ab-angle->ABCF A

Percentage Accurate: 87.0% → 86.5%

Time: 36.8s

Alternatives: 8

Speedup: 1.0×

• 29.1% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. a = -2.8596904483816153e-171
  2. b = -9.035647969315671e+193
  3. angle = -1.309056001303669e+92

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: 87.0% 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: 86.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, -1.54320987654321 \cdot 10^{-5} \cdot {\left(angle \cdot \pi\right)}^{2}\right)\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow
   (*
    a
    (sin
     (expm1
      (fma
       0.005555555555555556
       (* angle PI)
       (* -1.54320987654321e-5 (pow (* angle PI) 2.0))))))
   2.0)
  (pow (* b (cos (* PI (/ angle 180.0)))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * sin(expm1(fma(0.005555555555555556, (angle * ((double) M_PI)), (-1.54320987654321e-5 * pow((angle * ((double) M_PI)), 2.0)))))), 2.0) + pow((b * cos((((double) M_PI) * (angle / 180.0)))), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(expm1(fma(0.005555555555555556, Float64(angle * pi), Float64(-1.54320987654321e-5 * (Float64(angle * pi) ^ 2.0)))))) ^ 2.0) + (Float64(b * cos(Float64(pi * Float64(angle / 180.0)))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(Exp[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(-1.54320987654321e-5 * N[Power[N[(angle * Pi), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. associate-*l/ 86.7%

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

   2. associate-*r/ 86.7%

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

   3. expm1-log1p-u 74.2%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \frac{\pi}{180}\right)\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   4. *-commutative 74.2%

      \[\leadsto {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\pi}{180} \cdot angle}\right)\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. associate-*l/ 74.1%

      \[\leadsto {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\pi \cdot angle}{180}}\right)\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. associate-*r/ 74.1%

      \[\leadsto {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\pi \cdot \frac{angle}{180}}\right)\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   7. div-inv 74.1%

      \[\leadsto {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   8. metadata-eval 74.1%

      \[\leadsto {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

4. Applied egg-rr 74.1%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

5. Taylor expanded in angle around 0 86.9%

   \[\leadsto {\left(a \cdot \sin \left(\mathsf{expm1}\left(\color{blue}{-1.54320987654321 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right) + 0.005555555555555556 \cdot \left(angle \cdot \pi\right)}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
6. Step-by-step derivation

   1. +-commutative 86.9%

      \[\leadsto {\left(a \cdot \sin \left(\mathsf{expm1}\left(\color{blue}{0.005555555555555556 \cdot \left(angle \cdot \pi\right) + -1.54320987654321 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   2. *-commutative 86.9%

      \[\leadsto {\left(a \cdot \sin \left(\mathsf{expm1}\left(0.005555555555555556 \cdot \color{blue}{\left(\pi \cdot angle\right)} + -1.54320987654321 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   3. fma-define 86.9%

      \[\leadsto {\left(a \cdot \sin \left(\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(0.005555555555555556, \pi \cdot angle, -1.54320987654321 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   4. *-commutative 86.9%

      \[\leadsto {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{fma}\left(0.005555555555555556, \color{blue}{angle \cdot \pi}, -1.54320987654321 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. *-commutative 86.9%

      \[\leadsto {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, -1.54320987654321 \cdot 10^{-5} \cdot \color{blue}{\left({\pi}^{2} \cdot {angle}^{2}\right)}\right)\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. unpow2 86.9%

      \[\leadsto {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, -1.54320987654321 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\pi \cdot \pi\right)} \cdot {angle}^{2}\right)\right)\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   7. unpow2 86.9%

      \[\leadsto {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \color{blue}{\left(angle \cdot angle\right)}\right)\right)\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   8. swap-sqr 86.9%

      \[\leadsto {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, -1.54320987654321 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)\right)}\right)\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   9. unpow2 86.9%

      \[\leadsto {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, -1.54320987654321 \cdot 10^{-5} \cdot \color{blue}{{\left(\pi \cdot angle\right)}^{2}}\right)\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   10. *-commutative 86.9%

       \[\leadsto {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, -1.54320987654321 \cdot 10^{-5} \cdot {\color{blue}{\left(angle \cdot \pi\right)}}^{2}\right)\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

7. Simplified 86.9%

   \[\leadsto {\left(a \cdot \sin \left(\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, -1.54320987654321 \cdot 10^{-5} \cdot {\left(angle \cdot \pi\right)}^{2}\right)}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

8. Final simplification 86.9%

   \[\leadsto {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, -1.54320987654321 \cdot 10^{-5} \cdot {\left(angle \cdot \pi\right)}^{2}\right)\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
9. Add Preprocessing

Alternative 2: 74.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ {\left(b \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (cos (* PI (/ angle 180.0)))) 2.0)
  (pow (* a (sin (expm1 (log1p (* PI (* 0.005555555555555556 angle)))))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((b * cos((((double) M_PI) * (angle / 180.0)))), 2.0) + pow((a * sin(expm1(log1p((((double) M_PI) * (0.005555555555555556 * angle)))))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((b * Math.cos((Math.PI * (angle / 180.0)))), 2.0) + Math.pow((a * Math.sin(Math.expm1(Math.log1p((Math.PI * (0.005555555555555556 * angle)))))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((b * math.cos((math.pi * (angle / 180.0)))), 2.0) + math.pow((a * math.sin(math.expm1(math.log1p((math.pi * (0.005555555555555556 * angle)))))), 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(b * cos(Float64(pi * Float64(angle / 180.0)))) ^ 2.0) + (Float64(a * sin(expm1(log1p(Float64(pi * Float64(0.005555555555555556 * angle)))))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(b * N[Cos[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Sin[N[(Exp[N[Log[1 + N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. associate-*l/ 86.7%

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

   2. associate-*r/ 86.7%

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

   3. expm1-log1p-u 74.2%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \frac{\pi}{180}\right)\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   4. *-commutative 74.2%

      \[\leadsto {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\pi}{180} \cdot angle}\right)\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. associate-*l/ 74.1%

      \[\leadsto {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\pi \cdot angle}{180}}\right)\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. associate-*r/ 74.1%

      \[\leadsto {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\pi \cdot \frac{angle}{180}}\right)\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   7. div-inv 74.1%

      \[\leadsto {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   8. metadata-eval 74.1%

      \[\leadsto {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

4. Applied egg-rr 74.1%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

5. Final simplification 74.1%

   \[\leadsto {\left(b \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)}^{2} \]
6. Add Preprocessing

Alternative 3: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* 0.005555555555555556 (* angle PI)))) 2.0)
  (pow (* b (cos (* angle (/ PI 180.0)))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * sin((0.005555555555555556 * (angle * ((double) M_PI))))), 2.0) + pow((b * cos((angle * (((double) M_PI) / 180.0)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin((0.005555555555555556 * (angle * Math.PI)))), 2.0) + Math.pow((b * Math.cos((angle * (Math.PI / 180.0)))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((a * math.sin((0.005555555555555556 * (angle * math.pi)))), 2.0) + math.pow((b * math.cos((angle * (math.pi / 180.0)))), 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(0.005555555555555556 * Float64(angle * pi)))) ^ 2.0) + (Float64(b * cos(Float64(angle * Float64(pi / 180.0)))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * sin((0.005555555555555556 * (angle * pi)))) ^ 2.0) + ((b * cos((angle * (pi / 180.0)))) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(angle * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. associate-*l/ 86.7%

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

   2. associate-/l* 86.7%

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

   3. associate-*l/ 86.6%

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

   4. associate-/l* 86.7%

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

3. Simplified 86.7%

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

5. Taylor expanded in angle around inf 86.7%

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

6. Final simplification 86.7%

   \[\leadsto {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} \]
7. Add Preprocessing

Alternative 4: 87.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. associate-*l/ 86.7%

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

   2. associate-*r/ 86.7%

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

   3. clear-num 86.7%

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

   4. un-div-inv 86.8%

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

4. Applied egg-rr 86.8%

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

5. Final simplification 86.8%

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

Alternative 5: 86.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. associate-*l/ 86.7%

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

   2. associate-/l* 86.7%

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

   3. associate-*l/ 86.6%

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

   4. associate-/l* 86.7%

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

3. Simplified 86.7%

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

5. Taylor expanded in angle around 0 86.7%

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

6. Taylor expanded in a around 0 86.7%

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

7. Final simplification 86.7%

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

Alternative 6: 82.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 2.9 \cdot 10^{-13}:\\ \;\;\;\;{b}^{2} + 0.005555555555555556 \cdot \left(a \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + 0.005555555555555556 \cdot \left(a \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a \cdot \left(0.005555555555555556 \cdot \frac{\pi}{angle}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 2.9e-13)
   (+
    (pow b 2.0)
    (*
     0.005555555555555556
     (* a (* (* angle PI) (* a (* 0.005555555555555556 (* angle PI)))))))
   (+
    (pow b 2.0)
    (*
     0.005555555555555556
     (* a (* (* angle PI) (* a (* 0.005555555555555556 (/ PI angle)))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 2.9e-13) {
		tmp = pow(b, 2.0) + (0.005555555555555556 * (a * ((angle * ((double) M_PI)) * (a * (0.005555555555555556 * (angle * ((double) M_PI)))))));
	} else {
		tmp = pow(b, 2.0) + (0.005555555555555556 * (a * ((angle * ((double) M_PI)) * (a * (0.005555555555555556 * (((double) M_PI) / angle))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 2.9e-13) {
		tmp = Math.pow(b, 2.0) + (0.005555555555555556 * (a * ((angle * Math.PI) * (a * (0.005555555555555556 * (angle * Math.PI))))));
	} else {
		tmp = Math.pow(b, 2.0) + (0.005555555555555556 * (a * ((angle * Math.PI) * (a * (0.005555555555555556 * (Math.PI / angle))))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 2.9e-13:
		tmp = math.pow(b, 2.0) + (0.005555555555555556 * (a * ((angle * math.pi) * (a * (0.005555555555555556 * (angle * math.pi))))))
	else:
		tmp = math.pow(b, 2.0) + (0.005555555555555556 * (a * ((angle * math.pi) * (a * (0.005555555555555556 * (math.pi / angle))))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 2.9e-13)
		tmp = Float64((b ^ 2.0) + Float64(0.005555555555555556 * Float64(a * Float64(Float64(angle * pi) * Float64(a * Float64(0.005555555555555556 * Float64(angle * pi)))))));
	else
		tmp = Float64((b ^ 2.0) + Float64(0.005555555555555556 * Float64(a * Float64(Float64(angle * pi) * Float64(a * Float64(0.005555555555555556 * Float64(pi / angle)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 2.9e-13)
		tmp = (b ^ 2.0) + (0.005555555555555556 * (a * ((angle * pi) * (a * (0.005555555555555556 * (angle * pi))))));
	else
		tmp = (b ^ 2.0) + (0.005555555555555556 * (a * ((angle * pi) * (a * (0.005555555555555556 * (pi / angle))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 2.9e-13], N[(N[Power[b, 2.0], $MachinePrecision] + N[(0.005555555555555556 * N[(a * N[(N[(angle * Pi), $MachinePrecision] * N[(a * N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(0.005555555555555556 * N[(a * N[(N[(angle * Pi), $MachinePrecision] * N[(a * N[(0.005555555555555556 * N[(Pi / angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 2.8999999999999998e-13

   1. Initial program 92.1%

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

      1. associate-*l/ 92.1%

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

      2. associate-/l* 92.2%

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

      3. associate-*l/ 92.2%

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

      4. associate-/l* 92.2%

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

   3. Simplified 92.2%

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

   5. Taylor expanded in angle around 0 92.2%

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

   6. Taylor expanded in angle around 0 87.8%

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

      1. unpow2 87.8%

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

      2. associate-*l* 87.8%

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

      3. *-commutative 87.8%

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

      4. associate-*l* 87.8%

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

      5. *-commutative 87.8%

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

      6. associate-*l* 87.8%

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

      7. *-commutative 87.8%

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

      8. associate-*l* 87.8%

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

      9. *-commutative 87.8%

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

   8. Applied egg-rr 87.8%

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

      1. *-commutative 87.8%

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

      2. associate-*l* 87.8%

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

      3. *-commutative 87.8%

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

      4. associate-*r* 87.9%

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

      5. associate-*r* 87.8%

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

      6. *-commutative 87.8%

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

      7. associate-*r* 87.8%

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

   10. Simplified 87.8%

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

   if 2.8999999999999998e-13 < angle

   1. Initial program 65.4%

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

      1. associate-*l/ 65.4%

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

      2. associate-/l* 65.2%

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

      3. associate-*l/ 65.0%

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

      4. associate-/l* 65.1%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in angle around 0 65.0%

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

   6. Taylor expanded in angle around 0 52.3%

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

      1. unpow2 52.3%

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

      2. associate-*l* 52.3%

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

      3. *-commutative 52.3%

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

      4. associate-*l* 52.3%

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

      5. *-commutative 52.3%

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

      6. associate-*l* 52.3%

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

      7. *-commutative 52.3%

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

      8. associate-*l* 52.3%

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

      9. *-commutative 52.3%

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

   8. Applied egg-rr 52.3%

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

      1. *-commutative 52.3%

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

      2. associate-*l* 52.3%

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

      3. *-commutative 52.3%

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

      4. associate-*r* 52.3%

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

      5. associate-*r* 52.3%

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

      6. *-commutative 52.3%

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

      7. associate-*r* 52.3%

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

   10. Simplified 52.3%

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

       1. *-commutative 52.3%

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

       2. associate-*r* 52.3%

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

       3. add-exp-log 52.3%

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

       4. sum-log 52.3%

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

       5. remove-double-neg 52.3%

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

       6. unsub-neg 52.3%

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

       7. exp-diff 52.3%

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

       8. add-exp-log 52.3%

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

       9. add-sqr-sqrt 5.8%

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

       10. sqrt-unprod 63.6%

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

       11. sqr-neg 63.6%

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

       12. sqrt-unprod 57.9%

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

       13. add-sqr-sqrt 63.6%

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

       14. remove-double-neg 63.6%

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

       15. add-exp-log 63.6%

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

   12. Applied egg-rr 63.6%

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

       1. associate-/l* 63.6%

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

   14. Simplified 63.6%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 2.9 \cdot 10^{-13}:\\ \;\;\;\;{b}^{2} + 0.005555555555555556 \cdot \left(a \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + 0.005555555555555556 \cdot \left(a \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a \cdot \left(0.005555555555555556 \cdot \frac{\pi}{angle}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 2.4 \cdot 10^{+14}:\\ \;\;\;\;{b}^{2} + \left(a \cdot 0.005555555555555556\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + 0.005555555555555556 \cdot \left(a \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a \cdot \left(0.005555555555555556 \cdot \frac{\pi}{angle}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 2.4e+14)
   (+
    (pow b 2.0)
    (*
     (* a 0.005555555555555556)
     (* (* angle PI) (* a (* PI (* 0.005555555555555556 angle))))))
   (+
    (pow b 2.0)
    (*
     0.005555555555555556
     (* a (* (* angle PI) (* a (* 0.005555555555555556 (/ PI angle)))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 2.4e+14) {
		tmp = pow(b, 2.0) + ((a * 0.005555555555555556) * ((angle * ((double) M_PI)) * (a * (((double) M_PI) * (0.005555555555555556 * angle)))));
	} else {
		tmp = pow(b, 2.0) + (0.005555555555555556 * (a * ((angle * ((double) M_PI)) * (a * (0.005555555555555556 * (((double) M_PI) / angle))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 2.4e+14) {
		tmp = Math.pow(b, 2.0) + ((a * 0.005555555555555556) * ((angle * Math.PI) * (a * (Math.PI * (0.005555555555555556 * angle)))));
	} else {
		tmp = Math.pow(b, 2.0) + (0.005555555555555556 * (a * ((angle * Math.PI) * (a * (0.005555555555555556 * (Math.PI / angle))))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 2.4e+14:
		tmp = math.pow(b, 2.0) + ((a * 0.005555555555555556) * ((angle * math.pi) * (a * (math.pi * (0.005555555555555556 * angle)))))
	else:
		tmp = math.pow(b, 2.0) + (0.005555555555555556 * (a * ((angle * math.pi) * (a * (0.005555555555555556 * (math.pi / angle))))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 2.4e+14)
		tmp = Float64((b ^ 2.0) + Float64(Float64(a * 0.005555555555555556) * Float64(Float64(angle * pi) * Float64(a * Float64(pi * Float64(0.005555555555555556 * angle))))));
	else
		tmp = Float64((b ^ 2.0) + Float64(0.005555555555555556 * Float64(a * Float64(Float64(angle * pi) * Float64(a * Float64(0.005555555555555556 * Float64(pi / angle)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 2.4e+14)
		tmp = (b ^ 2.0) + ((a * 0.005555555555555556) * ((angle * pi) * (a * (pi * (0.005555555555555556 * angle)))));
	else
		tmp = (b ^ 2.0) + (0.005555555555555556 * (a * ((angle * pi) * (a * (0.005555555555555556 * (pi / angle))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 2.4e+14], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(a * 0.005555555555555556), $MachinePrecision] * N[(N[(angle * Pi), $MachinePrecision] * N[(a * N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(0.005555555555555556 * N[(a * N[(N[(angle * Pi), $MachinePrecision] * N[(a * N[(0.005555555555555556 * N[(Pi / angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 2.4e14

   1. Initial program 92.0%

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

      1. associate-*l/ 92.0%

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

      2. associate-/l* 92.0%

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

      3. associate-*l/ 92.0%

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

      4. associate-/l* 92.1%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in angle around 0 92.0%

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

   6. Taylor expanded in angle around 0 87.7%

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

      1. unpow2 87.7%

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

      2. associate-*r* 87.7%

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

      3. associate-*l* 87.7%

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

      4. *-commutative 87.7%

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

      5. associate-*l* 87.7%

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

      6. *-commutative 87.7%

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

      7. associate-*l* 87.7%

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

      8. *-commutative 87.7%

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

   8. Applied egg-rr 87.7%

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

   if 2.4e14 < angle

   1. Initial program 63.2%

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

      1. associate-*l/ 63.1%

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

      2. associate-/l* 63.0%

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

      3. associate-*l/ 62.7%

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

      4. associate-/l* 62.7%

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

   3. Simplified 62.7%

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

   5. Taylor expanded in angle around 0 63.0%

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

   6. Taylor expanded in angle around 0 49.0%

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

      1. unpow2 49.0%

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

      2. associate-*l* 49.0%

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

      3. *-commutative 49.0%

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

      4. associate-*l* 49.0%

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

      5. *-commutative 49.0%

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

      6. associate-*l* 49.0%

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

      7. *-commutative 49.0%

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

      8. associate-*l* 49.0%

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

      9. *-commutative 49.0%

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

   8. Applied egg-rr 49.0%

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

      1. *-commutative 49.0%

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

      2. associate-*l* 49.0%

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

      3. *-commutative 49.0%

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

      4. associate-*r* 49.0%

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

      5. associate-*r* 49.0%

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

      6. *-commutative 49.0%

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

      7. associate-*r* 49.0%

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

   10. Simplified 49.0%

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

       1. *-commutative 49.0%

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

       2. associate-*r* 49.0%

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

       3. add-exp-log 49.0%

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

       4. sum-log 49.0%

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

       5. remove-double-neg 49.0%

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

       6. unsub-neg 49.0%

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

       7. exp-diff 49.0%

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

       8. add-exp-log 49.0%

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

       9. add-sqr-sqrt 0.0%

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

       10. sqrt-unprod 61.5%

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

       11. sqr-neg 61.5%

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

       12. sqrt-unprod 61.5%

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

       13. add-sqr-sqrt 61.5%

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

       14. remove-double-neg 61.5%

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

       15. add-exp-log 61.5%

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

   12. Applied egg-rr 61.5%

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

       1. associate-/l* 61.5%

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

   14. Simplified 61.5%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 2.4 \cdot 10^{+14}:\\ \;\;\;\;{b}^{2} + \left(a \cdot 0.005555555555555556\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + 0.005555555555555556 \cdot \left(a \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a \cdot \left(0.005555555555555556 \cdot \frac{\pi}{angle}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.4% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. associate-*l/ 86.7%

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

   2. associate-/l* 86.7%

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

   3. associate-*l/ 86.6%

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

   4. associate-/l* 86.7%

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

3. Simplified 86.7%

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

5. Taylor expanded in angle around 0 86.7%

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

6. Taylor expanded in angle around 0 80.6%

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

   1. unpow2 80.6%

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

   2. associate-*l* 80.6%

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

   3. *-commutative 80.6%

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

   4. associate-*l* 80.6%

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

   5. *-commutative 80.6%

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

   6. associate-*l* 80.6%

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

   7. *-commutative 80.6%

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

   8. associate-*l* 80.6%

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

   9. *-commutative 80.6%

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

8. Applied egg-rr 80.6%

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

   1. *-commutative 80.6%

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

   2. associate-*l* 80.6%

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

   3. *-commutative 80.6%

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

   4. associate-*r* 80.6%

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

   5. associate-*r* 80.6%

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

   6. *-commutative 80.6%

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

   7. associate-*r* 80.6%

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

10. Simplified 80.6%

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

11. Final simplification 80.6%

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

Reproduce

herbie shell --seed 2024052 
(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)))