 71.5%

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

       11. associate-*l* 71.4%

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

   11. Simplified 71.4%

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

       1. add-sqr-sqrt 71.4%

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

       2. pow2 71.4%

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

   13. Applied egg-rr 71.4%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.4 \cdot 10^{-66}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(b, 0.005555555555555556 \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.5% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ {b}^{2} \end{array} \]

FPCore

(FPCore (a b angle) :precision binary64 (pow b 2.0))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, angle)
	return b ^ 2.0
end

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[Power[b, 2.0], $MachinePrecision]

Derivation

1. Initial program 79.4%

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

   1. *-commutative 79.4%

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

   2. associate-*r/ 79.5%

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

   3. associate-*l/ 79.6%

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

   4. *-commutative 79.6%

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

   5. *-commutative 79.6%

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

   6. associate-*r/ 79.4%

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

   7. associate-*l/ 79.5%

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

   8. *-commutative 79.5%

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

3. Simplified 79.5%

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

5. Taylor expanded in angle around 0 79.8%

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

6. Taylor expanded in angle around 0 75.6%

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

   1. *-commutative 75.6%

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

   2. associate-*l* 75.6%

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

8. Simplified 75.6%

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

9. Taylor expanded in angle around 0 62.2%

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

    1. *-commutative 62.2%

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

    2. associate-*l* 62.2%

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

    3. unpow2 62.2%

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

    4. unpow2 62.2%

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

    5. unpow2 62.2%

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

    6. swap-sqr 62.2%

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

    7. swap-sqr 75.6%

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

    8. unpow2 75.6%

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

    9. associate-*r* 75.6%

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

    10. *-commutative 75.6%

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

    11. associate-*l* 75.6%

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

11. Simplified 75.6%

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

12. Taylor expanded in angle around 0 52.9%

    \[\leadsto \color{blue}{{b}^{2}} \]

13. Final simplification 52.9%

    \[\leadsto {b}^{2} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))