ab-angle->ABCF A

Percentage Accurate: 78.8% → 78.9%

Time: 48.5s

Alternatives: 10

Speedup: 1.3×

• 35.5% 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: 78.8% 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: 78.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(b \cdot \left(e^{\mathsf{log1p}\left(\cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)} + -1\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* (* 0.005555555555555556 angle) PI))) 2.0)
  (pow (* b (+ (exp (log1p (cos (/ angle (/ 180.0 PI))))) -1.0)) 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 * (exp(log1p(cos((angle / (180.0 / ((double) M_PI)))))) + -1.0)), 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.exp(Math.log1p(Math.cos((angle / (180.0 / Math.PI))))) + -1.0)), 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.exp(math.log1p(math.cos((angle / (180.0 / math.pi))))) + -1.0)), 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(Float64(0.005555555555555556 * angle) * pi))) ^ 2.0) + (Float64(b * Float64(exp(log1p(cos(Float64(angle / Float64(180.0 / pi))))) + -1.0)) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[(N[Exp[N[Log[1 + N[Cos[N[(angle / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. add-sqr-sqrt 38.3%

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

   2. pow2 38.3%

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

   3. associate-*l/ 38.3%

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

   4. associate-*r/ 38.3%

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

   5. div-inv 38.3%

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

   6. metadata-eval 38.3%

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

4. Applied egg-rr 38.3%

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

   1. unpow2 38.3%

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

   2. add-sqr-sqrt 83.6%

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

   3. associate-*r* 83.6%

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

   4. *-commutative 83.6%

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

   5. associate-*r* 83.6%

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

6. Applied egg-rr 83.6%

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

   1. expm1-log1p-u 83.6%

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

   2. expm1-undefine 83.6%

      \[\leadsto {\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\frac{angle}{180} \cdot \pi\right)\right)} - 1\right)}\right)}^{2} \]

   3. associate-*l/ 83.6%

      \[\leadsto {\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(b \cdot \left(e^{\mathsf{log1p}\left(\cos \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)}\right)} - 1\right)\right)}^{2} \]

   4. associate-*r/ 83.6%

      \[\leadsto {\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(b \cdot \left(e^{\mathsf{log1p}\left(\cos \color{blue}{\left(angle \cdot \frac{\pi}{180}\right)}\right)} - 1\right)\right)}^{2} \]

   5. clear-num 83.6%

      \[\leadsto {\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(b \cdot \left(e^{\mathsf{log1p}\left(\cos \left(angle \cdot \color{blue}{\frac{1}{\frac{180}{\pi}}}\right)\right)} - 1\right)\right)}^{2} \]

   6. un-div-inv 83.7%

      \[\leadsto {\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(b \cdot \left(e^{\mathsf{log1p}\left(\cos \color{blue}{\left(\frac{angle}{\frac{180}{\pi}}\right)}\right)} - 1\right)\right)}^{2} \]

8. Applied egg-rr 83.7%

   \[\leadsto {\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)} - 1\right)}\right)}^{2} \]

9. Final simplification 83.7%

   \[\leadsto {\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(b \cdot \left(e^{\mathsf{log1p}\left(\cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)} + -1\right)\right)}^{2} \]
10. Add Preprocessing

Alternative 2: 78.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sqrt[3]{{\cos \left(\frac{angle}{\frac{180}{\pi}}\right)}^{3}}\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* (* 0.005555555555555556 angle) PI))) 2.0)
  (pow (* b (cbrt (pow (cos (/ angle (/ 180.0 PI))) 3.0))) 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 * cbrt(pow(cos((angle / (180.0 / ((double) M_PI)))), 3.0))), 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.cbrt(Math.pow(Math.cos((angle / (180.0 / Math.PI))), 3.0))), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(Float64(0.005555555555555556 * angle) * pi))) ^ 2.0) + (Float64(b * cbrt((cos(Float64(angle / Float64(180.0 / pi))) ^ 3.0))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Power[N[Power[N[Cos[N[(angle / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. add-sqr-sqrt 38.3%

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

   2. pow2 38.3%

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

   3. associate-*l/ 38.3%

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

   4. associate-*r/ 38.3%

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

   5. div-inv 38.3%

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

   6. metadata-eval 38.3%

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

4. Applied egg-rr 38.3%

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

   1. unpow2 38.3%

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

   2. add-sqr-sqrt 83.6%

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

   3. associate-*r* 83.6%

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

   4. *-commutative 83.6%

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

   5. associate-*r* 83.6%

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

6. Applied egg-rr 83.6%

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

   1. add-cbrt-cube 83.6%

      \[\leadsto {\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{\sqrt[3]{\left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}}\right)}^{2} \]

   2. pow3 83.6%

      \[\leadsto {\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sqrt[3]{\color{blue}{{\cos \left(\frac{angle}{180} \cdot \pi\right)}^{3}}}\right)}^{2} \]

   3. associate-*l/ 83.6%

      \[\leadsto {\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sqrt[3]{{\cos \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)}}^{3}}\right)}^{2} \]

   4. associate-*r/ 83.6%

      \[\leadsto {\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sqrt[3]{{\cos \color{blue}{\left(angle \cdot \frac{\pi}{180}\right)}}^{3}}\right)}^{2} \]

   5. rem-cbrt-cube 83.6%

      \[\leadsto {\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sqrt[3]{{\cos \left(angle \cdot \frac{\color{blue}{\sqrt[3]{{\pi}^{3}}}}{180}\right)}^{3}}\right)}^{2} \]

   6. rem-cbrt-cube 83.6%

      \[\leadsto {\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sqrt[3]{{\cos \left(angle \cdot \frac{\color{blue}{\pi}}{180}\right)}^{3}}\right)}^{2} \]

   7. clear-num 83.6%

      \[\leadsto {\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sqrt[3]{{\cos \left(angle \cdot \color{blue}{\frac{1}{\frac{180}{\pi}}}\right)}^{3}}\right)}^{2} \]

   8. un-div-inv 83.7%

      \[\leadsto {\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sqrt[3]{{\cos \color{blue}{\left(\frac{angle}{\frac{180}{\pi}}\right)}}^{3}}\right)}^{2} \]

8. Applied egg-rr 83.7%

   \[\leadsto {\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{\sqrt[3]{{\cos \left(\frac{angle}{\frac{180}{\pi}}\right)}^{3}}}\right)}^{2} \]

9. Final simplification 83.7%

   \[\leadsto {\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sqrt[3]{{\cos \left(\frac{angle}{\frac{180}{\pi}}\right)}^{3}}\right)}^{2} \]
10. Add Preprocessing

Alternative 3: 78.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* (* 0.005555555555555556 angle) PI))) 2.0)
  (pow (* b (log (exp (cos (/ angle (/ 180.0 PI)))))) 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 * log(exp(cos((angle / (180.0 / ((double) M_PI))))))), 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.log(Math.exp(Math.cos((angle / (180.0 / Math.PI)))))), 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.log(math.exp(math.cos((angle / (180.0 / math.pi)))))), 2.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. add-sqr-sqrt 38.3%

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

   2. pow2 38.3%

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

   3. associate-*l/ 38.3%

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

   4. associate-*r/ 38.3%

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

   5. div-inv 38.3%

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

   6. metadata-eval 38.3%

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

4. Applied egg-rr 38.3%

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

   1. unpow2 38.3%

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

   2. add-sqr-sqrt 83.6%

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

   3. associate-*r* 83.6%

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

   4. *-commutative 83.6%

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

   5. associate-*r* 83.6%

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

6. Applied egg-rr 83.6%

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

   1. add-log-exp 83.6%

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

   2. associate-*l/ 83.6%

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

   3. associate-*r/ 83.6%

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

   4. rem-cbrt-cube 83.6%

      \[\leadsto {\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(b \cdot \log \left(e^{\cos \left(angle \cdot \frac{\color{blue}{\sqrt[3]{{\pi}^{3}}}}{180}\right)}\right)\right)}^{2} \]

   5. rem-cbrt-cube 83.6%

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

   6. clear-num 83.6%

      \[\leadsto {\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(b \cdot \log \left(e^{\cos \left(angle \cdot \color{blue}{\frac{1}{\frac{180}{\pi}}}\right)}\right)\right)}^{2} \]

   7. un-div-inv 83.7%

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

8. Applied egg-rr 83.7%

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

9. Final simplification 83.7%

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

Alternative 4: 78.8% 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 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. associate-*l/ 83.5%

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

   2. associate-/l* 83.6%

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

   3. cos-neg 83.6%

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

   4. distribute-lft-neg-out 83.6%

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

   5. distribute-frac-neg 83.6%

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

   6. distribute-frac-neg 83.6%

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

   7. distribute-lft-neg-out 83.6%

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

   8. cos-neg 83.6%

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

   9. associate-*l/ 83.6%

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

   10. associate-/l* 83.6%

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

3. Simplified 83.6%

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

5. Taylor expanded in angle around inf 83.6%

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

6. Final simplification 83.6%

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

Alternative 5: 78.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* (* 0.005555555555555556 angle) PI))) 2.0)
  (pow (* b (cos (* PI (/ angle 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((((double) M_PI) * (angle / 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((Math.PI * (angle / 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((math.pi * (angle / 180.0)))), 2.0)

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $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 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. Step-by-step derivation

   1. add-sqr-sqrt 38.3%

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

   2. pow2 38.3%

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

   3. associate-*l/ 38.3%

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

   4. associate-*r/ 38.3%

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

   5. div-inv 38.3%

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

   6. metadata-eval 38.3%

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

4. Applied egg-rr 38.3%

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

   1. unpow2 38.3%

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

   2. add-sqr-sqrt 83.6%

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

   3. associate-*r* 83.6%

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

   4. *-commutative 83.6%

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

   5. associate-*r* 83.6%

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

6. Applied egg-rr 83.6%

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

7. Final simplification 83.6%

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

Alternative 6: 78.8% 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 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. associate-*l/ 83.5%

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

   2. associate-/l* 83.6%

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

   3. cos-neg 83.6%

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

   4. distribute-lft-neg-out 83.6%

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

   5. distribute-frac-neg 83.6%

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

   6. distribute-frac-neg 83.6%

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

   7. distribute-lft-neg-out 83.6%

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

   8. cos-neg 83.6%

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

   9. associate-*l/ 83.6%

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

   10. associate-/l* 83.6%

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

3. Simplified 83.6%

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

5. Taylor expanded in angle around inf 83.6%

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

6. Taylor expanded in angle around 0 83.3%

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

7. Final simplification 83.3%

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

Alternative 7: 78.9% accurate, 1.3× speedup

Math

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

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(Float64(0.005555555555555556 * 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[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. add-sqr-sqrt 38.3%

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

   2. pow2 38.3%

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

   3. associate-*l/ 38.3%

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

   4. associate-*r/ 38.3%

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

   5. div-inv 38.3%

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

   6. metadata-eval 38.3%

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

4. Applied egg-rr 38.3%

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

   1. unpow2 38.3%

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

   2. add-sqr-sqrt 83.6%

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

   3. associate-*r* 83.6%

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

   4. *-commutative 83.6%

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

   5. associate-*r* 83.6%

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

6. Applied egg-rr 83.6%

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

7. Taylor expanded in angle around 0 83.4%

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

8. Final simplification 83.4%

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

Alternative 8: 73.7% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. associate-*l/ 83.5%

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

   2. associate-/l* 83.6%

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

   3. cos-neg 83.6%

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

   4. distribute-lft-neg-out 83.6%

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

   5. distribute-frac-neg 83.6%

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

   6. distribute-frac-neg 83.6%

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

   7. distribute-lft-neg-out 83.6%

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

   8. cos-neg 83.6%

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

   9. associate-*l/ 83.6%

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

   10. associate-/l* 83.6%

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

3. Simplified 83.6%

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

5. Taylor expanded in angle around 0 83.3%

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

6. Taylor expanded in angle around 0 78.7%

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

   1. *-commutative 78.7%

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

   2. *-commutative 78.7%

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

   3. associate-*l* 78.7%

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

8. Simplified 78.7%

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

   1. unpow2 78.7%

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

   2. associate-*r* 78.7%

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

   3. *-commutative 78.7%

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

   4. associate-*l* 78.8%

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

   5. *-commutative 78.8%

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

   6. associate-*r* 78.8%

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

   7. *-commutative 78.8%

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

   8. associate-*l* 78.8%

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

10. Applied egg-rr 78.8%

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

11. Final simplification 78.8%

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

Alternative 9: 72.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ {b}^{2} + a \cdot \left(angle \cdot \left(\pi \cdot \left(\left(angle \cdot \left(a \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow b 2.0)
  (* a (* angle (* PI (* (* angle (* a PI)) 3.08641975308642e-5))))))

C

double code(double a, double b, double angle) {
	return pow(b, 2.0) + (a * (angle * (((double) M_PI) * ((angle * (a * ((double) M_PI))) * 3.08641975308642e-5))));
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow(b, 2.0) + (a * (angle * (Math.PI * ((angle * (a * Math.PI)) * 3.08641975308642e-5))));
}

Python

def code(a, b, angle):
	return math.pow(b, 2.0) + (a * (angle * (math.pi * ((angle * (a * math.pi)) * 3.08641975308642e-5))))

Julia

function code(a, b, angle)
	return Float64((b ^ 2.0) + Float64(a * Float64(angle * Float64(pi * Float64(Float64(angle * Float64(a * pi)) * 3.08641975308642e-5)))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (b ^ 2.0) + (a * (angle * (pi * ((angle * (a * pi)) * 3.08641975308642e-5))));
end

Wolfram

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

Derivation

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. associate-*l/ 83.5%

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

   2. associate-/l* 83.6%

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

   3. cos-neg 83.6%

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

   4. distribute-lft-neg-out 83.6%

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

   5. distribute-frac-neg 83.6%

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

   6. distribute-frac-neg 83.6%

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

   7. distribute-lft-neg-out 83.6%

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

   8. cos-neg 83.6%

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

   9. associate-*l/ 83.6%

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

   10. associate-/l* 83.6%

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

3. Simplified 83.6%

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

5. Taylor expanded in angle around 0 83.3%

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

6. Taylor expanded in angle around 0 78.7%

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

   1. *-commutative 78.7%

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

   2. *-commutative 78.7%

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

   3. associate-*l* 78.7%

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

8. Simplified 78.7%

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

   1. unpow2 78.7%

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

   2. associate-*r* 78.7%

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

   3. *-commutative 78.7%

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

   4. associate-*l* 78.8%

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

   5. *-commutative 78.8%

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

   6. associate-*r* 78.8%

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

   7. *-commutative 78.8%

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

   8. associate-*l* 78.8%

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

10. Applied egg-rr 78.8%

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

    1. associate-*r* 78.7%

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

    2. *-commutative 78.7%

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

    3. associate-*l* 78.8%

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

    4. associate-*r* 77.4%

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

    5. *-commutative 77.4%

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

    6. *-commutative 77.4%

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

    7. associate-*r* 77.4%

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

    8. *-commutative 77.4%

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

    9. *-commutative 77.4%

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

    10. associate-*r* 77.4%

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

    11. metadata-eval 77.4%

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

12. Simplified 77.4%

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

13. Final simplification 77.4%

    \[\leadsto {b}^{2} + a \cdot \left(angle \cdot \left(\pi \cdot \left(\left(angle \cdot \left(a \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right) \]
14. Add Preprocessing

Alternative 10: 72.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ {b}^{2} + a \cdot \left(angle \cdot \left(\pi \cdot \left(\pi \cdot \left(a \cdot \left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow b 2.0)
  (* a (* angle (* PI (* PI (* a (* angle 3.08641975308642e-5))))))))

C

double code(double a, double b, double angle) {
	return pow(b, 2.0) + (a * (angle * (((double) M_PI) * (((double) M_PI) * (a * (angle * 3.08641975308642e-5))))));
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow(b, 2.0) + (a * (angle * (Math.PI * (Math.PI * (a * (angle * 3.08641975308642e-5))))));
}

Python

def code(a, b, angle):
	return math.pow(b, 2.0) + (a * (angle * (math.pi * (math.pi * (a * (angle * 3.08641975308642e-5))))))

Julia

function code(a, b, angle)
	return Float64((b ^ 2.0) + Float64(a * Float64(angle * Float64(pi * Float64(pi * Float64(a * Float64(angle * 3.08641975308642e-5)))))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (b ^ 2.0) + (a * (angle * (pi * (pi * (a * (angle * 3.08641975308642e-5))))));
end

Wolfram

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

Derivation

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. associate-*l/ 83.5%

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

   2. associate-/l* 83.6%

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

   3. cos-neg 83.6%

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

   4. distribute-lft-neg-out 83.6%

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

   5. distribute-frac-neg 83.6%

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

   6. distribute-frac-neg 83.6%

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

   7. distribute-lft-neg-out 83.6%

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

   8. cos-neg 83.6%

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

   9. associate-*l/ 83.6%

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

   10. associate-/l* 83.6%

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

3. Simplified 83.6%

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

5. Taylor expanded in angle around 0 83.3%

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

6. Taylor expanded in angle around 0 78.7%

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

   1. *-commutative 78.7%

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

   2. *-commutative 78.7%

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

   3. associate-*l* 78.7%

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

8. Simplified 78.7%

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

   1. unpow2 78.7%

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

   2. associate-*r* 78.7%

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

   3. *-commutative 78.7%

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

   4. associate-*l* 78.8%

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

   5. *-commutative 78.8%

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

   6. associate-*r* 78.8%

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

   7. *-commutative 78.8%

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

   8. associate-*l* 78.8%

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

10. Applied egg-rr 78.8%

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

    1. associate-*r* 78.7%

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

    2. *-commutative 78.7%

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

    3. associate-*l* 78.8%

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

    4. associate-*r* 77.4%

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

    5. *-commutative 77.4%

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

    6. *-commutative 77.4%

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

    7. associate-*r* 77.4%

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

    8. *-commutative 77.4%

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

    9. *-commutative 77.4%

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

    10. associate-*r* 77.4%

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

    11. metadata-eval 77.4%

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

12. Simplified 77.4%

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

13. Taylor expanded in angle around 0 77.4%

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

    1. *-commutative 77.4%

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

    2. *-commutative 77.4%

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

    3. associate-*r* 77.4%

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

    4. *-commutative 77.4%

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

    5. associate-*r* 77.4%

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

    6. associate-*l* 77.4%

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

15. Simplified 77.4%

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

16. Final simplification 77.4%

    \[\leadsto {b}^{2} + a \cdot \left(angle \cdot \left(\pi \cdot \left(\pi \cdot \left(a \cdot \left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\right) \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024082 
(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)))