ab-angle->ABCF A

Percentage Accurate: 79.2% → 79.1%

Time: 14.8s

Alternatives: 12

Speedup: 2.0×

• 36.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Taylor expanded in angle around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(angle, 180\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \color{blue}{1}\right), 2\right)\right) \]
4. Step-by-step derivation

5. Simplified 80.2%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. frac-2neg N/A

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

   4. associate-/r/ N/A

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

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

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

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

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

   7. distribute-neg-frac N/A

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

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

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

   9. metadata-eval N/A

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

   10. neg-sub0 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-180, angle\right)\right), \left(0 - \mathsf{PI}\left(\right)\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, 1\right), 2\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-180, angle\right)\right), \mathsf{\_.f64}\left(0, \mathsf{PI}\left(\right)\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, 1\right), 2\right)\right) \]

   12. PI-lowering-PI.f64 80.1%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-180, angle\right)\right), \mathsf{\_.f64}\left(0, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, 1\right), 2\right)\right) \]

7. Applied egg-rr 80.1%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{1}{\frac{-180}{angle}} \cdot \left(0 - \pi\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
8. Step-by-step derivation

   1. add-cbrt-cube N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-180, angle\right)\right), \mathsf{\_.f64}\left(0, \left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}\right)\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, 1\right), 2\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-180, angle\right)\right), \mathsf{\_.f64}\left(0, \left(\sqrt[3]{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, 1\right), 2\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-180, angle\right)\right), \mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, 1\right), 2\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-180, angle\right)\right), \mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, 1\right), 2\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-180, angle\right)\right), \mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, 1\right), 2\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-180, angle\right)\right), \mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, 1\right), 2\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-180, angle\right)\right), \mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, 1\right), 2\right)\right) \]

   8. PI-lowering-PI.f64 80.2%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-180, angle\right)\right), \mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, 1\right), 2\right)\right) \]

9. Applied egg-rr 80.2%

   \[\leadsto {\left(a \cdot \sin \left(\frac{1}{\frac{-180}{angle}} \cdot \left(0 - \color{blue}{\sqrt[3]{\pi \cdot \left(\pi \cdot \pi\right)}}\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

10. Final simplification 80.2%

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

Alternative 2: 79.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Taylor expanded in angle around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(angle, 180\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \color{blue}{1}\right), 2\right)\right) \]
4. Step-by-step derivation

5. Simplified 80.2%

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

   1. add-cbrt-cube N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(angle, 180\right), \left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, 1\right), 2\right)\right) \]

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

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

   3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   8. PI-lowering-PI.f64 80.2%

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

7. Applied egg-rr 80.2%

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

   1. *-rgt-identity N/A

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

   2. pow2 N/A

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

   3. *-lowering-*.f64 80.2%

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

9. Applied egg-rr 80.2%

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

10. Final simplification 80.2%

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

Alternative 3: 79.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Taylor expanded in angle around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(angle, 180\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \color{blue}{1}\right), 2\right)\right) \]
4. Step-by-step derivation

5. Simplified 80.2%

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

   1. associate-/r/ N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. div-inv N/A

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

   5. associate-/l* N/A

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

   6. unpow-prod-down N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

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

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

   10. metadata-eval N/A

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

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

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

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

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

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

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

   14. PI-lowering-PI.f64 80.1%

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

7. Applied egg-rr 80.1%

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

   1. unpow-1 N/A

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

   2. div-inv N/A

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

   3. associate-/r* N/A

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

   4. clear-num N/A

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

   5. /-rgt-identity N/A

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

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

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

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

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

   8. /-lowering-/.f64 80.2%

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

9. Applied egg-rr 80.2%

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

10. Final simplification 80.2%

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

Alternative 4: 79.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Taylor expanded in angle around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(angle, 180\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \color{blue}{1}\right), 2\right)\right) \]
4. Step-by-step derivation

5. Simplified 80.2%

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

   1. *-rgt-identity N/A

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

   2. pow2 N/A

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

   3. *-lowering-*.f64 80.2%

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

7. Applied egg-rr 80.2%

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

8. Final simplification 80.2%

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

Alternative 5: 72.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 0.00067:\\ \;\;\;\;b \cdot b + a \cdot \left(a \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + a \cdot \left(a \cdot \left(0.5 + \cos \left(2 \cdot \frac{angle}{\frac{180}{\pi}}\right) \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 0.00067)
   (+
    (* b b)
    (* a (* a (* 3.08641975308642e-5 (* angle (* angle (* PI PI)))))))
   (+
    (* b b)
    (* a (* a (+ 0.5 (* (cos (* 2.0 (/ angle (/ 180.0 PI)))) -0.5)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 0.00067) {
		tmp = (b * b) + (a * (a * (3.08641975308642e-5 * (angle * (angle * (((double) M_PI) * ((double) M_PI)))))));
	} else {
		tmp = (b * b) + (a * (a * (0.5 + (cos((2.0 * (angle / (180.0 / ((double) M_PI))))) * -0.5))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 0.00067) {
		tmp = (b * b) + (a * (a * (3.08641975308642e-5 * (angle * (angle * (Math.PI * Math.PI))))));
	} else {
		tmp = (b * b) + (a * (a * (0.5 + (Math.cos((2.0 * (angle / (180.0 / Math.PI)))) * -0.5))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 0.00067:
		tmp = (b * b) + (a * (a * (3.08641975308642e-5 * (angle * (angle * (math.pi * math.pi))))))
	else:
		tmp = (b * b) + (a * (a * (0.5 + (math.cos((2.0 * (angle / (180.0 / math.pi)))) * -0.5))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 0.00067)
		tmp = Float64(Float64(b * b) + Float64(a * Float64(a * Float64(3.08641975308642e-5 * Float64(angle * Float64(angle * Float64(pi * pi)))))));
	else
		tmp = Float64(Float64(b * b) + Float64(a * Float64(a * Float64(0.5 + Float64(cos(Float64(2.0 * Float64(angle / Float64(180.0 / pi)))) * -0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 0.00067)
		tmp = (b * b) + (a * (a * (3.08641975308642e-5 * (angle * (angle * (pi * pi))))));
	else
		tmp = (b * b) + (a * (a * (0.5 + (cos((2.0 * (angle / (180.0 / pi)))) * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 0.00067], N[(N[(b * b), $MachinePrecision] + N[(a * N[(a * N[(3.08641975308642e-5 * N[(angle * N[(angle * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] + N[(a * N[(a * N[(0.5 + N[(N[Cos[N[(2.0 * N[(angle / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 6.7000000000000002e-4

   1. Initial program 86.3%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(angle, 180\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \color{blue}{1}\right), 2\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 86.7%

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

   6. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   8. Simplified 71.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

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

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

      12. PI-lowering-PI.f64 77.5%

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

   10. Applied egg-rr 77.5%

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

   if 6.7000000000000002e-4 < angle

   1. Initial program 56.9%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(angle, 180\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \color{blue}{1}\right), 2\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 60.3%

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

      1. add-cbrt-cube N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(angle, 180\right), \left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, 1\right), 2\right)\right) \]

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

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      8. PI-lowering-PI.f64 60.6%

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

   7. Applied egg-rr 60.6%

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

      1. +-commutative N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left({\left(b \cdot 1\right)}^{2}\right), \color{blue}{\left({\left(a \cdot \sin \left(\frac{angle}{180} \cdot \sqrt[3]{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}^{2}\right)}\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\left({b}^{2}\right), \left({\left(\color{blue}{a} \cdot \sin \left(\frac{angle}{180} \cdot \sqrt[3]{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}^{2}\right)\right) \]

      4. pow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(b \cdot b\right), \left({\color{blue}{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \sqrt[3]{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}}^{2}\right)\right) \]

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

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

      6. unpow-prod-down N/A

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

      7. pow2 N/A

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

      8. associate-*r* N/A

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

      9. pow2 N/A

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

      10. associate-/r/ N/A

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

   9. Applied egg-rr 60.4%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 0.00067:\\ \;\;\;\;b \cdot b + a \cdot \left(a \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + a \cdot \left(a \cdot \left(0.5 + \cos \left(2 \cdot \frac{angle}{\frac{180}{\pi}}\right) \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 0.00067:\\ \;\;\;\;b \cdot b + a \cdot \left(a \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + a \cdot \left(a \cdot \left(0.5 + -0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 0.00067)
   (+
    (* b b)
    (* a (* a (* 3.08641975308642e-5 (* angle (* angle (* PI PI)))))))
   (+
    (* b b)
    (* a (* a (+ 0.5 (* -0.5 (cos (* (* angle PI) 0.011111111111111112)))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 0.00067) {
		tmp = (b * b) + (a * (a * (3.08641975308642e-5 * (angle * (angle * (((double) M_PI) * ((double) M_PI)))))));
	} else {
		tmp = (b * b) + (a * (a * (0.5 + (-0.5 * cos(((angle * ((double) M_PI)) * 0.011111111111111112))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 0.00067) {
		tmp = (b * b) + (a * (a * (3.08641975308642e-5 * (angle * (angle * (Math.PI * Math.PI))))));
	} else {
		tmp = (b * b) + (a * (a * (0.5 + (-0.5 * Math.cos(((angle * Math.PI) * 0.011111111111111112))))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 0.00067:
		tmp = (b * b) + (a * (a * (3.08641975308642e-5 * (angle * (angle * (math.pi * math.pi))))))
	else:
		tmp = (b * b) + (a * (a * (0.5 + (-0.5 * math.cos(((angle * math.pi) * 0.011111111111111112))))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 0.00067)
		tmp = Float64(Float64(b * b) + Float64(a * Float64(a * Float64(3.08641975308642e-5 * Float64(angle * Float64(angle * Float64(pi * pi)))))));
	else
		tmp = Float64(Float64(b * b) + Float64(a * Float64(a * Float64(0.5 + Float64(-0.5 * cos(Float64(Float64(angle * pi) * 0.011111111111111112)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 0.00067)
		tmp = (b * b) + (a * (a * (3.08641975308642e-5 * (angle * (angle * (pi * pi))))));
	else
		tmp = (b * b) + (a * (a * (0.5 + (-0.5 * cos(((angle * pi) * 0.011111111111111112))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 0.00067], N[(N[(b * b), $MachinePrecision] + N[(a * N[(a * N[(3.08641975308642e-5 * N[(angle * N[(angle * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] + N[(a * N[(a * N[(0.5 + N[(-0.5 * N[Cos[N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 6.7000000000000002e-4

   1. Initial program 86.3%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(angle, 180\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \color{blue}{1}\right), 2\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 86.7%

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

   6. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   8. Simplified 71.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

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

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

      12. PI-lowering-PI.f64 77.5%

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

   10. Applied egg-rr 77.5%

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

   if 6.7000000000000002e-4 < angle

   1. Initial program 56.9%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(angle, 180\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \color{blue}{1}\right), 2\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 60.3%

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

      1. +-commutative N/A

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

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

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

      3. *-rgt-identity N/A

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

      4. pow2 N/A

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

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

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

      6. unpow-prod-down N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. pow2 N/A

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

      10. associate-/r/ N/A

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

      11. associate-/r/ N/A

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

      12. sqr-sin-a N/A

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

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

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

      14. sqr-sin-a N/A

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

   7. Applied egg-rr 60.2%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 0.00067:\\ \;\;\;\;b \cdot b + a \cdot \left(a \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + a \cdot \left(a \cdot \left(0.5 + -0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.8% accurate, 18.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(\pi \cdot \pi\right)\\ \mathbf{if}\;a \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;b \cdot b + t\_0 \cdot \left(angle \cdot \left(a \cdot \left(a \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + a \cdot \left(a \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot t\_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* PI PI))))
   (if (<= a 7.5e+153)
     (+ (* b b) (* t_0 (* angle (* a (* a 3.08641975308642e-5)))))
     (+ (* b b) (* a (* a (* 3.08641975308642e-5 (* angle t_0))))))))

C

double code(double a, double b, double angle) {
	double t_0 = angle * (((double) M_PI) * ((double) M_PI));
	double tmp;
	if (a <= 7.5e+153) {
		tmp = (b * b) + (t_0 * (angle * (a * (a * 3.08641975308642e-5))));
	} else {
		tmp = (b * b) + (a * (a * (3.08641975308642e-5 * (angle * t_0))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = angle * (Math.PI * Math.PI);
	double tmp;
	if (a <= 7.5e+153) {
		tmp = (b * b) + (t_0 * (angle * (a * (a * 3.08641975308642e-5))));
	} else {
		tmp = (b * b) + (a * (a * (3.08641975308642e-5 * (angle * t_0))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = angle * (math.pi * math.pi)
	tmp = 0
	if a <= 7.5e+153:
		tmp = (b * b) + (t_0 * (angle * (a * (a * 3.08641975308642e-5))))
	else:
		tmp = (b * b) + (a * (a * (3.08641975308642e-5 * (angle * t_0))))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(angle * Float64(pi * pi))
	tmp = 0.0
	if (a <= 7.5e+153)
		tmp = Float64(Float64(b * b) + Float64(t_0 * Float64(angle * Float64(a * Float64(a * 3.08641975308642e-5)))));
	else
		tmp = Float64(Float64(b * b) + Float64(a * Float64(a * Float64(3.08641975308642e-5 * Float64(angle * t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = angle * (pi * pi);
	tmp = 0.0;
	if (a <= 7.5e+153)
		tmp = (b * b) + (t_0 * (angle * (a * (a * 3.08641975308642e-5))));
	else
		tmp = (b * b) + (a * (a * (3.08641975308642e-5 * (angle * t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 7.5e+153], N[(N[(b * b), $MachinePrecision] + N[(t$95$0 * N[(angle * N[(a * N[(a * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] + N[(a * N[(a * N[(3.08641975308642e-5 * N[(angle * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 7.50000000000000065e153

   1. Initial program 75.3%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(angle, 180\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \color{blue}{1}\right), 2\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 76.7%

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

   6. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   8. Simplified 65.1%

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

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

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

      7. pow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

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

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

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

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

      15. PI-lowering-PI.f64 69.7%

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

   10. Applied egg-rr 69.7%

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

   if 7.50000000000000065e153 < a

   1. Initial program 99.7%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(angle, 180\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \color{blue}{1}\right), 2\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 99.7%

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

   6. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   8. Simplified 67.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

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

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

      12. PI-lowering-PI.f64 87.6%

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

   10. Applied egg-rr 87.6%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;b \cdot b + \left(angle \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(angle \cdot \left(a \cdot \left(a \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + a \cdot \left(a \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.6% accurate, 18.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(\pi \cdot \pi\right)\\ \mathbf{if}\;a \leq 5.4 \cdot 10^{+137}:\\ \;\;\;\;b \cdot b + \left(3.08641975308642 \cdot 10^{-5} \cdot t\_0\right) \cdot \left(angle \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + a \cdot \left(a \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot t\_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* PI PI))))
   (if (<= a 5.4e+137)
     (+ (* b b) (* (* 3.08641975308642e-5 t_0) (* angle (* a a))))
     (+ (* b b) (* a (* a (* 3.08641975308642e-5 (* angle t_0))))))))

C

double code(double a, double b, double angle) {
	double t_0 = angle * (((double) M_PI) * ((double) M_PI));
	double tmp;
	if (a <= 5.4e+137) {
		tmp = (b * b) + ((3.08641975308642e-5 * t_0) * (angle * (a * a)));
	} else {
		tmp = (b * b) + (a * (a * (3.08641975308642e-5 * (angle * t_0))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = angle * (Math.PI * Math.PI);
	double tmp;
	if (a <= 5.4e+137) {
		tmp = (b * b) + ((3.08641975308642e-5 * t_0) * (angle * (a * a)));
	} else {
		tmp = (b * b) + (a * (a * (3.08641975308642e-5 * (angle * t_0))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = angle * (math.pi * math.pi)
	tmp = 0
	if a <= 5.4e+137:
		tmp = (b * b) + ((3.08641975308642e-5 * t_0) * (angle * (a * a)))
	else:
		tmp = (b * b) + (a * (a * (3.08641975308642e-5 * (angle * t_0))))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(angle * Float64(pi * pi))
	tmp = 0.0
	if (a <= 5.4e+137)
		tmp = Float64(Float64(b * b) + Float64(Float64(3.08641975308642e-5 * t_0) * Float64(angle * Float64(a * a))));
	else
		tmp = Float64(Float64(b * b) + Float64(a * Float64(a * Float64(3.08641975308642e-5 * Float64(angle * t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = angle * (pi * pi);
	tmp = 0.0;
	if (a <= 5.4e+137)
		tmp = (b * b) + ((3.08641975308642e-5 * t_0) * (angle * (a * a)));
	else
		tmp = (b * b) + (a * (a * (3.08641975308642e-5 * (angle * t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 5.4e+137], N[(N[(b * b), $MachinePrecision] + N[(N[(3.08641975308642e-5 * t$95$0), $MachinePrecision] * N[(angle * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] + N[(a * N[(a * N[(3.08641975308642e-5 * N[(angle * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 5.40000000000000034e137

   1. Initial program 75.4%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(angle, 180\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \color{blue}{1}\right), 2\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 76.8%

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

   6. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   8. Simplified 65.3%

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

      1. pow2 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      13. pow2 N/A

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

      14. *-lowering-*.f64 70.0%

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

   10. Applied egg-rr 70.0%

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

   if 5.40000000000000034e137 < a

   1. Initial program 96.9%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(angle, 180\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \color{blue}{1}\right), 2\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 96.9%

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

   6. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   8. Simplified 66.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

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

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

      12. PI-lowering-PI.f64 84.5%

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

   10. Applied egg-rr 84.5%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.4 \cdot 10^{+137}:\\ \;\;\;\;b \cdot b + \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(angle \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + a \cdot \left(a \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.2% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.65 \cdot 10^{+142}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.65e+142)
   (* b b)
   (* a (* a (* 3.08641975308642e-5 (* angle (* angle (* PI PI))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.65e+142) {
		tmp = b * b;
	} else {
		tmp = a * (a * (3.08641975308642e-5 * (angle * (angle * (((double) M_PI) * ((double) M_PI))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.65e+142) {
		tmp = b * b;
	} else {
		tmp = a * (a * (3.08641975308642e-5 * (angle * (angle * (Math.PI * Math.PI)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 1.65e+142:
		tmp = b * b
	else:
		tmp = a * (a * (3.08641975308642e-5 * (angle * (angle * (math.pi * math.pi)))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.65e+142)
		tmp = Float64(b * b);
	else
		tmp = Float64(a * Float64(a * Float64(3.08641975308642e-5 * Float64(angle * Float64(angle * Float64(pi * pi))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1.65e+142)
		tmp = b * b;
	else
		tmp = a * (a * (3.08641975308642e-5 * (angle * (angle * (pi * pi)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.65e+142], N[(b * b), $MachinePrecision], N[(a * N[(a * N[(3.08641975308642e-5 * N[(angle * N[(angle * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.6500000000000001e142

   1. Initial program 75.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 58.8%

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

   5. Simplified 58.8%

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

   if 1.6500000000000001e142 < a

   1. Initial program 96.7%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(angle, 180\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \color{blue}{1}\right), 2\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 96.7%

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

   6. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   8. Simplified 64.4%

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

   9. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. metadata-eval N/A

          \[\leadsto \left({a}^{2} \cdot {angle}^{2}\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{5400} \cdot \color{blue}{\frac{-1}{6}}\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \left({a}^{2} \cdot {angle}^{2}\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\left(\frac{-1}{8100} + \frac{-1}{16200}\right) \cdot \frac{-1}{6}\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \left({a}^{2} \cdot {angle}^{2}\right) \cdot \left(\left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{8100} + \frac{-1}{16200}\right)\right) \cdot \color{blue}{\frac{-1}{6}}\right) \]

       7. distribute-rgt-out N/A

          \[\leadsto \left({a}^{2} \cdot {angle}^{2}\right) \cdot \left(\left(\frac{-1}{8100} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-1}{16200} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{-1}{6}\right) \]

       8. associate-*l* N/A

          \[\leadsto \left(\left({a}^{2} \cdot {angle}^{2}\right) \cdot \left(\frac{-1}{8100} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-1}{16200} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{-1}{6}} \]

       9. associate-*r* N/A

          \[\leadsto \left({a}^{2} \cdot \left({angle}^{2} \cdot \left(\frac{-1}{8100} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-1}{16200} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \cdot \frac{-1}{6} \]

       10. associate-*l* N/A

           \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left({angle}^{2} \cdot \left(\frac{-1}{8100} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-1}{16200} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{-1}{6}\right)} \]

       11. associate-*r* N/A

           \[\leadsto {a}^{2} \cdot \left({angle}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{8100} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-1}{16200} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{-1}{6}\right)}\right) \]

       12. *-commutative N/A

           \[\leadsto {a}^{2} \cdot \left({angle}^{2} \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\frac{-1}{8100} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-1}{16200} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right) \]

   11. Simplified 64.4%

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

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

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

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

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

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

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

       6. associate-*l* N/A

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

       7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

       12. PI-lowering-PI.f64 69.6%

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

   13. Applied egg-rr 69.6%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.65 \cdot 10^{+142}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.4% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.5 \cdot 10^{+142}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \left(\pi \cdot \left(angle \cdot angle\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.5e+142)
   (* b b)
   (* (* a a) (* 3.08641975308642e-5 (* PI (* PI (* angle angle)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.5e+142) {
		tmp = b * b;
	} else {
		tmp = (a * a) * (3.08641975308642e-5 * (((double) M_PI) * (((double) M_PI) * (angle * angle))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.5e+142) {
		tmp = b * b;
	} else {
		tmp = (a * a) * (3.08641975308642e-5 * (Math.PI * (Math.PI * (angle * angle))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 1.5e+142:
		tmp = b * b
	else:
		tmp = (a * a) * (3.08641975308642e-5 * (math.pi * (math.pi * (angle * angle))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.5e+142)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(a * a) * Float64(3.08641975308642e-5 * Float64(pi * Float64(pi * Float64(angle * angle)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1.5e+142)
		tmp = b * b;
	else
		tmp = (a * a) * (3.08641975308642e-5 * (pi * (pi * (angle * angle))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.5e+142], N[(b * b), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(3.08641975308642e-5 * N[(Pi * N[(Pi * N[(angle * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.49999999999999987e142

   1. Initial program 75.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 58.8%

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

   5. Simplified 58.8%

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

   if 1.49999999999999987e142 < a

   1. Initial program 96.7%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(angle, 180\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \color{blue}{1}\right), 2\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 96.7%

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

   6. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   8. Simplified 64.4%

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

   9. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. metadata-eval N/A

          \[\leadsto \left({a}^{2} \cdot {angle}^{2}\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{5400} \cdot \color{blue}{\frac{-1}{6}}\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \left({a}^{2} \cdot {angle}^{2}\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\left(\frac{-1}{8100} + \frac{-1}{16200}\right) \cdot \frac{-1}{6}\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \left({a}^{2} \cdot {angle}^{2}\right) \cdot \left(\left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{8100} + \frac{-1}{16200}\right)\right) \cdot \color{blue}{\frac{-1}{6}}\right) \]

       7. distribute-rgt-out N/A

          \[\leadsto \left({a}^{2} \cdot {angle}^{2}\right) \cdot \left(\left(\frac{-1}{8100} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-1}{16200} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{-1}{6}\right) \]

       8. associate-*l* N/A

          \[\leadsto \left(\left({a}^{2} \cdot {angle}^{2}\right) \cdot \left(\frac{-1}{8100} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-1}{16200} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{-1}{6}} \]

       9. associate-*r* N/A

          \[\leadsto \left({a}^{2} \cdot \left({angle}^{2} \cdot \left(\frac{-1}{8100} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-1}{16200} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \cdot \frac{-1}{6} \]

       10. associate-*l* N/A

           \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left({angle}^{2} \cdot \left(\frac{-1}{8100} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-1}{16200} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{-1}{6}\right)} \]

       11. associate-*r* N/A

           \[\leadsto {a}^{2} \cdot \left({angle}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{8100} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-1}{16200} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{-1}{6}\right)}\right) \]

       12. *-commutative N/A

           \[\leadsto {a}^{2} \cdot \left({angle}^{2} \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\frac{-1}{8100} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-1}{16200} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right) \]

   11. Simplified 64.4%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.5 \cdot 10^{+142}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \left(\pi \cdot \left(angle \cdot angle\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.9% accurate, 24.5× speedup

Math

\[\begin{array}{l} \\ b \cdot b + \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(angle \cdot \left(a \cdot a\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+ (* b b) (* (* 3.08641975308642e-5 (* angle (* PI PI))) (* angle (* a a)))))

C

double code(double a, double b, double angle) {
	return (b * b) + ((3.08641975308642e-5 * (angle * (((double) M_PI) * ((double) M_PI)))) * (angle * (a * a)));
}

Java

public static double code(double a, double b, double angle) {
	return (b * b) + ((3.08641975308642e-5 * (angle * (Math.PI * Math.PI))) * (angle * (a * a)));
}

Python

def code(a, b, angle):
	return (b * b) + ((3.08641975308642e-5 * (angle * (math.pi * math.pi))) * (angle * (a * a)))

Julia

function code(a, b, angle)
	return Float64(Float64(b * b) + Float64(Float64(3.08641975308642e-5 * Float64(angle * Float64(pi * pi))) * Float64(angle * Float64(a * a))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (b * b) + ((3.08641975308642e-5 * (angle * (pi * pi))) * (angle * (a * a)));
end

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Taylor expanded in angle around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(angle, 180\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \color{blue}{1}\right), 2\right)\right) \]
4. Step-by-step derivation

5. Simplified 80.2%

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

6. Taylor expanded in angle around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*l* N/A

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

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

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

   6. unpow2 N/A

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

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

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

   8. associate-*l* N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. *-commutative N/A

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

   15. associate-*r* N/A

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

8. Simplified 65.5%

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

   1. pow2 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*l* N/A

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

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

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

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

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

   7. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

   13. pow2 N/A

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

   14. *-lowering-*.f64 69.9%

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

10. Applied egg-rr 69.9%

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

Alternative 12: 56.4% accurate, 139.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 55.0%

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

5. Simplified 55.0%

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

Reproduce

herbie shell --seed 2024154 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))