ab-angle->ABCF C

Percentage Accurate: 79.8% → 79.8%

Time: 14.8s

Alternatives: 12

Speedup: 2.0×

• 36.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

3. Taylor expanded in angle around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   2. *-lowering-*.f64 78.0%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

5. Simplified 78.0%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot b\right), 2\right)\right) \]

   2. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot b\right), 2\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right), b\right), 2\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\sin \left(\frac{\mathsf{PI}\left(\right)}{180} \cdot angle\right), b\right), 2\right)\right) \]

   5. associate-/r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right), b\right), 2\right)\right) \]

   6. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), b\right), 2\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(\frac{180}{angle}\right)\right)\right), b\right), 2\right)\right) \]

   8. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{180}{angle}\right)\right)\right), b\right), 2\right)\right) \]

   9. /-lowering-/.f64 78.1%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), b\right), 2\right)\right) \]

7. Applied egg-rr 78.1%

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

   1. add-cbrt-cube N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), b\right), 2\right)\right) \]

   2. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), b\right), 2\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), b\right), 2\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), b\right), 2\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), b\right), 2\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), b\right), 2\right)\right) \]

   7. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), b\right), 2\right)\right) \]

   8. PI-lowering-PI.f64 78.1%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), b\right), 2\right)\right) \]

9. Applied egg-rr 78.1%

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

Alternative 2: 79.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

3. Taylor expanded in angle around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   2. *-lowering-*.f64 78.0%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

5. Simplified 78.0%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot b\right), 2\right)\right) \]

   2. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot b\right), 2\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right), b\right), 2\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\sin \left(\frac{\mathsf{PI}\left(\right)}{180} \cdot angle\right), b\right), 2\right)\right) \]

   5. associate-/r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right), b\right), 2\right)\right) \]

   6. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), b\right), 2\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(\frac{180}{angle}\right)\right)\right), b\right), 2\right)\right) \]

   8. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{180}{angle}\right)\right)\right), b\right), 2\right)\right) \]

   9. /-lowering-/.f64 78.1%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), b\right), 2\right)\right) \]

7. Applied egg-rr 78.1%

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

8. Final simplification 78.1%

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

Alternative 3: 79.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

3. Taylor expanded in angle around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   2. *-lowering-*.f64 78.0%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

5. Simplified 78.0%

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

   1. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), 2\right)\right) \]

   2. div-inv N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right), 2\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right), 2\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), \frac{1}{180}\right)\right)\right), 2\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), \frac{1}{180}\right)\right)\right), 2\right)\right) \]

   6. PI-lowering-PI.f64 78.1%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), \frac{1}{180}\right)\right)\right), 2\right)\right) \]

7. Applied egg-rr 78.1%

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

8. Final simplification 78.1%

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

Alternative 4: 79.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

3. Taylor expanded in angle around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   2. *-lowering-*.f64 78.0%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

5. Simplified 78.0%

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

Alternative 5: 77.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 9 \cdot 10^{-7}:\\ \;\;\;\;a \cdot a + \left(angle \cdot b\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(angle \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + b \cdot \left(b \cdot \left(0.5 + \cos \left(\frac{\pi \cdot 2}{\frac{180}{angle}}\right) \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 9e-7)
   (+
    (* a a)
    (* (* angle b) (* (* PI (* PI 3.08641975308642e-5)) (* angle b))))
   (+
    (* a a)
    (* b (* b (+ 0.5 (* (cos (/ (* PI 2.0) (/ 180.0 angle))) -0.5)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 9e-7) {
		tmp = (a * a) + ((angle * b) * ((((double) M_PI) * (((double) M_PI) * 3.08641975308642e-5)) * (angle * b)));
	} else {
		tmp = (a * a) + (b * (b * (0.5 + (cos(((((double) M_PI) * 2.0) / (180.0 / angle))) * -0.5))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 9e-7) {
		tmp = (a * a) + ((angle * b) * ((Math.PI * (Math.PI * 3.08641975308642e-5)) * (angle * b)));
	} else {
		tmp = (a * a) + (b * (b * (0.5 + (Math.cos(((Math.PI * 2.0) / (180.0 / angle))) * -0.5))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 9e-7:
		tmp = (a * a) + ((angle * b) * ((math.pi * (math.pi * 3.08641975308642e-5)) * (angle * b)))
	else:
		tmp = (a * a) + (b * (b * (0.5 + (math.cos(((math.pi * 2.0) / (180.0 / angle))) * -0.5))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 9e-7)
		tmp = Float64(Float64(a * a) + Float64(Float64(angle * b) * Float64(Float64(pi * Float64(pi * 3.08641975308642e-5)) * Float64(angle * b))));
	else
		tmp = Float64(Float64(a * a) + Float64(b * Float64(b * Float64(0.5 + Float64(cos(Float64(Float64(pi * 2.0) / Float64(180.0 / angle))) * -0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 9e-7)
		tmp = (a * a) + ((angle * b) * ((pi * (pi * 3.08641975308642e-5)) * (angle * b)));
	else
		tmp = (a * a) + (b * (b * (0.5 + (cos(((pi * 2.0) / (180.0 / angle))) * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 9e-7], N[(N[(a * a), $MachinePrecision] + N[(N[(angle * b), $MachinePrecision] * N[(N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] * N[(angle * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(b * N[(b * N[(0.5 + N[(N[Cos[N[(N[(Pi * 2.0), $MachinePrecision] / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 8.99999999999999959e-7

   1. Initial program 84.2%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 84.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 84.2%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{32400} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) + {a}^{2} \]

      2. associate-*r* N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{32400} \cdot \color{blue}{\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \color{blue}{\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({angle}^{2} \cdot {b}^{2}\right), \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({b}^{2} \cdot {angle}^{2}\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left({angle}^{2}\right)\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{\frac{1}{32400}}\right)\right)\right) \]

   8. Simplified 64.7%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right) \cdot \color{blue}{\left(\left(b \cdot b\right) \cdot \left(angle \cdot angle\right)\right)}\right)\right) \]

      2. unswap-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right) \cdot \left(\left(b \cdot angle\right) \cdot \color{blue}{\left(b \cdot angle\right)}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right) \cdot \left(b \cdot angle\right)\right) \cdot \color{blue}{\left(b \cdot angle\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right) \cdot \left(b \cdot angle\right)\right), \color{blue}{\left(b \cdot angle\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right), \left(b \cdot angle\right)\right), \left(\color{blue}{b} \cdot angle\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right), \left(b \cdot angle\right)\right), \left(b \cdot angle\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right), \left(b \cdot angle\right)\right), \left(b \cdot angle\right)\right)\right) \]

      8. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right), \left(b \cdot angle\right)\right), \left(b \cdot angle\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \frac{1}{32400}\right)\right), \left(b \cdot angle\right)\right), \left(b \cdot angle\right)\right)\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right)\right), \left(b \cdot angle\right)\right), \left(b \cdot angle\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right)\right), \left(angle \cdot b\right)\right), \left(b \cdot angle\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right)\right), \mathsf{*.f64}\left(angle, b\right)\right), \left(b \cdot angle\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right)\right), \mathsf{*.f64}\left(angle, b\right)\right), \left(angle \cdot \color{blue}{b}\right)\right)\right) \]

      14. *-lowering-*.f64 80.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right)\right), \mathsf{*.f64}\left(angle, b\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{b}\right)\right)\right) \]

   10. Applied egg-rr 80.3%

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

   if 8.99999999999999959e-7 < angle

   1. Initial program 60.4%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 61.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 61.0%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot b\right), 2\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot b\right), 2\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right), b\right), 2\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\sin \left(\frac{\mathsf{PI}\left(\right)}{180} \cdot angle\right), b\right), 2\right)\right) \]

      5. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right), b\right), 2\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), b\right), 2\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(\frac{180}{angle}\right)\right)\right), b\right), 2\right)\right) \]

      8. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{180}{angle}\right)\right)\right), b\right), 2\right)\right) \]

      9. /-lowering-/.f64 61.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), b\right), 2\right)\right) \]

   7. Applied egg-rr 61.2%

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

      1. add-cbrt-cube N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), b\right), 2\right)\right) \]

      2. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), b\right), 2\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), b\right), 2\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), b\right), 2\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), b\right), 2\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), b\right), 2\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), b\right), 2\right)\right) \]

      8. PI-lowering-PI.f64 61.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), b\right), 2\right)\right) \]

   9. Applied egg-rr 61.3%

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

       1. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\sin \left(\frac{\sqrt[3]{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}}{\frac{180}{angle}}\right) \cdot b\right) \cdot \color{blue}{\left(\sin \left(\frac{\sqrt[3]{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}}{\frac{180}{angle}}\right) \cdot b\right)}\right)\right) \]

       2. swap-sqr N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\sin \left(\frac{\sqrt[3]{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}}{\frac{180}{angle}}\right) \cdot \sin \left(\frac{\sqrt[3]{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}}{\frac{180}{angle}}\right)\right) \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\sin \left(\frac{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}{\frac{180}{angle}}\right) \cdot \sin \left(\frac{\sqrt[3]{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right)\right)\right) \]

       4. add-cbrt-cube N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \sin \left(\frac{\sqrt[3]{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right)\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \sin \left(\frac{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right)\right)\right) \]

       6. add-cbrt-cube N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right)\right)\right) \]

       7. sqr-sin-a N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\color{blue}{b} \cdot b\right)\right)\right) \]

   11. Applied egg-rr 61.2%

       \[\leadsto a \cdot a + \color{blue}{\left(b \cdot \left(0.5 + \cos \left(\frac{\pi \cdot 2}{\frac{180}{angle}}\right) \cdot -0.5\right)\right) \cdot b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 9 \cdot 10^{-7}:\\ \;\;\;\;a \cdot a + \left(angle \cdot b\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(angle \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + b \cdot \left(b \cdot \left(0.5 + \cos \left(\frac{\pi \cdot 2}{\frac{180}{angle}}\right) \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 9 \cdot 10^{-7}:\\ \;\;\;\;a \cdot a + \left(angle \cdot b\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(angle \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(b \cdot b\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 9e-7)
   (+
    (* a a)
    (* (* angle b) (* (* PI (* PI 3.08641975308642e-5)) (* angle b))))
   (+
    (* a a)
    (* (* b b) (+ 0.5 (* -0.5 (cos (* (* PI angle) 0.011111111111111112))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 9e-7) {
		tmp = (a * a) + ((angle * b) * ((((double) M_PI) * (((double) M_PI) * 3.08641975308642e-5)) * (angle * b)));
	} else {
		tmp = (a * a) + ((b * b) * (0.5 + (-0.5 * cos(((((double) M_PI) * angle) * 0.011111111111111112)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 9e-7) {
		tmp = (a * a) + ((angle * b) * ((Math.PI * (Math.PI * 3.08641975308642e-5)) * (angle * b)));
	} else {
		tmp = (a * a) + ((b * b) * (0.5 + (-0.5 * Math.cos(((Math.PI * angle) * 0.011111111111111112)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 9e-7:
		tmp = (a * a) + ((angle * b) * ((math.pi * (math.pi * 3.08641975308642e-5)) * (angle * b)))
	else:
		tmp = (a * a) + ((b * b) * (0.5 + (-0.5 * math.cos(((math.pi * angle) * 0.011111111111111112)))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 9e-7)
		tmp = Float64(Float64(a * a) + Float64(Float64(angle * b) * Float64(Float64(pi * Float64(pi * 3.08641975308642e-5)) * Float64(angle * b))));
	else
		tmp = Float64(Float64(a * a) + Float64(Float64(b * b) * Float64(0.5 + Float64(-0.5 * cos(Float64(Float64(pi * angle) * 0.011111111111111112))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 9e-7)
		tmp = (a * a) + ((angle * b) * ((pi * (pi * 3.08641975308642e-5)) * (angle * b)));
	else
		tmp = (a * a) + ((b * b) * (0.5 + (-0.5 * cos(((pi * angle) * 0.011111111111111112)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 9e-7], N[(N[(a * a), $MachinePrecision] + N[(N[(angle * b), $MachinePrecision] * N[(N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] * N[(angle * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(0.5 + N[(-0.5 * N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 8.99999999999999959e-7

   1. Initial program 84.2%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 84.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 84.2%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{32400} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) + {a}^{2} \]

      2. associate-*r* N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{32400} \cdot \color{blue}{\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \color{blue}{\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({angle}^{2} \cdot {b}^{2}\right), \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({b}^{2} \cdot {angle}^{2}\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left({angle}^{2}\right)\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{\frac{1}{32400}}\right)\right)\right) \]

   8. Simplified 64.7%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right) \cdot \color{blue}{\left(\left(b \cdot b\right) \cdot \left(angle \cdot angle\right)\right)}\right)\right) \]

      2. unswap-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right) \cdot \left(\left(b \cdot angle\right) \cdot \color{blue}{\left(b \cdot angle\right)}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right) \cdot \left(b \cdot angle\right)\right) \cdot \color{blue}{\left(b \cdot angle\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right) \cdot \left(b \cdot angle\right)\right), \color{blue}{\left(b \cdot angle\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right), \left(b \cdot angle\right)\right), \left(\color{blue}{b} \cdot angle\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right), \left(b \cdot angle\right)\right), \left(b \cdot angle\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right), \left(b \cdot angle\right)\right), \left(b \cdot angle\right)\right)\right) \]

      8. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right), \left(b \cdot angle\right)\right), \left(b \cdot angle\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \frac{1}{32400}\right)\right), \left(b \cdot angle\right)\right), \left(b \cdot angle\right)\right)\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right)\right), \left(b \cdot angle\right)\right), \left(b \cdot angle\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right)\right), \left(angle \cdot b\right)\right), \left(b \cdot angle\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right)\right), \mathsf{*.f64}\left(angle, b\right)\right), \left(b \cdot angle\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right)\right), \mathsf{*.f64}\left(angle, b\right)\right), \left(angle \cdot \color{blue}{b}\right)\right)\right) \]

      14. *-lowering-*.f64 80.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right)\right), \mathsf{*.f64}\left(angle, b\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{b}\right)\right)\right) \]

   10. Applied egg-rr 80.3%

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

   if 8.99999999999999959e-7 < angle

   1. Initial program 60.4%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 61.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 61.0%

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

   7. Applied egg-rr 2.9%

      \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{6}\right) \cdot \frac{1}{\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(\left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\pi}{\frac{180}{angle}}\right)\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\pi}{\frac{180}{angle}}\right)\right) - a \cdot a\right)}} \]

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + {a}^{2}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {a}^{2} + \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left({b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{{b}^{2}} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{{b}^{2}} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right), \frac{1}{90}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right), \frac{1}{90}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      17. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), \frac{1}{90}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      18. metadata-eval 61.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), \frac{1}{90}\right)\right), \frac{-1}{2}\right)\right)\right)\right) \]

   10. Simplified 61.1%

       \[\leadsto \color{blue}{a \cdot a + \left(b \cdot b\right) \cdot \left(0.5 + \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot -0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 9 \cdot 10^{-7}:\\ \;\;\;\;a \cdot a + \left(angle \cdot b\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(angle \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(b \cdot b\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.6% accurate, 18.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.8 \cdot 10^{-60}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(angle \cdot b\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(angle \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 8.8e-60)
   (* a a)
   (+
    (* a a)
    (* (* angle b) (* (* PI (* PI 3.08641975308642e-5)) (* angle b))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 8.8e-60) {
		tmp = a * a;
	} else {
		tmp = (a * a) + ((angle * b) * ((((double) M_PI) * (((double) M_PI) * 3.08641975308642e-5)) * (angle * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 8.8e-60) {
		tmp = a * a;
	} else {
		tmp = (a * a) + ((angle * b) * ((Math.PI * (Math.PI * 3.08641975308642e-5)) * (angle * b)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 8.8e-60:
		tmp = a * a
	else:
		tmp = (a * a) + ((angle * b) * ((math.pi * (math.pi * 3.08641975308642e-5)) * (angle * b)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 8.8e-60)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + Float64(Float64(angle * b) * Float64(Float64(pi * Float64(pi * 3.08641975308642e-5)) * Float64(angle * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 8.8e-60)
		tmp = a * a;
	else
		tmp = (a * a) + ((angle * b) * ((pi * (pi * 3.08641975308642e-5)) * (angle * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 8.8e-60], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(N[(angle * b), $MachinePrecision] * N[(N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] * N[(angle * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.7999999999999995e-60

   1. Initial program 76.5%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{a}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto a \cdot \color{blue}{a} \]

      2. *-lowering-*.f64 60.8%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

   5. Simplified 60.8%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 8.7999999999999995e-60 < b

   1. Initial program 80.4%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 79.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 79.9%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{32400} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) + {a}^{2} \]

      2. associate-*r* N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{32400} \cdot \color{blue}{\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \color{blue}{\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({angle}^{2} \cdot {b}^{2}\right), \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({b}^{2} \cdot {angle}^{2}\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left({angle}^{2}\right)\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{\frac{1}{32400}}\right)\right)\right) \]

   8. Simplified 61.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right) \cdot \color{blue}{\left(\left(b \cdot b\right) \cdot \left(angle \cdot angle\right)\right)}\right)\right) \]

      2. unswap-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right) \cdot \left(\left(b \cdot angle\right) \cdot \color{blue}{\left(b \cdot angle\right)}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right) \cdot \left(b \cdot angle\right)\right) \cdot \color{blue}{\left(b \cdot angle\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right) \cdot \left(b \cdot angle\right)\right), \color{blue}{\left(b \cdot angle\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right), \left(b \cdot angle\right)\right), \left(\color{blue}{b} \cdot angle\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right), \left(b \cdot angle\right)\right), \left(b \cdot angle\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right), \left(b \cdot angle\right)\right), \left(b \cdot angle\right)\right)\right) \]

      8. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right), \left(b \cdot angle\right)\right), \left(b \cdot angle\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \frac{1}{32400}\right)\right), \left(b \cdot angle\right)\right), \left(b \cdot angle\right)\right)\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right)\right), \left(b \cdot angle\right)\right), \left(b \cdot angle\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right)\right), \left(angle \cdot b\right)\right), \left(b \cdot angle\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right)\right), \mathsf{*.f64}\left(angle, b\right)\right), \left(b \cdot angle\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right)\right), \mathsf{*.f64}\left(angle, b\right)\right), \left(angle \cdot \color{blue}{b}\right)\right)\right) \]

      14. *-lowering-*.f64 77.2%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right)\right), \mathsf{*.f64}\left(angle, b\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{b}\right)\right)\right) \]

   10. Applied egg-rr 77.2%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.8 \cdot 10^{-60}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(angle \cdot b\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(angle \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.6% accurate, 18.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-60}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(\left(angle \cdot b\right) \cdot \left(angle \cdot b\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.05e-60)
   (* a a)
   (+
    (* a a)
    (* (* (* angle b) (* angle b)) (* (* PI PI) 3.08641975308642e-5)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.05e-60) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (((angle * b) * (angle * b)) * ((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.05e-60) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (((angle * b) * (angle * b)) * ((Math.PI * Math.PI) * 3.08641975308642e-5));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.05e-60:
		tmp = a * a
	else:
		tmp = (a * a) + (((angle * b) * (angle * b)) * ((math.pi * math.pi) * 3.08641975308642e-5))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.05e-60)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + Float64(Float64(Float64(angle * b) * Float64(angle * b)) * Float64(Float64(pi * pi) * 3.08641975308642e-5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.05e-60)
		tmp = a * a;
	else
		tmp = (a * a) + (((angle * b) * (angle * b)) * ((pi * pi) * 3.08641975308642e-5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.05e-60], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(N[(N[(angle * b), $MachinePrecision] * N[(angle * b), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.04999999999999996e-60

   1. Initial program 76.5%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{a}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto a \cdot \color{blue}{a} \]

      2. *-lowering-*.f64 60.8%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

   5. Simplified 60.8%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 1.04999999999999996e-60 < b

   1. Initial program 80.4%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 79.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 79.9%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{32400} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) + {a}^{2} \]

      2. associate-*r* N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{32400} \cdot \color{blue}{\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \color{blue}{\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({angle}^{2} \cdot {b}^{2}\right), \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({b}^{2} \cdot {angle}^{2}\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left({angle}^{2}\right)\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{\frac{1}{32400}}\right)\right)\right) \]

   8. Simplified 61.1%

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

      1. unswap-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left(\left(b \cdot angle\right) \cdot \left(b \cdot angle\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)}, \frac{1}{32400}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot angle\right), \left(b \cdot angle\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)}, \frac{1}{32400}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(angle \cdot b\right), \left(b \cdot angle\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}, \mathsf{PI.f64}\left(\right)\right), \frac{1}{32400}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \left(b \cdot angle\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}, \mathsf{PI.f64}\left(\right)\right), \frac{1}{32400}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \left(angle \cdot b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \color{blue}{\mathsf{PI.f64}\left(\right)}\right), \frac{1}{32400}\right)\right)\right) \]

      6. *-lowering-*.f64 76.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(angle, b\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \color{blue}{\mathsf{PI.f64}\left(\right)}\right), \frac{1}{32400}\right)\right)\right) \]

   10. Applied egg-rr 76.4%

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

Alternative 9: 67.1% accurate, 18.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.1 \cdot 10^{-61}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(b \cdot \left(angle \cdot \left(angle \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 6.1e-61)
   (* a a)
   (+
    (* a a)
    (* (* (* PI PI) 3.08641975308642e-5) (* b (* angle (* angle b)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 6.1e-61) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5) * (b * (angle * (angle * b))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 6.1e-61) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (((Math.PI * Math.PI) * 3.08641975308642e-5) * (b * (angle * (angle * b))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 6.1e-61:
		tmp = a * a
	else:
		tmp = (a * a) + (((math.pi * math.pi) * 3.08641975308642e-5) * (b * (angle * (angle * b))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 6.1e-61)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + Float64(Float64(Float64(pi * pi) * 3.08641975308642e-5) * Float64(b * Float64(angle * Float64(angle * b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 6.1e-61)
		tmp = a * a;
	else
		tmp = (a * a) + (((pi * pi) * 3.08641975308642e-5) * (b * (angle * (angle * b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 6.1e-61], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] * N[(b * N[(angle * N[(angle * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.1000000000000001e-61

   1. Initial program 76.5%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{a}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto a \cdot \color{blue}{a} \]

      2. *-lowering-*.f64 60.8%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

   5. Simplified 60.8%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 6.1000000000000001e-61 < b

   1. Initial program 80.4%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 79.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 79.9%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{32400} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) + {a}^{2} \]

      2. associate-*r* N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{32400} \cdot \color{blue}{\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \color{blue}{\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({angle}^{2} \cdot {b}^{2}\right), \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({b}^{2} \cdot {angle}^{2}\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left({angle}^{2}\right)\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{\frac{1}{32400}}\right)\right)\right) \]

   8. Simplified 61.1%

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

      1. unswap-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left(\left(b \cdot angle\right) \cdot \left(b \cdot angle\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)}, \frac{1}{32400}\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left(b \cdot \left(angle \cdot \left(b \cdot angle\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)}, \frac{1}{32400}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \left(angle \cdot \left(b \cdot angle\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)}, \frac{1}{32400}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, \left(b \cdot angle\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \color{blue}{\mathsf{PI.f64}\left(\right)}\right), \frac{1}{32400}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, \left(angle \cdot b\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1}{32400}\right)\right)\right) \]

      6. *-lowering-*.f64 75.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, b\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1}{32400}\right)\right)\right) \]

   10. Applied egg-rr 75.9%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.1 \cdot 10^{-61}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(b \cdot \left(angle \cdot \left(angle \cdot b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.8% accurate, 18.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.4 \cdot 10^{-60}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + b \cdot \left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(b \cdot \left(angle \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 6.4e-60)
   (* a a)
   (+
    (* a a)
    (* b (* (* PI (* PI 3.08641975308642e-5)) (* b (* angle angle)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 6.4e-60) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (b * ((((double) M_PI) * (((double) M_PI) * 3.08641975308642e-5)) * (b * (angle * angle))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 6.4e-60) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (b * ((Math.PI * (Math.PI * 3.08641975308642e-5)) * (b * (angle * angle))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 6.4e-60:
		tmp = a * a
	else:
		tmp = (a * a) + (b * ((math.pi * (math.pi * 3.08641975308642e-5)) * (b * (angle * angle))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 6.4e-60)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + Float64(b * Float64(Float64(pi * Float64(pi * 3.08641975308642e-5)) * Float64(b * Float64(angle * angle)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 6.4e-60)
		tmp = a * a;
	else
		tmp = (a * a) + (b * ((pi * (pi * 3.08641975308642e-5)) * (b * (angle * angle))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 6.4e-60], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(b * N[(N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] * N[(b * N[(angle * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.4000000000000003e-60

   1. Initial program 76.5%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{a}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto a \cdot \color{blue}{a} \]

      2. *-lowering-*.f64 60.8%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

   5. Simplified 60.8%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 6.4000000000000003e-60 < b

   1. Initial program 80.4%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 79.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 79.9%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{32400} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) + {a}^{2} \]

      2. associate-*r* N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{32400} \cdot \color{blue}{\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \color{blue}{\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({angle}^{2} \cdot {b}^{2}\right), \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({b}^{2} \cdot {angle}^{2}\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left({angle}^{2}\right)\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{\frac{1}{32400}}\right)\right)\right) \]

   8. Simplified 61.1%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(b \cdot \left(b \cdot \left(angle \cdot angle\right)\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{32400}\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(b \cdot \color{blue}{\left(\left(b \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, \color{blue}{\left(\left(b \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left(b \cdot \left(angle \cdot angle\right)\right), \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \left(angle \cdot angle\right)\right), \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{32400}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, angle\right)\right), \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{32400}\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, angle\right)\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{1}{32400}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\frac{1}{32400}}\right)\right)\right)\right)\right) \]

      11. PI-lowering-PI.f64 69.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{32400}\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 69.9%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.4 \cdot 10^{-60}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + b \cdot \left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(b \cdot \left(angle \cdot angle\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.9% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{+137}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(angle \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 9.5e+137)
   (* a a)
   (* (* angle (* (* PI PI) 3.08641975308642e-5)) (* angle (* b b)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 9.5e+137) {
		tmp = a * a;
	} else {
		tmp = (angle * ((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5)) * (angle * (b * b));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 9.5e+137) {
		tmp = a * a;
	} else {
		tmp = (angle * ((Math.PI * Math.PI) * 3.08641975308642e-5)) * (angle * (b * b));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 9.5e+137:
		tmp = a * a
	else:
		tmp = (angle * ((math.pi * math.pi) * 3.08641975308642e-5)) * (angle * (b * b))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 9.5e+137)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(angle * Float64(Float64(pi * pi) * 3.08641975308642e-5)) * Float64(angle * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 9.5e+137)
		tmp = a * a;
	else
		tmp = (angle * ((pi * pi) * 3.08641975308642e-5)) * (angle * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 9.5e+137], N[(a * a), $MachinePrecision], N[(N[(angle * N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.50000000000000031e137

   1. Initial program 74.9%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{a}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto a \cdot \color{blue}{a} \]

      2. *-lowering-*.f64 59.1%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

   5. Simplified 59.1%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 9.50000000000000031e137 < b

   1. Initial program 90.7%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 90.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 90.7%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{32400} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) + {a}^{2} \]

      2. associate-*r* N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{32400} \cdot \color{blue}{\left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \color{blue}{\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({angle}^{2} \cdot {b}^{2}\right), \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left({b}^{2} \cdot {angle}^{2}\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left({angle}^{2}\right)\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2} \cdot \frac{1}{32400}\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{\color{blue}{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{\frac{1}{32400}}\right)\right)\right) \]

   8. Simplified 61.9%

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

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}} \]

       2. *-commutative N/A

          \[\leadsto \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) \cdot \frac{1}{32400} \]

       3. associate-*l* N/A

          \[\leadsto \left({b}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)\right) \cdot \frac{1}{32400} \]

       4. *-commutative N/A

          \[\leadsto \left({b}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400} \]

       5. associate-*r* N/A

          \[\leadsto {b}^{2} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \]

       6. *-commutative N/A

          \[\leadsto {b}^{2} \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right) \]

       7. *-commutative N/A

          \[\leadsto {b}^{2} \cdot \left(\frac{1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{{angle}^{2}}\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto {b}^{2} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{{angle}^{2}}\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{\left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)}\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\color{blue}{\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \cdot {angle}^{2}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \cdot {angle}^{2}\right)\right) \]

       12. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

       15. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({angle}^{2} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{\frac{1}{32400}} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       18. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(angle \cdot \color{blue}{\left(angle \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right) \]

   11. Simplified 61.9%

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

       1. associate-*r* N/A

          \[\leadsto \left(\left(b \cdot b\right) \cdot angle\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \color{blue}{\left(\left(b \cdot b\right) \cdot angle\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \color{blue}{\left(\left(b \cdot b\right) \cdot angle\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(\color{blue}{\left(b \cdot b\right)} \cdot angle\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right), \left(\left(b \cdot \color{blue}{b}\right) \cdot angle\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{32400}\right)\right), \left(\left(b \cdot \color{blue}{b}\right) \cdot angle\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right), \frac{1}{32400}\right)\right), \left(\left(b \cdot b\right) \cdot angle\right)\right) \]

       8. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right), \frac{1}{32400}\right)\right), \left(\left(b \cdot b\right) \cdot angle\right)\right) \]

       9. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1}{32400}\right)\right), \left(\left(b \cdot b\right) \cdot angle\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1}{32400}\right)\right), \left(angle \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1}{32400}\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{\left(b \cdot b\right)}\right)\right) \]

       12. *-lowering-*.f64 66.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1}{32400}\right)\right), \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   13. Applied egg-rr 66.9%

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

Alternative 12: 57.1% accurate, 139.0× speedup

Math

\[\begin{array}{l} \\ a \cdot a \end{array} \]

FPCore

(FPCore (a b angle) :precision binary64 (* a a))

C

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

Fortran

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

Java

public static double code(double a, double b, double angle) {
	return a * a;
}

Python

def code(a, b, angle):
	return a * a

Julia

function code(a, b, angle)
	return Float64(a * a)
end

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[(a * a), $MachinePrecision]

Derivation

1. Initial program 77.8%

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

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{{a}^{2}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto a \cdot \color{blue}{a} \]

   2. *-lowering-*.f64 55.1%

      \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

5. Simplified 55.1%

   \[\leadsto \color{blue}{a \cdot a} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))