ab-angle->ABCF C

Percentage Accurate: 79.1% → 78.1%

Time: 4.4s

Alternatives: 10

Speedup: N/A×

• 35.9% 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: 79.1% 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: 78.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.6 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left({\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}^{2} \cdot b\_m, b\_m, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(\left(\left(0.005555555555555556 \cdot b\_m\right) \cdot \pi\right) \cdot angle\right)}^{2}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= b_m 1.6e+151)
   (fma
    (* (pow (sin (* (* angle PI) 0.005555555555555556)) 2.0) b_m)
    b_m
    (* a a))
   (+ (* a a) (pow (* (* (* 0.005555555555555556 b_m) PI) angle) 2.0))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (b_m <= 1.6e+151) {
		tmp = fma((pow(sin(((angle * ((double) M_PI)) * 0.005555555555555556)), 2.0) * b_m), b_m, (a * a));
	} else {
		tmp = (a * a) + pow((((0.005555555555555556 * b_m) * ((double) M_PI)) * angle), 2.0);
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (b_m <= 1.6e+151)
		tmp = fma(Float64((sin(Float64(Float64(angle * pi) * 0.005555555555555556)) ^ 2.0) * b_m), b_m, Float64(a * a));
	else
		tmp = Float64(Float64(a * a) + (Float64(Float64(Float64(0.005555555555555556 * b_m) * pi) * angle) ^ 2.0));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[b$95$m, 1.6e+151], N[(N[(N[Power[N[Sin[N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * b$95$m), $MachinePrecision] * b$95$m + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[Power[N[(N[(N[(0.005555555555555556 * b$95$m), $MachinePrecision] * Pi), $MachinePrecision] * angle), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.59999999999999997e151

   1. Initial program 72.8%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 72.6

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

   5. Applied rewrites 72.6%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow-prod-down N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

   7. Applied rewrites 71.5%

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

   8. Taylor expanded in angle around inf

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

      1. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left({\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{\color{blue}{2}} \cdot b, b, a \cdot a\right) \]

      2. lower-sin.f64 N/A

         \[\leadsto \mathsf{fma}\left({\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot b, b, a \cdot a\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left({\sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}^{2} \cdot b, b, a \cdot a\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left({\sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}^{2} \cdot b, b, a \cdot a\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left({\sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}^{2} \cdot b, b, a \cdot a\right) \]

      6. lift-PI.f64 71.5

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

   10. Applied rewrites 71.5%

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

   if 1.59999999999999997e151 < b

   1. Initial program 97.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in angle around 0

      \[\leadsto a \cdot a + {\color{blue}{\left(angle \cdot \left(\frac{1}{180} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right) + {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{22674816000000} \cdot \left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)\right)}}^{2} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto a \cdot a + {\left(\left(\frac{1}{180} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right) + {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{22674816000000} \cdot \left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \color{blue}{angle}\right)}^{2} \]

      2. lower-*.f64 N/A

         \[\leadsto a \cdot a + {\left(\left(\frac{1}{180} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right) + {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{22674816000000} \cdot \left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \color{blue}{angle}\right)}^{2} \]

   8. Applied rewrites 96.3%

      \[\leadsto a \cdot a + {\color{blue}{\left(\mathsf{fma}\left(0.005555555555555556 \cdot b, \pi, \mathsf{fma}\left(4.410179116778721 \cdot 10^{-14} \cdot \left(angle \cdot angle\right), {\pi}^{5} \cdot b, \left({\pi}^{3} \cdot b\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right) \cdot \left(angle \cdot angle\right)\right) \cdot angle\right)}}^{2} \]

   9. Taylor expanded in angle around 0

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

       1. associate-*r* N/A

          \[\leadsto a \cdot a + {\left(\left(\left(\frac{1}{180} \cdot b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \]

       2. lower-*.f64 N/A

          \[\leadsto a \cdot a + {\left(\left(\left(\frac{1}{180} \cdot b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \]

       3. lift-*.f64 N/A

          \[\leadsto a \cdot a + {\left(\left(\left(\frac{1}{180} \cdot b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \]

       4. lift-PI.f64 97.9

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

   11. Applied rewrites 97.9%

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

Alternative 2: 79.0% accurate, 1.9× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (+ (* a a) (pow (* b_m (sin (* angle (/ PI 180.0)))) 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := N[(N[(a * a), $MachinePrecision] + N[Power[N[(b$95$m * N[Sin[N[(angle * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.1%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 79.0

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

5. Applied rewrites 79.0%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. *-commutative N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 79.0

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

7. Applied rewrites 79.0%

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

   1. lift-/.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lift-PI.f64 79.0

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

9. Applied rewrites 79.0%

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

Alternative 3: 76.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 0.00135:\\ \;\;\;\;a \cdot a + \left({\left(angle \cdot b\_m\right)}^{2} \cdot \left(\pi \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\frac{angle}{180} \cdot \pi\right)\right)\right) \cdot b\_m, b\_m, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= angle 0.00135)
   (+ (* a a) (* (* (pow (* angle b_m) 2.0) (* PI PI)) 3.08641975308642e-5))
   (fma
    (* (- 0.5 (* 0.5 (cos (* 2.0 (* (/ angle 180.0) PI))))) b_m)
    b_m
    (* a a))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (angle <= 0.00135) {
		tmp = (a * a) + ((pow((angle * b_m), 2.0) * (((double) M_PI) * ((double) M_PI))) * 3.08641975308642e-5);
	} else {
		tmp = fma(((0.5 - (0.5 * cos((2.0 * ((angle / 180.0) * ((double) M_PI)))))) * b_m), b_m, (a * a));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (angle <= 0.00135)
		tmp = Float64(Float64(a * a) + Float64(Float64((Float64(angle * b_m) ^ 2.0) * Float64(pi * pi)) * 3.08641975308642e-5));
	else
		tmp = fma(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(angle / 180.0) * pi))))) * b_m), b_m, Float64(a * a));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[angle, 0.00135], N[(N[(a * a), $MachinePrecision] + N[(N[(N[Power[N[(angle * b$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b$95$m), $MachinePrecision] * b$95$m + N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 0.0013500000000000001

   1. Initial program 85.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 85.8

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

   5. Applied rewrites 85.8%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

         \[\leadsto a \cdot a + \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}} \]

      2. lower-*.f64 N/A

         \[\leadsto a \cdot a + \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}} \]

      3. *-commutative N/A

         \[\leadsto a \cdot a + \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {b}^{2}\right)\right) \cdot \frac{1}{32400} \]

      4. unpow-prod-down N/A

         \[\leadsto a \cdot a + \left({angle}^{2} \cdot {\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2}\right) \cdot \frac{1}{32400} \]

      5. pow-prod-down N/A

         \[\leadsto a \cdot a + {\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot b\right)\right)}^{2} \cdot \frac{1}{32400} \]

      6. *-commutative N/A

         \[\leadsto a \cdot a + {\left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \frac{1}{32400} \]

      7. lower-pow.f64 N/A

         \[\leadsto a \cdot a + {\left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \frac{1}{32400} \]

      8. *-commutative N/A

         \[\leadsto a \cdot a + {\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot b\right)\right)}^{2} \cdot \frac{1}{32400} \]

      9. *-commutative N/A

         \[\leadsto a \cdot a + {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      10. lower-*.f64 N/A

          \[\leadsto a \cdot a + {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      11. *-commutative N/A

          \[\leadsto a \cdot a + {\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      12. lower-*.f64 N/A

          \[\leadsto a \cdot a + {\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      13. lift-PI.f64 82.1

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

   8. Applied rewrites 82.1%

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

      1. lift-pow.f64 N/A

         \[\leadsto a \cdot a + {\left(\left(b \cdot \pi\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      2. lift-*.f64 N/A

         \[\leadsto a \cdot a + {\left(\left(b \cdot \pi\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      3. lift-PI.f64 N/A

         \[\leadsto a \cdot a + {\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      4. lift-*.f64 N/A

         \[\leadsto a \cdot a + {\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      5. unpow-prod-down N/A

         \[\leadsto a \cdot a + \left({\left(b \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot {angle}^{2}\right) \cdot \frac{1}{32400} \]

      6. pow-prod-down N/A

         \[\leadsto a \cdot a + \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) \cdot \frac{1}{32400} \]

      7. *-commutative N/A

         \[\leadsto a \cdot a + \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400} \]

      8. associate-*r* N/A

         \[\leadsto a \cdot a + \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      9. lower-*.f64 N/A

         \[\leadsto a \cdot a + \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      10. pow-prod-down N/A

          \[\leadsto a \cdot a + \left({\left(angle \cdot b\right)}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      11. lower-pow.f64 N/A

          \[\leadsto a \cdot a + \left({\left(angle \cdot b\right)}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      12. lower-*.f64 N/A

          \[\leadsto a \cdot a + \left({\left(angle \cdot b\right)}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      13. unpow2 N/A

          \[\leadsto a \cdot a + \left({\left(angle \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} \]

      14. lower-*.f64 N/A

          \[\leadsto a \cdot a + \left({\left(angle \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} \]

      15. lift-PI.f64 N/A

          \[\leadsto a \cdot a + \left({\left(angle \cdot b\right)}^{2} \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} \]

      16. lift-PI.f64 82.2

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

   10. Applied rewrites 82.2%

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

   if 0.0013500000000000001 < angle

   1. Initial program 58.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 58.3

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

   5. Applied rewrites 58.3%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow-prod-down N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

   7. Applied rewrites 58.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-PI.f64 N/A

         \[\leadsto \mathsf{fma}\left({\sin \left(\frac{angle}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}^{2} \cdot b, b, a \cdot a\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left({\sin \color{blue}{\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}}^{2} \cdot b, b, a \cdot a\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left({\sin \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot b, b, a \cdot a\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot b, b, a \cdot a\right) \]

      7. sqr-sin-a N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot b, b, a \cdot a\right) \]

      8. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot b, b, a \cdot a\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot b, b, a \cdot a\right) \]

      10. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot b, b, a \cdot a\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot b, b, a \cdot a\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot b, b, a \cdot a\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \color{blue}{\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \cdot b, b, a \cdot a\right) \]

      14. lift-PI.f64 58.3

          \[\leadsto \mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\pi}\right)\right)\right) \cdot b, b, a \cdot a\right) \]

   9. Applied rewrites 58.3%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\frac{angle}{180} \cdot \pi\right)\right)\right)} \cdot b, b, a \cdot a\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 76.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 7.6 \cdot 10^{-96}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left({\left(angle \cdot b\_m\right)}^{2} \cdot \left(\pi \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= b_m 7.6e-96)
   (* a a)
   (+ (* a a) (* (* (pow (* angle b_m) 2.0) (* PI PI)) 3.08641975308642e-5))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (b_m <= 7.6e-96) {
		tmp = a * a;
	} else {
		tmp = (a * a) + ((pow((angle * b_m), 2.0) * (((double) M_PI) * ((double) M_PI))) * 3.08641975308642e-5);
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double tmp;
	if (b_m <= 7.6e-96) {
		tmp = a * a;
	} else {
		tmp = (a * a) + ((Math.pow((angle * b_m), 2.0) * (Math.PI * Math.PI)) * 3.08641975308642e-5);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	tmp = 0
	if b_m <= 7.6e-96:
		tmp = a * a
	else:
		tmp = (a * a) + ((math.pow((angle * b_m), 2.0) * (math.pi * math.pi)) * 3.08641975308642e-5)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (b_m <= 7.6e-96)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + Float64(Float64((Float64(angle * b_m) ^ 2.0) * Float64(pi * pi)) * 3.08641975308642e-5));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	tmp = 0.0;
	if (b_m <= 7.6e-96)
		tmp = a * a;
	else
		tmp = (a * a) + ((((angle * b_m) ^ 2.0) * (pi * pi)) * 3.08641975308642e-5);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[b$95$m, 7.6e-96], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(N[(N[Power[N[(angle * b$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.6000000000000001e-96

   1. Initial program 77.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

   if 7.6000000000000001e-96 < 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. Add Preprocessing

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 79.9

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

   5. Applied rewrites 79.9%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

         \[\leadsto a \cdot a + \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}} \]

      2. lower-*.f64 N/A

         \[\leadsto a \cdot a + \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}} \]

      3. *-commutative N/A

         \[\leadsto a \cdot a + \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {b}^{2}\right)\right) \cdot \frac{1}{32400} \]

      4. unpow-prod-down N/A

         \[\leadsto a \cdot a + \left({angle}^{2} \cdot {\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2}\right) \cdot \frac{1}{32400} \]

      5. pow-prod-down N/A

         \[\leadsto a \cdot a + {\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot b\right)\right)}^{2} \cdot \frac{1}{32400} \]

      6. *-commutative N/A

         \[\leadsto a \cdot a + {\left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \frac{1}{32400} \]

      7. lower-pow.f64 N/A

         \[\leadsto a \cdot a + {\left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \frac{1}{32400} \]

      8. *-commutative N/A

         \[\leadsto a \cdot a + {\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot b\right)\right)}^{2} \cdot \frac{1}{32400} \]

      9. *-commutative N/A

         \[\leadsto a \cdot a + {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      10. lower-*.f64 N/A

          \[\leadsto a \cdot a + {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      11. *-commutative N/A

          \[\leadsto a \cdot a + {\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      12. lower-*.f64 N/A

          \[\leadsto a \cdot a + {\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      13. lift-PI.f64 76.2

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

   8. Applied rewrites 76.2%

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

      1. lift-pow.f64 N/A

         \[\leadsto a \cdot a + {\left(\left(b \cdot \pi\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      2. lift-*.f64 N/A

         \[\leadsto a \cdot a + {\left(\left(b \cdot \pi\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      3. lift-PI.f64 N/A

         \[\leadsto a \cdot a + {\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      4. lift-*.f64 N/A

         \[\leadsto a \cdot a + {\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      5. unpow-prod-down N/A

         \[\leadsto a \cdot a + \left({\left(b \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot {angle}^{2}\right) \cdot \frac{1}{32400} \]

      6. pow-prod-down N/A

         \[\leadsto a \cdot a + \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) \cdot \frac{1}{32400} \]

      7. *-commutative N/A

         \[\leadsto a \cdot a + \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400} \]

      8. associate-*r* N/A

         \[\leadsto a \cdot a + \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      9. lower-*.f64 N/A

         \[\leadsto a \cdot a + \left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      10. pow-prod-down N/A

          \[\leadsto a \cdot a + \left({\left(angle \cdot b\right)}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      11. lower-pow.f64 N/A

          \[\leadsto a \cdot a + \left({\left(angle \cdot b\right)}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      12. lower-*.f64 N/A

          \[\leadsto a \cdot a + \left({\left(angle \cdot b\right)}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400} \]

      13. unpow2 N/A

          \[\leadsto a \cdot a + \left({\left(angle \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} \]

      14. lower-*.f64 N/A

          \[\leadsto a \cdot a + \left({\left(angle \cdot b\right)}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} \]

      15. lift-PI.f64 N/A

          \[\leadsto a \cdot a + \left({\left(angle \cdot b\right)}^{2} \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} \]

      16. lift-PI.f64 76.3

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

   10. Applied rewrites 76.3%

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

Alternative 5: 76.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 7.6 \cdot 10^{-96}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(\left(\left(0.005555555555555556 \cdot b\_m\right) \cdot \pi\right) \cdot angle\right)}^{2}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= b_m 7.6e-96)
   (* a a)
   (+ (* a a) (pow (* (* (* 0.005555555555555556 b_m) PI) angle) 2.0))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (b_m <= 7.6e-96) {
		tmp = a * a;
	} else {
		tmp = (a * a) + pow((((0.005555555555555556 * b_m) * ((double) M_PI)) * angle), 2.0);
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double tmp;
	if (b_m <= 7.6e-96) {
		tmp = a * a;
	} else {
		tmp = (a * a) + Math.pow((((0.005555555555555556 * b_m) * Math.PI) * angle), 2.0);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	tmp = 0
	if b_m <= 7.6e-96:
		tmp = a * a
	else:
		tmp = (a * a) + math.pow((((0.005555555555555556 * b_m) * math.pi) * angle), 2.0)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (b_m <= 7.6e-96)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + (Float64(Float64(Float64(0.005555555555555556 * b_m) * pi) * angle) ^ 2.0));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	tmp = 0.0;
	if (b_m <= 7.6e-96)
		tmp = a * a;
	else
		tmp = (a * a) + ((((0.005555555555555556 * b_m) * pi) * angle) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[b$95$m, 7.6e-96], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[Power[N[(N[(N[(0.005555555555555556 * b$95$m), $MachinePrecision] * Pi), $MachinePrecision] * angle), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.6000000000000001e-96

   1. Initial program 77.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

   if 7.6000000000000001e-96 < 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. Add Preprocessing

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 79.9

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

   5. Applied rewrites 79.9%

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

   6. Taylor expanded in angle around 0

      \[\leadsto a \cdot a + {\color{blue}{\left(angle \cdot \left(\frac{1}{180} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right) + {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{22674816000000} \cdot \left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)\right)}}^{2} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto a \cdot a + {\left(\left(\frac{1}{180} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right) + {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{22674816000000} \cdot \left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \color{blue}{angle}\right)}^{2} \]

      2. lower-*.f64 N/A

         \[\leadsto a \cdot a + {\left(\left(\frac{1}{180} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right) + {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{22674816000000} \cdot \left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \color{blue}{angle}\right)}^{2} \]

   8. Applied rewrites 75.5%

      \[\leadsto a \cdot a + {\color{blue}{\left(\mathsf{fma}\left(0.005555555555555556 \cdot b, \pi, \mathsf{fma}\left(4.410179116778721 \cdot 10^{-14} \cdot \left(angle \cdot angle\right), {\pi}^{5} \cdot b, \left({\pi}^{3} \cdot b\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right) \cdot \left(angle \cdot angle\right)\right) \cdot angle\right)}}^{2} \]

   9. Taylor expanded in angle around 0

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

       1. associate-*r* N/A

          \[\leadsto a \cdot a + {\left(\left(\left(\frac{1}{180} \cdot b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \]

       2. lower-*.f64 N/A

          \[\leadsto a \cdot a + {\left(\left(\left(\frac{1}{180} \cdot b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \]

       3. lift-*.f64 N/A

          \[\leadsto a \cdot a + {\left(\left(\left(\frac{1}{180} \cdot b\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \]

       4. lift-PI.f64 76.4

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

   11. Applied rewrites 76.4%

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

Alternative 6: 76.3% accurate, 10.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(b\_m \cdot \pi\right) \cdot angle\\ \mathbf{if}\;b\_m \leq 7.6 \cdot 10^{-96}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(t\_0 \cdot t\_0\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* (* b_m PI) angle)))
   (if (<= b_m 7.6e-96)
     (* a a)
     (+ (* a a) (* (* t_0 t_0) 3.08641975308642e-5)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = (b_m * ((double) M_PI)) * angle;
	double tmp;
	if (b_m <= 7.6e-96) {
		tmp = a * a;
	} else {
		tmp = (a * a) + ((t_0 * t_0) * 3.08641975308642e-5);
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double t_0 = (b_m * Math.PI) * angle;
	double tmp;
	if (b_m <= 7.6e-96) {
		tmp = a * a;
	} else {
		tmp = (a * a) + ((t_0 * t_0) * 3.08641975308642e-5);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	t_0 = (b_m * math.pi) * angle
	tmp = 0
	if b_m <= 7.6e-96:
		tmp = a * a
	else:
		tmp = (a * a) + ((t_0 * t_0) * 3.08641975308642e-5)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(Float64(b_m * pi) * angle)
	tmp = 0.0
	if (b_m <= 7.6e-96)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + Float64(Float64(t_0 * t_0) * 3.08641975308642e-5));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	t_0 = (b_m * pi) * angle;
	tmp = 0.0;
	if (b_m <= 7.6e-96)
		tmp = a * a;
	else
		tmp = (a * a) + ((t_0 * t_0) * 3.08641975308642e-5);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(N[(b$95$m * Pi), $MachinePrecision] * angle), $MachinePrecision]}, If[LessEqual[b$95$m, 7.6e-96], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.6000000000000001e-96

   1. Initial program 77.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

   if 7.6000000000000001e-96 < 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. Add Preprocessing

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 79.9

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

   5. Applied rewrites 79.9%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

         \[\leadsto a \cdot a + \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}} \]

      2. lower-*.f64 N/A

         \[\leadsto a \cdot a + \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}} \]

      3. *-commutative N/A

         \[\leadsto a \cdot a + \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {b}^{2}\right)\right) \cdot \frac{1}{32400} \]

      4. unpow-prod-down N/A

         \[\leadsto a \cdot a + \left({angle}^{2} \cdot {\left(\mathsf{PI}\left(\right) \cdot b\right)}^{2}\right) \cdot \frac{1}{32400} \]

      5. pow-prod-down N/A

         \[\leadsto a \cdot a + {\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot b\right)\right)}^{2} \cdot \frac{1}{32400} \]

      6. *-commutative N/A

         \[\leadsto a \cdot a + {\left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \frac{1}{32400} \]

      7. lower-pow.f64 N/A

         \[\leadsto a \cdot a + {\left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \frac{1}{32400} \]

      8. *-commutative N/A

         \[\leadsto a \cdot a + {\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot b\right)\right)}^{2} \cdot \frac{1}{32400} \]

      9. *-commutative N/A

         \[\leadsto a \cdot a + {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      10. lower-*.f64 N/A

          \[\leadsto a \cdot a + {\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      11. *-commutative N/A

          \[\leadsto a \cdot a + {\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      12. lower-*.f64 N/A

          \[\leadsto a \cdot a + {\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      13. lift-PI.f64 76.2

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

   8. Applied rewrites 76.2%

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

      1. lift-pow.f64 N/A

         \[\leadsto a \cdot a + {\left(\left(b \cdot \pi\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      2. lift-*.f64 N/A

         \[\leadsto a \cdot a + {\left(\left(b \cdot \pi\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      3. lift-PI.f64 N/A

         \[\leadsto a \cdot a + {\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      4. lift-*.f64 N/A

         \[\leadsto a \cdot a + {\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}^{2} \cdot \frac{1}{32400} \]

      5. unpow2 N/A

         \[\leadsto a \cdot a + \left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \cdot \frac{1}{32400} \]

      6. lower-*.f64 N/A

         \[\leadsto a \cdot a + \left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \cdot \frac{1}{32400} \]

      7. lift-*.f64 N/A

         \[\leadsto a \cdot a + \left(\left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \cdot \frac{1}{32400} \]

      8. lift-PI.f64 N/A

         \[\leadsto a \cdot a + \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \cdot \frac{1}{32400} \]

      9. lift-*.f64 N/A

         \[\leadsto a \cdot a + \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \cdot \frac{1}{32400} \]

      10. lift-*.f64 N/A

          \[\leadsto a \cdot a + \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right) \cdot \frac{1}{32400} \]

      11. lift-PI.f64 N/A

          \[\leadsto a \cdot a + \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \left(\left(b \cdot \pi\right) \cdot angle\right)\right) \cdot \frac{1}{32400} \]

      12. lift-*.f64 76.2

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

   10. Applied rewrites 76.2%

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

Alternative 7: 72.6% accurate, 10.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 7.6 \cdot 10^{-96}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(angle \cdot angle\right) \cdot b\_m\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}, b\_m, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= b_m 7.6e-96)
   (* a a)
   (fma
    (* (* (* (* angle angle) b_m) (* PI PI)) 3.08641975308642e-5)
    b_m
    (* a a))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (b_m <= 7.6e-96) {
		tmp = a * a;
	} else {
		tmp = fma(((((angle * angle) * b_m) * (((double) M_PI) * ((double) M_PI))) * 3.08641975308642e-5), b_m, (a * a));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (b_m <= 7.6e-96)
		tmp = Float64(a * a);
	else
		tmp = fma(Float64(Float64(Float64(Float64(angle * angle) * b_m) * Float64(pi * pi)) * 3.08641975308642e-5), b_m, Float64(a * a));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[b$95$m, 7.6e-96], N[(a * a), $MachinePrecision], N[(N[(N[(N[(N[(angle * angle), $MachinePrecision] * b$95$m), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] * b$95$m + N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.6000000000000001e-96

   1. Initial program 77.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

   if 7.6000000000000001e-96 < 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. Add Preprocessing

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 79.9

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

   5. Applied rewrites 79.9%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow-prod-down N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

   7. Applied rewrites 74.1%

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

   8. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot {angle}^{2}\right) \cdot \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, b, a \cdot a\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot {angle}^{2}\right) \cdot \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, b, a \cdot a\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot {angle}^{2}\right) \cdot \left(\color{blue}{b} \cdot {\mathsf{PI}\left(\right)}^{2}\right), b, a \cdot a\right) \]

      4. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right), b, a \cdot a\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right), b, a \cdot a\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{b}\right), b, a \cdot a\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{b}\right), b, a \cdot a\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right), b, a \cdot a\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right), b, a \cdot a\right) \]

      10. lift-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot b\right), b, a \cdot a\right) \]

      11. lift-PI.f64 70.5

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

   10. Applied rewrites 70.5%

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

       1. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \color{blue}{\left(\left(\pi \cdot \pi\right) \cdot b\right)}, b, a \cdot a\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\color{blue}{\left(\pi \cdot \pi\right)} \cdot b\right), b, a \cdot a\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \color{blue}{\pi}\right) \cdot b\right), b, a \cdot a\right) \]

       4. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot {angle}^{2}\right) \cdot \left(\left(\pi \cdot \color{blue}{\pi}\right) \cdot b\right), b, a \cdot a\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left(\left(\pi \cdot \pi\right) \cdot b\right)\right)}, b, a \cdot a\right) \]

       6. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left(\left(\pi \cdot \pi\right) \cdot \color{blue}{b}\right)\right), b, a \cdot a\right) \]

       7. lift-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \pi\right) \cdot b\right)\right), b, a \cdot a\right) \]

       8. lift-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)\right), b, a \cdot a\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right)\right), b, a \cdot a\right) \]

       10. pow2 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot b\right)\right), b, a \cdot a\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left(b \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right), b, a \cdot a\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}}, b, a \cdot a\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}}, b, a \cdot a\right) \]

   12. Applied rewrites 70.6%

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

Alternative 8: 72.5% accurate, 10.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 7.6 \cdot 10^{-96}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot b\_m\right) \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot angle\right) \cdot angle\right), b\_m, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= b_m 7.6e-96)
   (* a a)
   (fma
    (* (* (* PI PI) b_m) (* (* 3.08641975308642e-5 angle) angle))
    b_m
    (* a a))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (b_m <= 7.6e-96) {
		tmp = a * a;
	} else {
		tmp = fma((((((double) M_PI) * ((double) M_PI)) * b_m) * ((3.08641975308642e-5 * angle) * angle)), b_m, (a * a));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (b_m <= 7.6e-96)
		tmp = Float64(a * a);
	else
		tmp = fma(Float64(Float64(Float64(pi * pi) * b_m) * Float64(Float64(3.08641975308642e-5 * angle) * angle)), b_m, Float64(a * a));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[b$95$m, 7.6e-96], N[(a * a), $MachinePrecision], N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * b$95$m), $MachinePrecision] * N[(N[(3.08641975308642e-5 * angle), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision] * b$95$m + N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.6000000000000001e-96

   1. Initial program 77.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

   if 7.6000000000000001e-96 < 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. Add Preprocessing

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 79.9

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

   5. Applied rewrites 79.9%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow-prod-down N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

   7. Applied rewrites 74.1%

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

   8. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot {angle}^{2}\right) \cdot \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, b, a \cdot a\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot {angle}^{2}\right) \cdot \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, b, a \cdot a\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot {angle}^{2}\right) \cdot \left(\color{blue}{b} \cdot {\mathsf{PI}\left(\right)}^{2}\right), b, a \cdot a\right) \]

      4. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right), b, a \cdot a\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right), b, a \cdot a\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{b}\right), b, a \cdot a\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{b}\right), b, a \cdot a\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right), b, a \cdot a\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right), b, a \cdot a\right) \]

      10. lift-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot b\right), b, a \cdot a\right) \]

      11. lift-PI.f64 70.5

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

   10. Applied rewrites 70.5%

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

       1. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \color{blue}{\left(\left(\pi \cdot \pi\right) \cdot b\right)}, b, a \cdot a\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot b\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right)}, b, a \cdot a\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot b\right) \cdot \left(\color{blue}{\frac{1}{32400}} \cdot \left(angle \cdot angle\right)\right), b, a \cdot a\right) \]

       4. lift-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \pi\right) \cdot b\right) \cdot \left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right), b, a \cdot a\right) \]

       5. lift-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot \left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right), b, a \cdot a\right) \]

       6. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot \left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right), b, a \cdot a\right) \]

       7. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\left({\mathsf{PI}\left(\right)}^{2} \cdot b\right) \cdot \left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right), b, a \cdot a\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \left(\color{blue}{\frac{1}{32400}} \cdot \left(angle \cdot angle\right)\right), b, a \cdot a\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right)}, b, a \cdot a\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\left({\mathsf{PI}\left(\right)}^{2} \cdot b\right) \cdot \left(\color{blue}{\frac{1}{32400}} \cdot \left(angle \cdot angle\right)\right), b, a \cdot a\right) \]

       11. pow2 N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot \left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right), b, a \cdot a\right) \]

       12. lift-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot \left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right), b, a \cdot a\right) \]

       13. lift-PI.f64 N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot \left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right), b, a \cdot a\right) \]

       14. lift-PI.f64 N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot b\right) \cdot \left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right), b, a \cdot a\right) \]

       15. lift-*.f64 70.5

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

       16. lift-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot b\right) \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(angle \cdot angle\right)}\right), b, a \cdot a\right) \]

       17. lift-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot b\right) \cdot \left(\frac{1}{32400} \cdot \left(angle \cdot \color{blue}{angle}\right)\right), b, a \cdot a\right) \]

       18. associate-*r* N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot b\right) \cdot \left(\left(\frac{1}{32400} \cdot angle\right) \cdot \color{blue}{angle}\right), b, a \cdot a\right) \]

       19. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot b\right) \cdot \left(\left(\frac{1}{32400} \cdot angle\right) \cdot \color{blue}{angle}\right), b, a \cdot a\right) \]

       20. lower-*.f64 70.5

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

   12. Applied rewrites 70.5%

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

Alternative 9: 72.5% accurate, 10.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 7.6 \cdot 10^{-96}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot b\_m\right), b\_m, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= b_m 7.6e-96)
   (* a a)
   (fma
    (* (* 3.08641975308642e-5 (* angle angle)) (* (* PI PI) b_m))
    b_m
    (* a a))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (b_m <= 7.6e-96) {
		tmp = a * a;
	} else {
		tmp = fma(((3.08641975308642e-5 * (angle * angle)) * ((((double) M_PI) * ((double) M_PI)) * b_m)), b_m, (a * a));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (b_m <= 7.6e-96)
		tmp = Float64(a * a);
	else
		tmp = fma(Float64(Float64(3.08641975308642e-5 * Float64(angle * angle)) * Float64(Float64(pi * pi) * b_m)), b_m, Float64(a * a));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[b$95$m, 7.6e-96], N[(a * a), $MachinePrecision], N[(N[(N[(3.08641975308642e-5 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision] * b$95$m + N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.6000000000000001e-96

   1. Initial program 77.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

   if 7.6000000000000001e-96 < 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. Add Preprocessing

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 79.9

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

   5. Applied rewrites 79.9%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow-prod-down N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

   7. Applied rewrites 74.1%

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

   8. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot {angle}^{2}\right) \cdot \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, b, a \cdot a\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot {angle}^{2}\right) \cdot \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, b, a \cdot a\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot {angle}^{2}\right) \cdot \left(\color{blue}{b} \cdot {\mathsf{PI}\left(\right)}^{2}\right), b, a \cdot a\right) \]

      4. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right), b, a \cdot a\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right), b, a \cdot a\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{b}\right), b, a \cdot a\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{b}\right), b, a \cdot a\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right), b, a \cdot a\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot b\right), b, a \cdot a\right) \]

      10. lift-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot b\right), b, a \cdot a\right) \]

      11. lift-PI.f64 70.5

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

   10. Applied rewrites 70.5%

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

Alternative 10: 57.1% accurate, 74.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a \cdot a \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle) :precision binary64 (* a a))

C

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

Fortran

b_m =     private
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_m, angle)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    code = a * a
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	return a * a;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	return a * a

Julia

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

MATLAB

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := N[(a * a), $MachinePrecision]

Derivation

1. Initial program 79.1%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 57.1

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

5. Applied rewrites 57.1%

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

Reproduce

herbie shell --seed 2025089 
(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)))