ab-angle->ABCF A

Percentage Accurate: 79.5% → 79.4%

Time: 24.5s

Alternatives: 5

Speedup: N/A×

• 36.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.4% 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 75.3%

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

   1. associate-*l/ 75.2%

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

   2. associate-*r/ 75.3%

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

   3. associate-*l/ 75.2%

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

   4. associate-*r/ 75.2%

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

3. Simplified 75.2%

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

   1. pow1 75.2%

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

   2. metadata-eval 75.2%

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

   3. sqrt-pow1 75.2%

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

   4. expm1-log1p-u 74.9%

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

   5. sqrt-pow1 63.0%

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

   6. metadata-eval 63.0%

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

   7. pow1 63.0%

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

   8. *-commutative 63.0%

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

   9. associate-*r/ 63.8%

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

   10. associate-*r/ 63.0%

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

   11. div-inv 63.0%

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

   12. metadata-eval 63.0%

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

5. Applied egg-rr 63.0%

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

6. Taylor expanded in angle around 0 63.1%

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

7. Taylor expanded in angle around inf 75.3%

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

8. Final simplification 75.3%

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

Alternative 2: 74.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

   1. associate-*l/ 75.2%

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

   2. associate-*r/ 75.3%

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

   3. associate-*l/ 75.2%

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

   4. associate-*r/ 75.2%

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

3. Simplified 75.2%

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

4. Taylor expanded in angle around 0 75.2%

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

5. Taylor expanded in angle around 0 69.8%

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

   1. *-commutative 69.8%

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

7. Simplified 69.8%

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

   1. unpow2 69.8%

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

   2. *-commutative 69.8%

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

   3. associate-*r* 69.8%

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

   4. associate-*r* 69.8%

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

   5. *-commutative 69.8%

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

9. Applied egg-rr 69.8%

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

10. Final simplification 69.8%

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

Alternative 3: 74.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

   1. associate-*l/ 75.2%

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

   2. associate-*r/ 75.3%

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

   3. associate-*l/ 75.2%

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

   4. associate-*r/ 75.2%

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

3. Simplified 75.2%

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

   1. pow1 75.2%

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

   2. metadata-eval 75.2%

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

   3. sqrt-pow1 75.2%

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

   4. expm1-log1p-u 74.9%

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

   5. sqrt-pow1 63.0%

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

   6. metadata-eval 63.0%

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

   7. pow1 63.0%

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

   8. *-commutative 63.0%

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

   9. associate-*r/ 63.8%

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

   10. associate-*r/ 63.0%

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

   11. div-inv 63.0%

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

   12. metadata-eval 63.0%

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

5. Applied egg-rr 63.0%

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

6. Taylor expanded in angle around 0 63.1%

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

7. Taylor expanded in angle around inf 75.3%

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

8. Taylor expanded in angle around 0 61.0%

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

   1. unpow2 61.0%

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

   2. unpow2 61.0%

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

   3. unpow2 61.0%

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

   4. swap-sqr 61.1%

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

   5. swap-sqr 69.8%

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

   6. unpow2 69.8%

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

   7. associate-*r* 69.8%

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

   8. *-commutative 69.8%

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

   9. *-commutative 69.8%

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

10. Simplified 69.8%

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

11. Final simplification 69.8%

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

Alternative 4: 74.5% 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 75.3%

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

   1. associate-*l/ 75.2%

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

   2. associate-*r/ 75.3%

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

   3. associate-*l/ 75.2%

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

   4. associate-*r/ 75.2%

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

3. Simplified 75.2%

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

   1. pow1 75.2%

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

   2. metadata-eval 75.2%

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

   3. sqrt-pow1 75.2%

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

   4. expm1-log1p-u 74.9%

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

   5. sqrt-pow1 63.0%

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

   6. metadata-eval 63.0%

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

   7. pow1 63.0%

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

   8. *-commutative 63.0%

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

   9. associate-*r/ 63.8%

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

   10. associate-*r/ 63.0%

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

   11. div-inv 63.0%

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

   12. metadata-eval 63.0%

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

5. Applied egg-rr 63.0%

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

6. Taylor expanded in angle around 0 63.1%

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

7. Taylor expanded in angle around inf 75.3%

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

8. Taylor expanded in angle around 0 69.8%

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

9. Final simplification 69.8%

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

Alternative 5: 74.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

   1. associate-*l/ 75.2%

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

   2. associate-*r/ 75.3%

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

   3. associate-*l/ 75.2%

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

   4. associate-*r/ 75.2%

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

3. Simplified 75.2%

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

4. Taylor expanded in angle around 0 75.2%

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

5. Taylor expanded in angle around 0 69.8%

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

   1. *-commutative 69.8%

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

7. Simplified 69.8%

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

   1. *-commutative 69.8%

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

   2. unpow-prod-down 69.8%

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

   3. metadata-eval 69.8%

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

9. Applied egg-rr 69.8%

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

10. Final simplification 69.8%

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

Reproduce

herbie shell --seed 2023178 
(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)))