ab-angle->ABCF C

Percentage Accurate: 80.5% → 80.4%

Time: 12.3s

Alternatives: 8

Speedup: N/A×

• 36.3% 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: 80.5% 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: 80.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.5%

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

3. Simplified 80.5%

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

5. Taylor expanded in angle around 0 80.6%

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

   1. unpow2 80.6%

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

   2. fma-define 80.6%

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

   3. add-sqr-sqrt 80.6%

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

   4. pow2 80.6%

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

   5. sqrt-pow1 80.6%

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

   6. metadata-eval 80.6%

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

   7. pow1 80.6%

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

   8. *-commutative 80.6%

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

   9. associate-*r* 80.6%

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

7. Applied egg-rr 80.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, a, {\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2}\right)} \]
8. Add Preprocessing

Alternative 2: 59.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.8 \cdot 10^{-160}:\\ \;\;\;\;{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(b \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 2.8e-160)
   (pow (* b (sin (* 0.005555555555555556 (* angle PI)))) 2.0)
   (+ (* a a) (pow (* b (* angle (* 0.005555555555555556 PI))) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.8e-160) {
		tmp = pow((b * sin((0.005555555555555556 * (angle * ((double) M_PI))))), 2.0);
	} else {
		tmp = (a * a) + pow((b * (angle * (0.005555555555555556 * ((double) M_PI)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.8e-160) {
		tmp = Math.pow((b * Math.sin((0.005555555555555556 * (angle * Math.PI)))), 2.0);
	} else {
		tmp = (a * a) + Math.pow((b * (angle * (0.005555555555555556 * Math.PI))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 2.8e-160:
		tmp = math.pow((b * math.sin((0.005555555555555556 * (angle * math.pi)))), 2.0)
	else:
		tmp = (a * a) + math.pow((b * (angle * (0.005555555555555556 * math.pi))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 2.8e-160)
		tmp = Float64(b * sin(Float64(0.005555555555555556 * Float64(angle * pi)))) ^ 2.0;
	else
		tmp = Float64(Float64(a * a) + (Float64(b * Float64(angle * Float64(0.005555555555555556 * pi))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 2.8e-160)
		tmp = (b * sin((0.005555555555555556 * (angle * pi)))) ^ 2.0;
	else
		tmp = (a * a) + ((b * (angle * (0.005555555555555556 * pi))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 2.8e-160], N[Power[N[(b * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[Power[N[(b * N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.80000000000000016e-160

   1. Initial program 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in angle around 0 80.8%

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

   6. Taylor expanded in a around 0 44.7%

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

      1. unpow2 44.7%

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

      2. *-commutative 44.7%

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

      3. associate-*l* 44.7%

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

      4. *-commutative 44.7%

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

      5. unpow2 44.7%

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

      6. swap-sqr 51.0%

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

      7. unpow2 51.0%

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

      8. *-commutative 51.0%

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

      9. associate-*l* 51.0%

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

      10. *-commutative 51.0%

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

   8. Simplified 51.0%

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

   if 2.80000000000000016e-160 < a

   1. Initial program 80.1%

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

   3. Simplified 80.2%

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

   5. Taylor expanded in angle around 0 80.3%

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

      1. unpow2 80.3%

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

   7. Applied egg-rr 80.3%

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

   8. Taylor expanded in angle around 0 77.9%

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

      1. associate-*r* 77.9%

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

      2. *-commutative 77.9%

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

      3. associate-*r* 77.9%

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

      4. *-commutative 77.9%

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

   10. Simplified 77.9%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.8 \cdot 10^{-160}:\\ \;\;\;\;{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(b \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\ \mathbf{if}\;b \leq 1.2 \cdot 10^{-90}:\\ \;\;\;\;{\left(a \cdot \cos t\_0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(b \cdot t\_0\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* 0.005555555555555556 PI))))
   (if (<= b 1.2e-90)
     (pow (* a (cos t_0)) 2.0)
     (+ (* a a) (pow (* b t_0) 2.0)))))

C

double code(double a, double b, double angle) {
	double t_0 = angle * (0.005555555555555556 * ((double) M_PI));
	double tmp;
	if (b <= 1.2e-90) {
		tmp = pow((a * cos(t_0)), 2.0);
	} else {
		tmp = (a * a) + pow((b * t_0), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = angle * (0.005555555555555556 * Math.PI);
	double tmp;
	if (b <= 1.2e-90) {
		tmp = Math.pow((a * Math.cos(t_0)), 2.0);
	} else {
		tmp = (a * a) + Math.pow((b * t_0), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = angle * (0.005555555555555556 * math.pi)
	tmp = 0
	if b <= 1.2e-90:
		tmp = math.pow((a * math.cos(t_0)), 2.0)
	else:
		tmp = (a * a) + math.pow((b * t_0), 2.0)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(angle * Float64(0.005555555555555556 * pi))
	tmp = 0.0
	if (b <= 1.2e-90)
		tmp = Float64(a * cos(t_0)) ^ 2.0;
	else
		tmp = Float64(Float64(a * a) + (Float64(b * t_0) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = angle * (0.005555555555555556 * pi);
	tmp = 0.0;
	if (b <= 1.2e-90)
		tmp = (a * cos(t_0)) ^ 2.0;
	else
		tmp = (a * a) + ((b * t_0) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.2e-90], N[Power[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[Power[N[(b * t$95$0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.2000000000000001e-90

   1. Initial program 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in a around inf 59.8%

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

      1. *-commutative 59.8%

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

      2. associate-*r* 59.9%

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

      3. *-commutative 59.9%

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

      4. associate-*r* 59.9%

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

      5. unpow2 59.9%

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

      6. unpow2 59.9%

         \[\leadsto \left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

      7. swap-sqr 59.9%

         \[\leadsto \color{blue}{\left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot a\right) \cdot \left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot a\right)} \]

      8. unpow2 59.9%

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

   7. Simplified 59.9%

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

   if 1.2000000000000001e-90 < b

   1. Initial program 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in angle around 0 80.9%

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

      1. unpow2 80.9%

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

   7. Applied egg-rr 80.9%

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

   8. Taylor expanded in angle around 0 78.2%

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

      1. associate-*r* 78.2%

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

      2. *-commutative 78.2%

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

      3. associate-*r* 78.2%

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

      4. *-commutative 78.2%

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

   10. Simplified 78.2%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.2 \cdot 10^{-90}:\\ \;\;\;\;{\left(a \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(b \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.5%

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

3. Simplified 80.5%

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

5. Taylor expanded in angle around 0 80.6%

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

   1. unpow2 80.6%

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

7. Applied egg-rr 80.6%

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

Alternative 5: 67.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.5 \cdot 10^{-90}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(b \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 4.5e-90)
   (* a a)
   (+ (* a a) (pow (* b (* angle (* 0.005555555555555556 PI))) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 4.5e-90) {
		tmp = a * a;
	} else {
		tmp = (a * a) + pow((b * (angle * (0.005555555555555556 * ((double) M_PI)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 4.5e-90) {
		tmp = a * a;
	} else {
		tmp = (a * a) + Math.pow((b * (angle * (0.005555555555555556 * Math.PI))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 4.5e-90:
		tmp = a * a
	else:
		tmp = (a * a) + math.pow((b * (angle * (0.005555555555555556 * math.pi))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 4.5e-90)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + (Float64(b * Float64(angle * Float64(0.005555555555555556 * pi))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 4.5e-90)
		tmp = a * a;
	else
		tmp = (a * a) + ((b * (angle * (0.005555555555555556 * pi))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 4.5e-90], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[Power[N[(b * N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.50000000000000009e-90

   1. Initial program 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in angle around 0 60.0%

      \[\leadsto \color{blue}{{a}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 80.5%

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

   7. Applied egg-rr 60.0%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 4.50000000000000009e-90 < b

   1. Initial program 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in angle around 0 80.9%

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

      1. unpow2 80.9%

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

   7. Applied egg-rr 80.9%

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

   8. Taylor expanded in angle around 0 78.2%

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

      1. associate-*r* 78.2%

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

      2. *-commutative 78.2%

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

      3. associate-*r* 78.2%

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

      4. *-commutative 78.2%

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

   10. Simplified 78.2%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.5 \cdot 10^{-90}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(b \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-91}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(angle \cdot \left(b \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 5e-91)
   (* a a)
   (+ (* a a) (pow (* angle (* b (* 0.005555555555555556 PI))) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 5e-91) {
		tmp = a * a;
	} else {
		tmp = (a * a) + pow((angle * (b * (0.005555555555555556 * ((double) M_PI)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 5e-91) {
		tmp = a * a;
	} else {
		tmp = (a * a) + Math.pow((angle * (b * (0.005555555555555556 * Math.PI))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 5e-91:
		tmp = a * a
	else:
		tmp = (a * a) + math.pow((angle * (b * (0.005555555555555556 * math.pi))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 5e-91)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + (Float64(angle * Float64(b * Float64(0.005555555555555556 * pi))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 5e-91)
		tmp = a * a;
	else
		tmp = (a * a) + ((angle * (b * (0.005555555555555556 * pi))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 5e-91], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[Power[N[(angle * N[(b * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.99999999999999997e-91

   1. Initial program 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in angle around 0 60.0%

      \[\leadsto \color{blue}{{a}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 80.5%

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

   7. Applied egg-rr 60.0%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 4.99999999999999997e-91 < b

   1. Initial program 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in angle around 0 80.9%

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

      1. unpow2 80.9%

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

   7. Applied egg-rr 80.9%

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

   8. Taylor expanded in angle around 0 78.2%

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

      1. associate-*r* 78.1%

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

      2. *-commutative 78.1%

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

      3. associate-*r* 78.1%

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

      4. *-commutative 78.1%

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

      5. associate-*r* 78.2%

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

      6. *-commutative 78.2%

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

   10. Simplified 78.2%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-91}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(angle \cdot \left(b \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.5 \cdot 10^{-92}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(0.005555555555555556 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 8.5e-92)
   (* a a)
   (+ (* a a) (pow (* 0.005555555555555556 (* angle (* b PI))) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 8.5e-92) {
		tmp = a * a;
	} else {
		tmp = (a * a) + pow((0.005555555555555556 * (angle * (b * ((double) M_PI)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 8.5e-92) {
		tmp = a * a;
	} else {
		tmp = (a * a) + Math.pow((0.005555555555555556 * (angle * (b * Math.PI))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 8.5e-92:
		tmp = a * a
	else:
		tmp = (a * a) + math.pow((0.005555555555555556 * (angle * (b * math.pi))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 8.5e-92)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + (Float64(0.005555555555555556 * Float64(angle * Float64(b * pi))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 8.5e-92)
		tmp = a * a;
	else
		tmp = (a * a) + ((0.005555555555555556 * (angle * (b * pi))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 8.5e-92], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[Power[N[(0.005555555555555556 * N[(angle * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.50000000000000067e-92

   1. Initial program 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in angle around 0 60.0%

      \[\leadsto \color{blue}{{a}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 80.5%

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

   7. Applied egg-rr 60.0%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 8.50000000000000067e-92 < b

   1. Initial program 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in angle around 0 80.9%

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

      1. unpow2 80.9%

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

   7. Applied egg-rr 80.9%

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

   8. Taylor expanded in angle around 0 78.2%

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

      1. *-commutative 78.2%

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

   10. Simplified 78.2%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.5 \cdot 10^{-92}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(0.005555555555555556 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.7% accurate, 139.0× speedup

Math

\[\begin{array}{l} \\ a \cdot a \end{array} \]

FPCore

(FPCore (a b angle) :precision binary64 (* a a))

C

double code(double a, double b, double angle) {
	return a * a;
}

Fortran

real(8) function code(a, b, angle)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    code = a * a
end function

Java

public static double code(double a, double b, double angle) {
	return a * a;
}

Python

def code(a, b, angle):
	return a * a

Julia

function code(a, b, angle)
	return Float64(a * a)
end

MATLAB

function tmp = code(a, b, angle)
	tmp = a * a;
end

Wolfram

code[a_, b_, angle_] := N[(a * a), $MachinePrecision]

Derivation

1. Initial program 80.5%

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

3. Simplified 80.5%

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

5. Taylor expanded in angle around 0 58.9%

   \[\leadsto \color{blue}{{a}^{2}} \]
6. Step-by-step derivation

   1. unpow2 80.6%

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

7. Applied egg-rr 58.9%

   \[\leadsto \color{blue}{a \cdot a} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024123 
(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)))