ab-angle->ABCF A

Percentage Accurate: 80.4% → 80.4%

Time: 52.6s

Alternatives: 11

Speedup: 1.0×

• 37.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.4% 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: 80.4% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* (/ angle 180.0) PI))) 2.0)
  (pow (* b (cos (pow (/ 1.0 (cbrt (/ (/ 180.0 PI) angle))) 3.0))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * sin(((angle / 180.0) * ((double) M_PI)))), 2.0) + pow((b * cos(pow((1.0 / cbrt(((180.0 / ((double) M_PI)) / angle))), 3.0))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin(((angle / 180.0) * Math.PI))), 2.0) + Math.pow((b * Math.cos(Math.pow((1.0 / Math.cbrt(((180.0 / Math.PI) / angle))), 3.0))), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(b * cos((Float64(1.0 / cbrt(Float64(Float64(180.0 / pi) / angle))) ^ 3.0))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[Power[N[(1.0 / N[Power[N[(N[(180.0 / Pi), $MachinePrecision] / angle), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.7%

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

   1. associate-*l/ 81.2%

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

   2. associate-*r/ 81.7%

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

   3. add-cube-cbrt 81.7%

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

   4. pow3 81.7%

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

   5. associate-*r/ 81.3%

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

   6. associate-/l* 81.7%

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

   7. div-inv 81.7%

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

   8. clear-num 81.7%

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

   9. div-inv 81.7%

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

   10. metadata-eval 81.7%

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

3. Applied egg-rr 81.7%

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

   1. associate-*r* 81.3%

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

   2. metadata-eval 81.3%

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

   3. div-inv 81.3%

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

   4. clear-num 81.3%

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

   5. cbrt-div 81.2%

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

   6. metadata-eval 81.2%

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

5. Applied egg-rr 81.2%

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

   1. associate-/l/ 81.7%

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

7. Simplified 81.7%

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

8. Final simplification 81.7%

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

Alternative 2: 66.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. associate-*l/ 81.2%

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

   2. expm1-log1p-u 63.9%

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

   3. associate-*r/ 64.0%

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

   4. div-inv 64.0%

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

   5. metadata-eval 64.0%

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

3. Applied egg-rr 64.0%

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

4. Final simplification 64.0%

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

Alternative 3: 80.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. associate-*l/ 81.2%

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

   2. associate-*r/ 81.7%

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

   3. associate-*l/ 81.2%

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

   4. associate-*r/ 81.7%

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

3. Simplified 81.7%

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

4. Taylor expanded in angle around inf 81.2%

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

5. Final simplification 81.2%

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

Alternative 4: 80.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (/ PI 180.0))))
   (+ (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 * (((double) M_PI) / 180.0);
	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 * (Math.PI / 180.0);
	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 * (math.pi / 180.0)
	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(angle * Float64(pi / 180.0))
	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 * (pi / 180.0);
	tmp = ((a * sin(t_0)) ^ 2.0) + ((b * cos(t_0)) ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. associate-*l/ 81.2%

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

   2. associate-*r/ 81.7%

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

   3. associate-*l/ 81.2%

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

   4. associate-*r/ 81.7%

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

3. Simplified 81.7%

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

4. Final simplification 81.7%

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

Alternative 5: 80.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. associate-*l/ 81.2%

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

   2. associate-*r/ 81.7%

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

   3. add-cube-cbrt 81.7%

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

   4. pow3 81.7%

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

   5. associate-*r/ 81.3%

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

   6. associate-/l* 81.7%

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

   7. div-inv 81.7%

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

   8. clear-num 81.7%

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

   9. div-inv 81.7%

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

   10. metadata-eval 81.7%

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

3. Applied egg-rr 81.7%

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

   1. rem-cube-cbrt 81.7%

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

   2. *-commutative 81.7%

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

   3. associate-*r* 81.7%

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

5. Applied egg-rr 81.7%

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

6. Final simplification 81.7%

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

Alternative 6: 65.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. associate-*l/ 81.2%

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

   2. expm1-log1p-u 63.9%

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

   3. associate-*r/ 64.0%

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

   4. div-inv 64.0%

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

   5. metadata-eval 64.0%

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

3. Applied egg-rr 64.0%

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

4. Taylor expanded in angle around 0 63.3%

   \[\leadsto {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]

5. Final simplification 63.3%

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

Alternative 7: 80.3% accurate, 1.5× 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 81.7%

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

   1. associate-*l/ 81.2%

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

   2. associate-*r/ 81.7%

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

   3. associate-*l/ 81.2%

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

   4. associate-*r/ 81.7%

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

3. Simplified 81.7%

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

4. Taylor expanded in angle around 0 81.0%

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

5. Taylor expanded in angle around inf 80.6%

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

6. Final simplification 80.6%

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

Alternative 8: 80.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. associate-*l/ 81.2%

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

   2. associate-*r/ 81.7%

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

   3. associate-*l/ 81.2%

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

   4. associate-*r/ 81.7%

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

3. Simplified 81.7%

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

4. Taylor expanded in angle around 0 81.0%

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

5. Final simplification 81.0%

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

Alternative 9: 75.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. associate-*l/ 81.2%

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

   2. associate-*r/ 81.7%

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

   3. associate-*l/ 81.2%

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

   4. associate-*r/ 81.7%

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

3. Simplified 81.7%

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

4. Taylor expanded in angle around 0 81.0%

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

5. Taylor expanded in angle around 0 77.7%

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

   1. *-commutative 77.7%

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

   2. associate-*r* 77.6%

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

   3. associate-*l* 77.7%

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

   4. *-commutative 77.7%

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

7. Simplified 77.7%

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

   1. unpow2 77.7%

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

   2. associate-*l* 77.7%

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

   3. associate-*l* 74.8%

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

   4. *-commutative 74.8%

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

   5. associate-*r* 74.8%

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

   6. *-commutative 74.8%

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

   7. associate-*l* 74.9%

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

9. Applied egg-rr 74.9%

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

    1. associate-*l* 74.8%

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

    2. associate-*l* 77.7%

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

    3. *-commutative 77.7%

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

    4. associate-*r* 77.7%

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

    5. *-commutative 77.7%

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

    6. *-commutative 77.7%

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

    7. associate-*r* 77.7%

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

    8. unpow2 77.7%

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

11. Simplified 77.7%

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

12. Final simplification 77.7%

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

Alternative 10: 75.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. associate-*l/ 81.2%

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

   2. associate-*r/ 81.7%

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

   3. associate-*l/ 81.2%

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

   4. associate-*r/ 81.7%

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

3. Simplified 81.7%

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

4. Taylor expanded in angle around 0 81.0%

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

5. Taylor expanded in angle around 0 77.7%

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

6. Final simplification 77.7%

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

Alternative 11: 75.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. associate-*l/ 81.2%

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

   2. associate-*r/ 81.7%

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

   3. associate-*l/ 81.2%

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

   4. associate-*r/ 81.7%

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

3. Simplified 81.7%

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

4. Taylor expanded in angle around 0 81.0%

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

5. Taylor expanded in angle around 0 77.7%

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

   1. *-commutative 77.7%

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

   2. associate-*r* 77.6%

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

   3. associate-*l* 77.7%

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

   4. *-commutative 77.7%

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

7. Simplified 77.7%

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

8. Final simplification 77.7%

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

Reproduce

herbie shell --seed 2023224 
(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)))