ab-angle->ABCF C

Percentage Accurate: 80.1% → 80.1%

Time: 7.6s

Alternatives: 13

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: 80.1% 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: 80.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 78.1%

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

6. Taylor expanded in angle around inf

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

8. Applied rewrites 78.1%

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

10. Applied rewrites 78.1%

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

Alternative 2: 67.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.45 \cdot 10^{-54}:\\ \;\;\;\;\left({\cos \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}^{2} \cdot a\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.45e-54)
   (* (* (pow (cos (* -0.005555555555555556 (* PI angle))) 2.0) a) a)
   (+ (* a a) (pow (* (* (* b PI) angle) 0.005555555555555556) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.45e-54) {
		tmp = (pow(cos((-0.005555555555555556 * (((double) M_PI) * angle))), 2.0) * a) * a;
	} else {
		tmp = (a * a) + pow((((b * ((double) M_PI)) * angle) * 0.005555555555555556), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.45e-54) {
		tmp = (Math.pow(Math.cos((-0.005555555555555556 * (Math.PI * angle))), 2.0) * a) * a;
	} else {
		tmp = (a * a) + Math.pow((((b * Math.PI) * angle) * 0.005555555555555556), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.45e-54:
		tmp = (math.pow(math.cos((-0.005555555555555556 * (math.pi * angle))), 2.0) * a) * a
	else:
		tmp = (a * a) + math.pow((((b * math.pi) * angle) * 0.005555555555555556), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.45e-54)
		tmp = Float64(Float64((cos(Float64(-0.005555555555555556 * Float64(pi * angle))) ^ 2.0) * a) * a);
	else
		tmp = Float64(Float64(a * a) + (Float64(Float64(Float64(b * pi) * angle) * 0.005555555555555556) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.45e-54)
		tmp = ((cos((-0.005555555555555556 * (pi * angle))) ^ 2.0) * a) * a;
	else
		tmp = (a * a) + ((((b * pi) * angle) * 0.005555555555555556) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.45e-54], N[(N[(N[Power[N[Cos[N[(-0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[Power[N[(N[(N[(b * Pi), $MachinePrecision] * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.45000000000000007e-54

   1. Initial program 73.4%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 59.9%

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

   7. Applied rewrites 67.9%

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

   8. Taylor expanded in a around inf

      \[\leadsto \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot a\right) \cdot a \]
   9. Step-by-step derivation

   10. Applied rewrites 54.2%

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

   if 1.45000000000000007e-54 < b

   1. Initial program 88.1%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 88.4%

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

   9. Taylor expanded in angle around 0

      \[\leadsto a \cdot a + {\color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 86.0%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.45 \cdot 10^{-54}:\\ \;\;\;\;\left({\cos \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}^{2} \cdot a\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.45 \cdot 10^{-54}:\\ \;\;\;\;{\cos \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}^{2} \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.45e-54)
   (* (pow (cos (* -0.005555555555555556 (* PI angle))) 2.0) (* a a))
   (+ (* a a) (pow (* (* (* b PI) angle) 0.005555555555555556) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.45e-54) {
		tmp = pow(cos((-0.005555555555555556 * (((double) M_PI) * angle))), 2.0) * (a * a);
	} else {
		tmp = (a * a) + pow((((b * ((double) M_PI)) * angle) * 0.005555555555555556), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.45e-54) {
		tmp = Math.pow(Math.cos((-0.005555555555555556 * (Math.PI * angle))), 2.0) * (a * a);
	} else {
		tmp = (a * a) + Math.pow((((b * Math.PI) * angle) * 0.005555555555555556), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.45e-54:
		tmp = math.pow(math.cos((-0.005555555555555556 * (math.pi * angle))), 2.0) * (a * a)
	else:
		tmp = (a * a) + math.pow((((b * math.pi) * angle) * 0.005555555555555556), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.45e-54)
		tmp = Float64((cos(Float64(-0.005555555555555556 * Float64(pi * angle))) ^ 2.0) * Float64(a * a));
	else
		tmp = Float64(Float64(a * a) + (Float64(Float64(Float64(b * pi) * angle) * 0.005555555555555556) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.45e-54)
		tmp = (cos((-0.005555555555555556 * (pi * angle))) ^ 2.0) * (a * a);
	else
		tmp = (a * a) + ((((b * pi) * angle) * 0.005555555555555556) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.45e-54], N[(N[Power[N[Cos[N[(-0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[Power[N[(N[(N[(b * Pi), $MachinePrecision] * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.45000000000000007e-54

   1. Initial program 73.4%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 59.9%

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

   6. Taylor expanded in a around inf

      \[\leadsto {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(\color{blue}{a} \cdot a\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 54.2%

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

   if 1.45000000000000007e-54 < b

   1. Initial program 88.1%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 88.4%

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

   9. Taylor expanded in angle around 0

      \[\leadsto a \cdot a + {\color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 86.0%

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

Alternative 4: 80.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 78.1%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 78.1%

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

Alternative 5: 67.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.15 \cdot 10^{-55}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 2.15e-55)
   (* a a)
   (+ (* a a) (pow (* (* (* b PI) angle) 0.005555555555555556) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2.15e-55) {
		tmp = a * a;
	} else {
		tmp = (a * a) + pow((((b * ((double) M_PI)) * angle) * 0.005555555555555556), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2.15e-55) {
		tmp = a * a;
	} else {
		tmp = (a * a) + Math.pow((((b * Math.PI) * angle) * 0.005555555555555556), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 2.15e-55:
		tmp = a * a
	else:
		tmp = (a * a) + math.pow((((b * math.pi) * angle) * 0.005555555555555556), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 2.15e-55)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + (Float64(Float64(Float64(b * pi) * angle) * 0.005555555555555556) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 2.15e-55)
		tmp = a * a;
	else
		tmp = (a * a) + ((((b * pi) * angle) * 0.005555555555555556) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 2.15e-55], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[Power[N[(N[(N[(b * Pi), $MachinePrecision] * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.15000000000000005e-55

   1. Initial program 73.4%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 54.6%

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

   if 2.15000000000000005e-55 < b

   1. Initial program 88.1%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 88.4%

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

   9. Taylor expanded in angle around 0

      \[\leadsto a \cdot a + {\color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 86.0%

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

Alternative 6: 54.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.45 \cdot 10^{-115}:\\ \;\;\;\;{\left(\left(b \cdot \pi\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right), angle \cdot angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.45e-115)
   (* (pow (* (* b PI) angle) 2.0) 3.08641975308642e-5)
   (if (<= a 1.1e+116)
     (fma
      (* (* PI PI) (* 3.08641975308642e-5 (* b b)))
      (* angle angle)
      (* a a))
     (* a a))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.45e-115) {
		tmp = pow(((b * ((double) M_PI)) * angle), 2.0) * 3.08641975308642e-5;
	} else if (a <= 1.1e+116) {
		tmp = fma(((((double) M_PI) * ((double) M_PI)) * (3.08641975308642e-5 * (b * b))), (angle * angle), (a * a));
	} else {
		tmp = a * a;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.45e-115)
		tmp = Float64((Float64(Float64(b * pi) * angle) ^ 2.0) * 3.08641975308642e-5);
	elseif (a <= 1.1e+116)
		tmp = fma(Float64(Float64(pi * pi) * Float64(3.08641975308642e-5 * Float64(b * b))), Float64(angle * angle), Float64(a * a));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.45e-115], N[(N[Power[N[(N[(b * Pi), $MachinePrecision] * angle), $MachinePrecision], 2.0], $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision], If[LessEqual[a, 1.1e+116], N[(N[(N[(Pi * Pi), $MachinePrecision] * N[(3.08641975308642e-5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle * angle), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.4499999999999999e-115

   1. Initial program 75.6%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   4. Step-by-step derivation

   5. Applied rewrites 41.8%

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

   6. Taylor expanded in a around 0

      \[\leadsto \frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 36.5%

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

   10. Applied rewrites 45.3%

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

   if 1.4499999999999999e-115 < a < 1.1e116

   1. Initial program 69.8%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   4. Step-by-step derivation

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \left(\frac{1}{32400} \cdot {b}^{2}\right), angle \cdot angle, a \cdot a\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 64.4%

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

   if 1.1e116 < a

   1. Initial program 98.1%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 91.8%

      \[\leadsto \color{blue}{a \cdot a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 57.1% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.1 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot angle, angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.1e+116)
   (fma
    (* (* (* PI PI) (* 3.08641975308642e-5 (- (* b b) (* a a)))) angle)
    angle
    (* a a))
   (* a a)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.1e+116) {
		tmp = fma((((((double) M_PI) * ((double) M_PI)) * (3.08641975308642e-5 * ((b * b) - (a * a)))) * angle), angle, (a * a));
	} else {
		tmp = a * a;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.1e+116)
		tmp = fma(Float64(Float64(Float64(pi * pi) * Float64(3.08641975308642e-5 * Float64(Float64(b * b) - Float64(a * a)))) * angle), angle, Float64(a * a));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.1e+116], N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * N[(3.08641975308642e-5 * N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * angle + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.1e116

   1. Initial program 74.1%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   4. Step-by-step derivation

   5. Applied rewrites 47.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, b \cdot b, -3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right), angle \cdot angle, a \cdot a\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.9%

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

   if 1.1e116 < a

   1. Initial program 98.1%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 91.8%

      \[\leadsto \color{blue}{a \cdot a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 61.8% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 2.25 \cdot 10^{-154}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right), angle \cdot angle, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 2.25e-154)
   (* a a)
   (fma
    (* (* PI PI) (* 3.08641975308642e-5 (* b b)))
    (* angle angle)
    (* a a))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 2.25e-154) {
		tmp = a * a;
	} else {
		tmp = fma(((((double) M_PI) * ((double) M_PI)) * (3.08641975308642e-5 * (b * b))), (angle * angle), (a * a));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 2.25e-154)
		tmp = Float64(a * a);
	else
		tmp = fma(Float64(Float64(pi * pi) * Float64(3.08641975308642e-5 * Float64(b * b))), Float64(angle * angle), Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 2.25e-154], N[(a * a), $MachinePrecision], N[(N[(N[(Pi * Pi), $MachinePrecision] * N[(3.08641975308642e-5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle * angle), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 2.2499999999999999e-154

   1. Initial program 84.3%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 58.4%

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

   if 2.2499999999999999e-154 < angle

   1. Initial program 68.2%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   4. Step-by-step derivation

   5. Applied rewrites 34.0%

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

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \left(\frac{1}{32400} \cdot {b}^{2}\right), angle \cdot angle, a \cdot a\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 59.7%

      \[\leadsto \mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right), angle \cdot angle, a \cdot a\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 61.7% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.6 \cdot 10^{+137}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \pi\right) \cdot b\right) \cdot \pi\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 6.6e+137)
   (* a a)
   (* (* (* (* (* (* angle angle) 3.08641975308642e-5) PI) b) PI) b)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 6.6e+137) {
		tmp = a * a;
	} else {
		tmp = (((((angle * angle) * 3.08641975308642e-5) * ((double) M_PI)) * b) * ((double) M_PI)) * b;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 6.6e+137) {
		tmp = a * a;
	} else {
		tmp = (((((angle * angle) * 3.08641975308642e-5) * Math.PI) * b) * Math.PI) * b;
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 6.6e+137:
		tmp = a * a
	else:
		tmp = (((((angle * angle) * 3.08641975308642e-5) * math.pi) * b) * math.pi) * b
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 6.6e+137)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(angle * angle) * 3.08641975308642e-5) * pi) * b) * pi) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 6.6e+137)
		tmp = a * a;
	else
		tmp = (((((angle * angle) * 3.08641975308642e-5) * pi) * b) * pi) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 6.6e+137], N[(a * a), $MachinePrecision], N[(N[(N[(N[(N[(N[(angle * angle), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] * Pi), $MachinePrecision] * b), $MachinePrecision] * Pi), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.60000000000000005e137

   1. Initial program 74.5%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 55.5%

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

   if 6.60000000000000005e137 < b

   1. Initial program 99.7%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   4. Step-by-step derivation

   5. Applied rewrites 47.7%

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

   6. Taylor expanded in a around 0

      \[\leadsto \frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 70.0%

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

   10. Applied rewrites 70.4%

       \[\leadsto \left(\left(\left(\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \pi\right) \cdot b\right) \cdot \pi\right) \cdot b \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 61.7% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.6 \cdot 10^{+137}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot \left(\left(b \cdot \pi\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 6.6e+137)
   (* a a)
   (* (* (* (* angle angle) 3.08641975308642e-5) b) (* (* b PI) PI))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 6.6e+137) {
		tmp = a * a;
	} else {
		tmp = (((angle * angle) * 3.08641975308642e-5) * b) * ((b * ((double) M_PI)) * ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 6.6e+137) {
		tmp = a * a;
	} else {
		tmp = (((angle * angle) * 3.08641975308642e-5) * b) * ((b * Math.PI) * Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 6.6e+137:
		tmp = a * a
	else:
		tmp = (((angle * angle) * 3.08641975308642e-5) * b) * ((b * math.pi) * math.pi)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 6.6e+137)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(Float64(Float64(angle * angle) * 3.08641975308642e-5) * b) * Float64(Float64(b * pi) * pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 6.6e+137)
		tmp = a * a;
	else
		tmp = (((angle * angle) * 3.08641975308642e-5) * b) * ((b * pi) * pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 6.6e+137], N[(a * a), $MachinePrecision], N[(N[(N[(N[(angle * angle), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] * b), $MachinePrecision] * N[(N[(b * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.60000000000000005e137

   1. Initial program 74.5%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 55.5%

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

   if 6.60000000000000005e137 < b

   1. Initial program 99.7%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   4. Step-by-step derivation

   5. Applied rewrites 47.7%

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

   6. Taylor expanded in a around 0

      \[\leadsto \frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 70.0%

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

   10. Applied rewrites 70.3%

       \[\leadsto \left(\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot \left(\left(b \cdot \pi\right) \cdot \color{blue}{\pi}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 60.8% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.6 \cdot 10^{+137}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\left(b \cdot \pi\right) \cdot \left(b \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 6.6e+137)
   (* a a)
   (* (* (* angle angle) 3.08641975308642e-5) (* (* b PI) (* b PI)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 6.6e+137) {
		tmp = a * a;
	} else {
		tmp = ((angle * angle) * 3.08641975308642e-5) * ((b * ((double) M_PI)) * (b * ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 6.6e+137) {
		tmp = a * a;
	} else {
		tmp = ((angle * angle) * 3.08641975308642e-5) * ((b * Math.PI) * (b * Math.PI));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 6.6e+137:
		tmp = a * a
	else:
		tmp = ((angle * angle) * 3.08641975308642e-5) * ((b * math.pi) * (b * math.pi))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 6.6e+137)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(Float64(angle * angle) * 3.08641975308642e-5) * Float64(Float64(b * pi) * Float64(b * pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 6.6e+137)
		tmp = a * a;
	else
		tmp = ((angle * angle) * 3.08641975308642e-5) * ((b * pi) * (b * pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 6.6e+137], N[(a * a), $MachinePrecision], N[(N[(N[(angle * angle), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] * N[(N[(b * Pi), $MachinePrecision] * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.60000000000000005e137

   1. Initial program 74.5%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 55.5%

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

   if 6.60000000000000005e137 < b

   1. Initial program 99.7%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   4. Step-by-step derivation

   5. Applied rewrites 47.7%

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

   6. Taylor expanded in a around 0

      \[\leadsto \frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 70.0%

      \[\leadsto \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \color{blue}{\left(\left(b \cdot \pi\right) \cdot \left(b \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 60.8% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.6 \cdot 10^{+137}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 6.6e+137)
   (* a a)
   (* (* b b) (* (* PI PI) (* (* angle angle) 3.08641975308642e-5)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 6.6e+137) {
		tmp = a * a;
	} else {
		tmp = (b * b) * ((((double) M_PI) * ((double) M_PI)) * ((angle * angle) * 3.08641975308642e-5));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 6.6e+137) {
		tmp = a * a;
	} else {
		tmp = (b * b) * ((Math.PI * Math.PI) * ((angle * angle) * 3.08641975308642e-5));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 6.6e+137:
		tmp = a * a
	else:
		tmp = (b * b) * ((math.pi * math.pi) * ((angle * angle) * 3.08641975308642e-5))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 6.6e+137)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(b * b) * Float64(Float64(pi * pi) * Float64(Float64(angle * angle) * 3.08641975308642e-5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 6.6e+137)
		tmp = a * a;
	else
		tmp = (b * b) * ((pi * pi) * ((angle * angle) * 3.08641975308642e-5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 6.6e+137], N[(a * a), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[(N[(angle * angle), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.60000000000000005e137

   1. Initial program 74.5%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 55.5%

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

   if 6.60000000000000005e137 < b

   1. Initial program 99.7%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   4. Step-by-step derivation

   5. Applied rewrites 47.7%

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

   6. Taylor expanded in a around 0

      \[\leadsto \frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 70.0%

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

   10. Applied rewrites 70.0%

       \[\leadsto \left(b \cdot b\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \color{blue}{\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 56.6% accurate, 74.7× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle)
use fmin_fmax_functions
    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 78.0%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 53.2%

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

Reproduce

herbie shell --seed 2025056 
(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)))