ab-angle->ABCF A

Percentage Accurate: 79.6% → 79.6%

Time: 16.1s

Alternatives: 12

Speedup: 1.0×

• 37.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

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

Alternative 2: 79.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
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 76.5%

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

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

   3. *-un-lft-identity N/A

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

   4. add-sqr-sqrt N/A

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

   5. div-inv N/A

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

   6. frac-times N/A

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

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

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

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

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

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

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

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

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

   11. associate-/r/ N/A

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

   12. /-rgt-identity N/A

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

   13. *-commutative N/A

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

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

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

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

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

   16. PI-lowering-PI.f64 75.9%

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

7. Applied egg-rr 75.9%

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. associate-*r* N/A

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

   4. add-sqr-sqrt N/A

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

   5. clear-num N/A

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

   6. inv-pow N/A

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

   7. associate-/r* N/A

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

   8. metadata-eval N/A

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

   9. associate-*l/ N/A

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

   10. *-un-lft-identity N/A

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

   11. times-frac N/A

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

   12. metadata-eval N/A

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

   13. distribute-neg-frac N/A

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

   14. unpow-prod-down N/A

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

   15. inv-pow N/A

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

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

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

9. Applied egg-rr 76.5%

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

    1. unpow2 N/A

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

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

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

    3. *-rgt-identity N/A

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

    4. *-rgt-identity 76.5%

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

11. Applied egg-rr 76.5%

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

12. Final simplification 76.5%

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

Alternative 3: 79.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b, angle)
	tmp = (b * b) + ((a * sin((pi / ((-1.0 / 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[(N[(-1.0 / angle), $MachinePrecision] * -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.6%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
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 76.5%

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

   1. *-commutative N/A

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

   2. clear-num N/A

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

   3. div-inv N/A

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

   4. add-sqr-sqrt N/A

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

   5. div-inv N/A

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

   6. *-commutative N/A

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

   7. frac-times N/A

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

   8. frac-2neg N/A

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

   9. frac-2neg N/A

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

   10. frac-times N/A

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

   11. sqr-neg N/A

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

   12. add-sqr-sqrt N/A

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

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

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

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

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

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

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

   16. distribute-neg-frac N/A

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

   17. metadata-eval N/A

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

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

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

   19. metadata-eval 76.5%

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

7. Applied egg-rr 76.5%

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

   1. *-rgt-identity N/A

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

   2. pow2 N/A

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

   3. *-lowering-*.f64 76.5%

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

9. Applied egg-rr 76.5%

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

10. Final simplification 76.5%

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

Alternative 4: 79.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
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 76.5%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
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 76.5%

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

7. Applied egg-rr 76.5%

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

Alternative 5: 74.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.005555555555555556 + angle \cdot \left(angle \cdot \left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(\pi \cdot \pi\right)\right)\right)\\ t_1 := 0.5 \cdot \cos \left(2 \cdot \frac{angle}{\frac{180}{\pi}}\right)\\ \mathbf{if}\;angle \leq 0.00012:\\ \;\;\;\;b \cdot \left(b \cdot \left(0.5 + t\_1\right)\right) + angle \cdot \left(\left(a \cdot \left(\pi \cdot t\_0\right)\right) \cdot \left(t\_0 \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + a \cdot \left(a \cdot \left(0.5 - t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0
         (+
          0.005555555555555556
          (* angle (* angle (* -2.8577960676726107e-8 (* PI PI))))))
        (t_1 (* 0.5 (cos (* 2.0 (/ angle (/ 180.0 PI)))))))
   (if (<= angle 0.00012)
     (+
      (* b (* b (+ 0.5 t_1)))
      (* angle (* (* a (* PI t_0)) (* t_0 (* angle (* a PI))))))
     (+ (* b b) (* a (* a (- 0.5 t_1)))))))

C

double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 + (angle * (angle * (-2.8577960676726107e-8 * (((double) M_PI) * ((double) M_PI)))));
	double t_1 = 0.5 * cos((2.0 * (angle / (180.0 / ((double) M_PI)))));
	double tmp;
	if (angle <= 0.00012) {
		tmp = (b * (b * (0.5 + t_1))) + (angle * ((a * (((double) M_PI) * t_0)) * (t_0 * (angle * (a * ((double) M_PI))))));
	} else {
		tmp = (b * b) + (a * (a * (0.5 - t_1)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 + (angle * (angle * (-2.8577960676726107e-8 * (Math.PI * Math.PI))));
	double t_1 = 0.5 * Math.cos((2.0 * (angle / (180.0 / Math.PI))));
	double tmp;
	if (angle <= 0.00012) {
		tmp = (b * (b * (0.5 + t_1))) + (angle * ((a * (Math.PI * t_0)) * (t_0 * (angle * (a * Math.PI)))));
	} else {
		tmp = (b * b) + (a * (a * (0.5 - t_1)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = 0.005555555555555556 + (angle * (angle * (-2.8577960676726107e-8 * (math.pi * math.pi))))
	t_1 = 0.5 * math.cos((2.0 * (angle / (180.0 / math.pi))))
	tmp = 0
	if angle <= 0.00012:
		tmp = (b * (b * (0.5 + t_1))) + (angle * ((a * (math.pi * t_0)) * (t_0 * (angle * (a * math.pi)))))
	else:
		tmp = (b * b) + (a * (a * (0.5 - t_1)))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(0.005555555555555556 + Float64(angle * Float64(angle * Float64(-2.8577960676726107e-8 * Float64(pi * pi)))))
	t_1 = Float64(0.5 * cos(Float64(2.0 * Float64(angle / Float64(180.0 / pi)))))
	tmp = 0.0
	if (angle <= 0.00012)
		tmp = Float64(Float64(b * Float64(b * Float64(0.5 + t_1))) + Float64(angle * Float64(Float64(a * Float64(pi * t_0)) * Float64(t_0 * Float64(angle * Float64(a * pi))))));
	else
		tmp = Float64(Float64(b * b) + Float64(a * Float64(a * Float64(0.5 - t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = 0.005555555555555556 + (angle * (angle * (-2.8577960676726107e-8 * (pi * pi))));
	t_1 = 0.5 * cos((2.0 * (angle / (180.0 / pi))));
	tmp = 0.0;
	if (angle <= 0.00012)
		tmp = (b * (b * (0.5 + t_1))) + (angle * ((a * (pi * t_0)) * (t_0 * (angle * (a * pi)))));
	else
		tmp = (b * b) + (a * (a * (0.5 - t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(0.005555555555555556 + N[(angle * N[(angle * N[(-2.8577960676726107e-8 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Cos[N[(2.0 * N[(angle / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[angle, 0.00012], N[(N[(b * N[(b * N[(0.5 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(angle * N[(N[(a * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(angle * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] + N[(a * N[(a * N[(0.5 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if angle < 1.20000000000000003e-4

   1. Initial program 83.6%

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

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

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

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

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

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

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

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

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

      5. associate-*l/ N/A

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

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

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

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

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

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

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

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

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

   3. Simplified 83.7%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\color{blue}{\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + \frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, 2\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), 2\right)\right) \]

   7. Simplified 78.0%

      \[\leadsto {\color{blue}{\left(angle \cdot \left(a \cdot \left(\pi \cdot \left(0.005555555555555556 + \left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} \]

   9. Applied egg-rr 77.1%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \frac{angle}{\frac{180}{\pi}}\right)\right)\right) + angle \cdot \left(\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 + angle \cdot \left(angle \cdot \left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\right) \cdot \left(\left(angle \cdot \left(a \cdot \pi\right)\right) \cdot \left(0.005555555555555556 + angle \cdot \left(angle \cdot \left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\right)} \]

   if 1.20000000000000003e-4 < angle

   1. Initial program 57.0%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   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 56.4%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   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. associate-*l/ N/A

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

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

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

      4. *-rgt-identity N/A

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

      5. pow2 N/A

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

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

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

      7. unpow-prod-down N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

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

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

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

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

      12. associate-*l/ N/A

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

      13. associate-/r/ N/A

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

      14. unpow2 N/A

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

   7. Applied egg-rr 56.3%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 0.00012:\\ \;\;\;\;b \cdot \left(b \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \frac{angle}{\frac{180}{\pi}}\right)\right)\right) + angle \cdot \left(\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 + angle \cdot \left(angle \cdot \left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\right) \cdot \left(\left(0.005555555555555556 + angle \cdot \left(angle \cdot \left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(\pi \cdot \pi\right)\right)\right)\right) \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + a \cdot \left(a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{angle}{\frac{180}{\pi}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 0.00012:\\ \;\;\;\;b \cdot b + a \cdot \left(a \cdot \left(\left(angle \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + a \cdot \left(a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{angle}{\frac{180}{\pi}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 0.00012)
   (+
    (* b b)
    (* a (* a (* (* angle (* PI PI)) (* angle 3.08641975308642e-5)))))
   (+
    (* b b)
    (* a (* a (- 0.5 (* 0.5 (cos (* 2.0 (/ angle (/ 180.0 PI)))))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 0.00012) {
		tmp = (b * b) + (a * (a * ((angle * (((double) M_PI) * ((double) M_PI))) * (angle * 3.08641975308642e-5))));
	} else {
		tmp = (b * b) + (a * (a * (0.5 - (0.5 * cos((2.0 * (angle / (180.0 / ((double) M_PI)))))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 0.00012) {
		tmp = (b * b) + (a * (a * ((angle * (Math.PI * Math.PI)) * (angle * 3.08641975308642e-5))));
	} else {
		tmp = (b * b) + (a * (a * (0.5 - (0.5 * Math.cos((2.0 * (angle / (180.0 / Math.PI))))))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 0.00012:
		tmp = (b * b) + (a * (a * ((angle * (math.pi * math.pi)) * (angle * 3.08641975308642e-5))))
	else:
		tmp = (b * b) + (a * (a * (0.5 - (0.5 * math.cos((2.0 * (angle / (180.0 / math.pi))))))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 0.00012)
		tmp = Float64(Float64(b * b) + Float64(a * Float64(a * Float64(Float64(angle * Float64(pi * pi)) * Float64(angle * 3.08641975308642e-5)))));
	else
		tmp = Float64(Float64(b * b) + Float64(a * Float64(a * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(angle / Float64(180.0 / pi)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 0.00012)
		tmp = (b * b) + (a * (a * ((angle * (pi * pi)) * (angle * 3.08641975308642e-5))));
	else
		tmp = (b * b) + (a * (a * (0.5 - (0.5 * cos((2.0 * (angle / (180.0 / pi))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 0.00012], N[(N[(b * b), $MachinePrecision] + N[(a * N[(a * N[(N[(angle * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * N[(angle * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] + N[(a * N[(a * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(angle / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 1.20000000000000003e-4

   1. Initial program 83.6%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   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 83.7%

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

   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

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      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. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

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

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

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

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

   8. Simplified 68.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      12. *-lowering-*.f64 74.0%

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

   10. Applied egg-rr 74.0%

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

   if 1.20000000000000003e-4 < angle

   1. Initial program 57.0%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   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 56.4%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   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. associate-*l/ N/A

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

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

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

      4. *-rgt-identity N/A

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

      5. pow2 N/A

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

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

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

      7. unpow-prod-down N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

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

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

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

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

      12. associate-*l/ N/A

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

      13. associate-/r/ N/A

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

      14. unpow2 N/A

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

   7. Applied egg-rr 56.3%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 0.00012:\\ \;\;\;\;b \cdot b + a \cdot \left(a \cdot \left(\left(angle \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + a \cdot \left(a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{angle}{\frac{180}{\pi}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.9% accurate, 18.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 12500000000000:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + a \cdot \left(a \cdot \left(\left(angle \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 12500000000000.0)
   (* b b)
   (+
    (* b b)
    (* a (* a (* (* angle (* PI PI)) (* angle 3.08641975308642e-5)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 12500000000000.0) {
		tmp = b * b;
	} else {
		tmp = (b * b) + (a * (a * ((angle * (((double) M_PI) * ((double) M_PI))) * (angle * 3.08641975308642e-5))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 12500000000000.0) {
		tmp = b * b;
	} else {
		tmp = (b * b) + (a * (a * ((angle * (Math.PI * Math.PI)) * (angle * 3.08641975308642e-5))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 12500000000000.0:
		tmp = b * b
	else:
		tmp = (b * b) + (a * (a * ((angle * (math.pi * math.pi)) * (angle * 3.08641975308642e-5))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 12500000000000.0)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(b * b) + Float64(a * Float64(a * Float64(Float64(angle * Float64(pi * pi)) * Float64(angle * 3.08641975308642e-5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 12500000000000.0)
		tmp = b * b;
	else
		tmp = (b * b) + (a * (a * ((angle * (pi * pi)) * (angle * 3.08641975308642e-5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 12500000000000.0], N[(b * b), $MachinePrecision], N[(N[(b * b), $MachinePrecision] + N[(a * N[(a * N[(N[(angle * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * N[(angle * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.25e13

   1. Initial program 73.6%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   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 60.1%

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

   5. Simplified 60.1%

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

   if 1.25e13 < a

   1. Initial program 86.7%

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

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      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. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

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

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

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

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

   8. Simplified 65.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      12. *-lowering-*.f64 72.3%

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

   10. Applied egg-rr 72.3%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 12500000000000:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + a \cdot \left(a \cdot \left(\left(angle \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.7% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.3 \cdot 10^{+162}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \pi\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 2.3e+162)
   (* b b)
   (* (* PI PI) (* 3.08641975308642e-5 (* angle (* angle (* a a)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.3e+162) {
		tmp = b * b;
	} else {
		tmp = (((double) M_PI) * ((double) M_PI)) * (3.08641975308642e-5 * (angle * (angle * (a * a))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.3e+162) {
		tmp = b * b;
	} else {
		tmp = (Math.PI * Math.PI) * (3.08641975308642e-5 * (angle * (angle * (a * a))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 2.3e+162:
		tmp = b * b
	else:
		tmp = (math.pi * math.pi) * (3.08641975308642e-5 * (angle * (angle * (a * a))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 2.3e+162)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(pi * pi) * Float64(3.08641975308642e-5 * Float64(angle * Float64(angle * Float64(a * a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 2.3e+162)
		tmp = b * b;
	else
		tmp = (pi * pi) * (3.08641975308642e-5 * (angle * (angle * (a * a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 2.3e+162], N[(b * b), $MachinePrecision], N[(N[(Pi * Pi), $MachinePrecision] * N[(3.08641975308642e-5 * N[(angle * N[(angle * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.29999999999999994e162

   1. Initial program 73.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 58.6%

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

   5. Simplified 58.6%

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

   if 2.29999999999999994e162 < 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

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      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. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

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

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

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

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

   8. Simplified 69.8%

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

   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

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

       3. *-commutative N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. metadata-eval N/A

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

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

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

       11. *-commutative N/A

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

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

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

       13. unpow2 N/A

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

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

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

       15. distribute-lft-neg-in N/A

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

       16. metadata-eval N/A

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

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

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

       18. unpow2 N/A

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

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

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

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

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

       21. PI-lowering-PI.f64 69.8%

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

   11. Simplified 69.8%

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

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

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

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

       6. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

       12. PI-lowering-PI.f64 74.3%

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

   13. Applied egg-rr 74.3%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.3 \cdot 10^{+162}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \pi\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.4% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.5 \cdot 10^{+162}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(angle \cdot angle\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 2.5e+162)
   (* b b)
   (* a (* (* a (* (* PI PI) 3.08641975308642e-5)) (* angle angle)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.5e+162) {
		tmp = b * b;
	} else {
		tmp = a * ((a * ((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5)) * (angle * angle));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.5e+162) {
		tmp = b * b;
	} else {
		tmp = a * ((a * ((Math.PI * Math.PI) * 3.08641975308642e-5)) * (angle * angle));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 2.5e+162:
		tmp = b * b
	else:
		tmp = a * ((a * ((math.pi * math.pi) * 3.08641975308642e-5)) * (angle * angle))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 2.5e+162)
		tmp = Float64(b * b);
	else
		tmp = Float64(a * Float64(Float64(a * Float64(Float64(pi * pi) * 3.08641975308642e-5)) * Float64(angle * angle)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 2.5e+162)
		tmp = b * b;
	else
		tmp = a * ((a * ((pi * pi) * 3.08641975308642e-5)) * (angle * angle));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 2.5e+162], N[(b * b), $MachinePrecision], N[(a * N[(N[(a * N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] * N[(angle * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.4999999999999998e162

   1. Initial program 73.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 58.6%

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

   5. Simplified 58.6%

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

   if 2.4999999999999998e162 < 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

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      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. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

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

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

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

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

   8. Simplified 69.8%

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

   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

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

       3. *-commutative N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. metadata-eval N/A

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

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

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

       11. *-commutative N/A

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

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

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

       13. unpow2 N/A

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

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

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

       15. distribute-lft-neg-in N/A

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

       16. metadata-eval N/A

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

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

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

       18. unpow2 N/A

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

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

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

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

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

       21. PI-lowering-PI.f64 69.8%

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

   11. Simplified 69.8%

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

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

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

       4. *-commutative N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

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

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

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

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

       9. *-commutative N/A

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

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

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

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

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

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

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

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

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

       14. *-lowering-*.f64 66.7%

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

   13. Applied egg-rr 66.7%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.5 \cdot 10^{+162}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(angle \cdot angle\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.7% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 6.2 \cdot 10^{+162}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(angle \cdot angle\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 6.2e+162)
   (* b b)
   (* (* a a) (* (* (* PI PI) 3.08641975308642e-5) (* angle angle)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 6.2e+162) {
		tmp = b * b;
	} else {
		tmp = (a * a) * (((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5) * (angle * angle));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 6.2e+162) {
		tmp = b * b;
	} else {
		tmp = (a * a) * (((Math.PI * Math.PI) * 3.08641975308642e-5) * (angle * angle));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 6.2e+162:
		tmp = b * b
	else:
		tmp = (a * a) * (((math.pi * math.pi) * 3.08641975308642e-5) * (angle * angle))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 6.2e+162)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(a * a) * Float64(Float64(Float64(pi * pi) * 3.08641975308642e-5) * Float64(angle * angle)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 6.2e+162)
		tmp = b * b;
	else
		tmp = (a * a) * (((pi * pi) * 3.08641975308642e-5) * (angle * angle));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 6.2e+162], N[(b * b), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] * N[(angle * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 6.1999999999999999e162

   1. Initial program 73.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 58.6%

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

   5. Simplified 58.6%

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

   if 6.1999999999999999e162 < 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

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      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. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

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

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

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

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

   8. Simplified 69.8%

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

   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

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

       3. *-commutative N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. metadata-eval N/A

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

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

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

       11. *-commutative N/A

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

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

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

       13. unpow2 N/A

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

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

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

       15. distribute-lft-neg-in N/A

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

       16. metadata-eval N/A

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

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

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

       18. unpow2 N/A

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

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

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

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

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

       21. PI-lowering-PI.f64 69.8%

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

   11. Simplified 69.8%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6.2 \cdot 10^{+162}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(angle \cdot angle\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.1% accurate, 24.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
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 76.5%

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

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

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

   4. +-commutative N/A

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

   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. *-lowering-*.f64 N/A

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

   14. unpow2 N/A

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

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

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

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

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

8. Simplified 62.0%

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. associate-*l* N/A

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

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

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

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

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

   6. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

   12. *-lowering-*.f64 67.3%

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

10. Applied egg-rr 67.3%

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

Alternative 12: 57.1% 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 76.6%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
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 56.3%

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

5. Simplified 56.3%

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

Reproduce

herbie shell --seed 2024161 
(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)))