ab-angle->ABCF C

Percentage Accurate: 80.4% → 80.5%

Time: 26.2s

Alternatives: 9

Speedup: 1.0×

• 37.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

   1. add-sqr-sqrt 40.5%

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

   2. sqrt-unprod 58.7%

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

   3. associate-*r/ 58.7%

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

   4. associate-*r/ 58.7%

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

   5. frac-times 58.3%

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

   6. *-commutative 58.3%

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

   7. *-commutative 58.3%

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

   8. metadata-eval 58.3%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\sqrt{\frac{\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)}{\color{blue}{32400}}}\right)\right)}^{2} \]

   9. metadata-eval 58.3%

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

   10. frac-times 58.7%

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

   11. associate-*r/ 58.7%

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

   12. associate-*r/ 58.8%

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

   13. sqrt-unprod 41.3%

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

   14. add-sqr-sqrt 81.9%

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

   15. clear-num 81.9%

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

   16. un-div-inv 81.9%

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

4. Applied egg-rr 81.9%

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

5. Final simplification 81.9%

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

Alternative 2: 80.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

3. Simplified 81.8%

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

   1. associate-/r/ 81.8%

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

   2. *-commutative 81.8%

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

   3. add-cube-cbrt 81.7%

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

   4. associate-*l* 81.6%

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

   5. pow2 81.6%

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

   6. div-inv 81.6%

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

   7. metadata-eval 81.6%

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

6. Applied egg-rr 81.6%

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

7. Taylor expanded in angle around inf 81.8%

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

8. Final simplification 81.8%

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

Alternative 3: 80.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

   1. add-sqr-sqrt 40.5%

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

   2. sqrt-unprod 58.7%

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

   3. associate-*r/ 58.7%

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

   4. associate-*r/ 58.7%

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

   5. frac-times 58.3%

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

   6. *-commutative 58.3%

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

   7. *-commutative 58.3%

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

   8. metadata-eval 58.3%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\sqrt{\frac{\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)}{\color{blue}{32400}}}\right)\right)}^{2} \]

   9. metadata-eval 58.3%

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

   10. frac-times 58.7%

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

   11. associate-*r/ 58.7%

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

   12. associate-*r/ 58.8%

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

   13. sqrt-unprod 41.3%

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

   14. add-sqr-sqrt 81.9%

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

   15. clear-num 81.9%

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

   16. un-div-inv 81.9%

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

4. Applied egg-rr 81.9%

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

5. Taylor expanded in angle around inf 81.9%

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

6. Final simplification 81.9%

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

Alternative 4: 80.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

3. Simplified 81.8%

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

5. Taylor expanded in angle around 0 81.7%

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

6. Taylor expanded in angle around inf 81.7%

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

7. Final simplification 81.7%

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

Alternative 5: 80.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

3. Simplified 81.8%

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

5. Taylor expanded in angle around 0 81.7%

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

6. Final simplification 81.7%

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

Alternative 6: 80.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

   1. add-sqr-sqrt 40.5%

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

   2. sqrt-unprod 58.7%

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

   3. associate-*r/ 58.7%

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

   4. associate-*r/ 58.7%

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

   5. frac-times 58.3%

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

   6. *-commutative 58.3%

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

   7. *-commutative 58.3%

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

   8. metadata-eval 58.3%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\sqrt{\frac{\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)}{\color{blue}{32400}}}\right)\right)}^{2} \]

   9. metadata-eval 58.3%

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

   10. frac-times 58.7%

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

   11. associate-*r/ 58.7%

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

   12. associate-*r/ 58.8%

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

   13. sqrt-unprod 41.3%

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

   14. add-sqr-sqrt 81.9%

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

   15. clear-num 81.9%

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

   16. un-div-inv 81.9%

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

4. Applied egg-rr 81.9%

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

   1. associate-*r/ 81.8%

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

   2. clear-num 81.9%

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

   3. *-commutative 81.9%

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

6. Applied egg-rr 81.9%

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

7. Taylor expanded in angle around 0 81.7%

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

8. Final simplification 81.7%

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

Alternative 7: 75.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)\\ {a}^{2} + t\_0 \cdot t\_0 \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* -0.005555555555555556 (* angle (* PI b)))))
   (+ (pow a 2.0) (* t_0 t_0))))

C

double code(double a, double b, double angle) {
	double t_0 = -0.005555555555555556 * (angle * (((double) M_PI) * b));
	return pow(a, 2.0) + (t_0 * t_0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = -0.005555555555555556 * (angle * (Math.PI * b));
	return Math.pow(a, 2.0) + (t_0 * t_0);
}

Python

def code(a, b, angle):
	t_0 = -0.005555555555555556 * (angle * (math.pi * b))
	return math.pow(a, 2.0) + (t_0 * t_0)

Julia

function code(a, b, angle)
	t_0 = Float64(-0.005555555555555556 * Float64(angle * Float64(pi * b)))
	return Float64((a ^ 2.0) + Float64(t_0 * t_0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = -0.005555555555555556 * (angle * (pi * b));
	tmp = (a ^ 2.0) + (t_0 * t_0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(-0.005555555555555556 * N[(angle * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[a, 2.0], $MachinePrecision] + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 81.8%

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

3. Simplified 81.8%

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

5. Taylor expanded in angle around 0 81.7%

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

6. Taylor expanded in angle around 0 74.5%

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

   1. unpow2 74.5%

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

   2. *-commutative 74.5%

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

   3. *-commutative 74.5%

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

8. Applied egg-rr 74.5%

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

9. Final simplification 74.5%

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

Alternative 8: 75.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(\pi \cdot b\right)\\ {a}^{2} + t\_0 \cdot \left(-0.005555555555555556 \cdot \left(-0.005555555555555556 \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* PI b))))
   (+
    (pow a 2.0)
    (* t_0 (* -0.005555555555555556 (* -0.005555555555555556 t_0))))))

C

double code(double a, double b, double angle) {
	double t_0 = angle * (((double) M_PI) * b);
	return pow(a, 2.0) + (t_0 * (-0.005555555555555556 * (-0.005555555555555556 * t_0)));
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = angle * (Math.PI * b);
	return Math.pow(a, 2.0) + (t_0 * (-0.005555555555555556 * (-0.005555555555555556 * t_0)));
}

Python

def code(a, b, angle):
	t_0 = angle * (math.pi * b)
	return math.pow(a, 2.0) + (t_0 * (-0.005555555555555556 * (-0.005555555555555556 * t_0)))

Julia

function code(a, b, angle)
	t_0 = Float64(angle * Float64(pi * b))
	return Float64((a ^ 2.0) + Float64(t_0 * Float64(-0.005555555555555556 * Float64(-0.005555555555555556 * t_0))))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = angle * (pi * b);
	tmp = (a ^ 2.0) + (t_0 * (-0.005555555555555556 * (-0.005555555555555556 * t_0)));
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(Pi * b), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[a, 2.0], $MachinePrecision] + N[(t$95$0 * N[(-0.005555555555555556 * N[(-0.005555555555555556 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 81.8%

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

3. Simplified 81.8%

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

5. Taylor expanded in angle around 0 81.7%

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

6. Taylor expanded in angle around 0 74.5%

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

   1. unpow2 74.5%

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

   2. associate-*r* 74.5%

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

   3. *-commutative 74.5%

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

   4. *-commutative 74.5%

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

8. Applied egg-rr 74.5%

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

9. Final simplification 74.5%

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

Alternative 9: 75.4% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

3. Simplified 81.8%

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

5. Taylor expanded in angle around 0 81.7%

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

6. Taylor expanded in angle around 0 74.5%

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

   1. unpow2 74.5%

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

   2. *-commutative 74.5%

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

   3. *-commutative 74.5%

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

8. Applied egg-rr 74.5%

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

   1. pow1 74.5%

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

   2. associate-*r* 74.5%

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

   3. *-commutative 74.5%

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

   4. associate-*l* 74.6%

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

10. Applied egg-rr 74.6%

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

11. Final simplification 74.6%

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

Reproduce

herbie shell --seed 2024027 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))