ab-angle->ABCF A

Percentage Accurate: 79.6% → 79.5%

Time: 13.7s

Alternatives: 13

Speedup: N/A×

• 37.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ {\left(a \cdot \sin t\_0\right)}^{2} + {\left(b \cdot \cos t\_0\right)}^{2} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ {\left(a \cdot \sin t\_0\right)}^{2} + {\left(b \cdot \cos t\_0\right)}^{2} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ b \cdot b + {\left(\sin \left(\frac{0.005555555555555556}{\frac{1}{angle \cdot \pi}}\right) \cdot a\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (* b b)
  (pow (* (sin (/ 0.005555555555555556 (/ 1.0 (* angle PI)))) a) 2.0)))

C

double code(double a, double b, double angle) {
	return (b * b) + pow((sin((0.005555555555555556 / (1.0 / (angle * ((double) M_PI))))) * a), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return (b * b) + Math.pow((Math.sin((0.005555555555555556 / (1.0 / (angle * Math.PI)))) * a), 2.0);
}

Python

def code(a, b, angle):
	return (b * b) + math.pow((math.sin((0.005555555555555556 / (1.0 / (angle * math.pi)))) * a), 2.0)

Julia

function code(a, b, angle)
	return Float64(Float64(b * b) + (Float64(sin(Float64(0.005555555555555556 / Float64(1.0 / Float64(angle * pi)))) * a) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (b * b) + ((sin((0.005555555555555556 / (1.0 / (angle * pi)))) * a) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[(b * b), $MachinePrecision] + N[Power[N[(N[Sin[N[(0.005555555555555556 / N[(1.0 / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 80.1%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lower-/.f64 80.1

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

7. Applied rewrites 80.1%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. div-inv N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 80.2

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

9. Applied rewrites 80.2%

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

10. Taylor expanded in angle around 0

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

    1. unpow2 N/A

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

    2. lower-*.f64 80.2

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

12. Applied rewrites 80.2%

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

13. Final simplification 80.2%

    \[\leadsto b \cdot b + {\left(\sin \left(\frac{0.005555555555555556}{\frac{1}{angle \cdot \pi}}\right) \cdot a\right)}^{2} \]
14. Add Preprocessing

Alternative 2: 76.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+14}:\\ \;\;\;\;{\left(1 \cdot b\right)}^{2} + {\left(\left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(\left(1 \cdot b\right) \cdot 1, b, \left(a \cdot a\right) \cdot \left(0.5 - \cos \left(\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot 2\right) \cdot 0.5\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 5e+14)
   (+
    (pow (* 1.0 b) 2.0)
    (pow (* (* (* PI 0.005555555555555556) angle) a) 2.0))
   (/
    1.0
    (/
     1.0
     (fma
      (* (* 1.0 b) 1.0)
      b
      (*
       (* a a)
       (-
        0.5
        (* (cos (* (* (* angle PI) 0.005555555555555556) 2.0)) 0.5))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 5e+14) {
		tmp = pow((1.0 * b), 2.0) + pow((((((double) M_PI) * 0.005555555555555556) * angle) * a), 2.0);
	} else {
		tmp = 1.0 / (1.0 / fma(((1.0 * b) * 1.0), b, ((a * a) * (0.5 - (cos((((angle * ((double) M_PI)) * 0.005555555555555556) * 2.0)) * 0.5)))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 5e+14)
		tmp = Float64((Float64(1.0 * b) ^ 2.0) + (Float64(Float64(Float64(pi * 0.005555555555555556) * angle) * a) ^ 2.0));
	else
		tmp = Float64(1.0 / Float64(1.0 / fma(Float64(Float64(1.0 * b) * 1.0), b, Float64(Float64(a * a) * Float64(0.5 - Float64(cos(Float64(Float64(Float64(angle * pi) * 0.005555555555555556) * 2.0)) * 0.5))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e+14], N[(N[Power[N[(1.0 * b), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle), $MachinePrecision] * a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(N[(N[(1.0 * b), $MachinePrecision] * 1.0), $MachinePrecision] * b + N[(N[(a * a), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5e14

   1. Initial program 85.5%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 82.3

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

   8. Applied rewrites 82.3%

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

   if 5e14 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 64.0%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 64.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-/.f64 64.0

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

   7. Applied rewrites 64.0%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

   9. Applied rewrites 64.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(\left(1 \cdot b\right) \cdot 1, b, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(a \cdot a\right)\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+14}:\\ \;\;\;\;{\left(1 \cdot b\right)}^{2} + {\left(\left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(\left(1 \cdot b\right) \cdot 1, b, \left(a \cdot a\right) \cdot \left(0.5 - \cos \left(\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot 2\right) \cdot 0.5\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.0%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 80.1%

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

6. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 80.1

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

8. Applied rewrites 80.1%

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

9. Final simplification 80.1%

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

Alternative 4: 76.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \left(\left(\left(angle \cdot a\right) \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right), a, \left(\left(\cos \left(\left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right) \cdot 2\right) \cdot 0.5 + 0.5\right) \cdot b\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(\left(1 \cdot b\right) \cdot 1, b, \left(a \cdot a\right) \cdot \left(0.5 - \cos \left(\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot 2\right) \cdot 0.5\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 5e+14)
   (fma
    (* (* PI PI) (* (* (* angle a) angle) 3.08641975308642e-5))
    a
    (*
     (* (+ (* (cos (* (* (* angle 0.005555555555555556) PI) 2.0)) 0.5) 0.5) b)
     b))
   (/
    1.0
    (/
     1.0
     (fma
      (* (* 1.0 b) 1.0)
      b
      (*
       (* a a)
       (-
        0.5
        (* (cos (* (* (* angle PI) 0.005555555555555556) 2.0)) 0.5))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 5e+14) {
		tmp = fma(((((double) M_PI) * ((double) M_PI)) * (((angle * a) * angle) * 3.08641975308642e-5)), a, ((((cos((((angle * 0.005555555555555556) * ((double) M_PI)) * 2.0)) * 0.5) + 0.5) * b) * b));
	} else {
		tmp = 1.0 / (1.0 / fma(((1.0 * b) * 1.0), b, ((a * a) * (0.5 - (cos((((angle * ((double) M_PI)) * 0.005555555555555556) * 2.0)) * 0.5)))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 5e+14)
		tmp = fma(Float64(Float64(pi * pi) * Float64(Float64(Float64(angle * a) * angle) * 3.08641975308642e-5)), a, Float64(Float64(Float64(Float64(cos(Float64(Float64(Float64(angle * 0.005555555555555556) * pi) * 2.0)) * 0.5) + 0.5) * b) * b));
	else
		tmp = Float64(1.0 / Float64(1.0 / fma(Float64(Float64(1.0 * b) * 1.0), b, Float64(Float64(a * a) * Float64(0.5 - Float64(cos(Float64(Float64(Float64(angle * pi) * 0.005555555555555556) * 2.0)) * 0.5))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e+14], N[(N[(N[(Pi * Pi), $MachinePrecision] * N[(N[(N[(angle * a), $MachinePrecision] * angle), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] * a + N[(N[(N[(N[(N[Cos[N[(N[(N[(angle * 0.005555555555555556), $MachinePrecision] * Pi), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(N[(N[(1.0 * b), $MachinePrecision] * 1.0), $MachinePrecision] * b + N[(N[(a * a), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5e14

   1. Initial program 85.5%

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

   4. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\log \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot a\right)}, a, \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right)} \]

   5. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{32400} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}, a, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{32400} \cdot \color{blue}{\left(\left(a \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, a, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{32400} \cdot \left(a \cdot {angle}^{2}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{2}}, a, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{32400} \cdot \left(a \cdot {angle}^{2}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{2}}, a, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{32400} \cdot \left(a \cdot {angle}^{2}\right)\right)} \cdot {\mathsf{PI}\left(\right)}^{2}, a, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(a \cdot \color{blue}{\left(angle \cdot angle\right)}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{2}, a, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \color{blue}{\left(\left(a \cdot angle\right) \cdot angle\right)}\right) \cdot {\mathsf{PI}\left(\right)}^{2}, a, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \color{blue}{\left(\left(a \cdot angle\right) \cdot angle\right)}\right) \cdot {\mathsf{PI}\left(\right)}^{2}, a, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(\color{blue}{\left(a \cdot angle\right)} \cdot angle\right)\right) \cdot {\mathsf{PI}\left(\right)}^{2}, a, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(\left(a \cdot angle\right) \cdot angle\right)\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, a, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(\left(a \cdot angle\right) \cdot angle\right)\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, a, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

      11. lower-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(\left(a \cdot angle\right) \cdot angle\right)\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), a, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

      12. lower-PI.f64 80.5

          \[\leadsto \mathsf{fma}\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(a \cdot angle\right) \cdot angle\right)\right) \cdot \left(\pi \cdot \color{blue}{\pi}\right), a, \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

   7. Applied rewrites 80.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(a \cdot angle\right) \cdot angle\right)\right) \cdot \left(\pi \cdot \pi\right)}, a, \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

   if 5e14 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 64.0%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 64.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-/.f64 64.0

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

   7. Applied rewrites 64.0%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

   9. Applied rewrites 64.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(\left(1 \cdot b\right) \cdot 1, b, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(a \cdot a\right)\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \left(\left(\left(angle \cdot a\right) \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right), a, \left(\left(\cos \left(\left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right) \cdot 2\right) \cdot 0.5 + 0.5\right) \cdot b\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(\left(1 \cdot b\right) \cdot 1, b, \left(a \cdot a\right) \cdot \left(0.5 - \cos \left(\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot 2\right) \cdot 0.5\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \left(\left(\left(angle \cdot a\right) \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right), a, \left(\left(\cos \left(\left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right) \cdot 2\right) \cdot 0.5 + 0.5\right) \cdot b\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5\right) \cdot a, a, \left(1 \cdot b\right) \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 5e+14)
   (fma
    (* (* PI PI) (* (* (* angle a) angle) 3.08641975308642e-5))
    a
    (*
     (* (+ (* (cos (* (* (* angle 0.005555555555555556) PI) 2.0)) 0.5) 0.5) b)
     b))
   (fma
    (* (fma -0.5 (cos (* 0.011111111111111112 (* angle PI))) 0.5) a)
    a
    (* (* 1.0 b) b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 5e+14) {
		tmp = fma(((((double) M_PI) * ((double) M_PI)) * (((angle * a) * angle) * 3.08641975308642e-5)), a, ((((cos((((angle * 0.005555555555555556) * ((double) M_PI)) * 2.0)) * 0.5) + 0.5) * b) * b));
	} else {
		tmp = fma((fma(-0.5, cos((0.011111111111111112 * (angle * ((double) M_PI)))), 0.5) * a), a, ((1.0 * b) * b));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 5e+14)
		tmp = fma(Float64(Float64(pi * pi) * Float64(Float64(Float64(angle * a) * angle) * 3.08641975308642e-5)), a, Float64(Float64(Float64(Float64(cos(Float64(Float64(Float64(angle * 0.005555555555555556) * pi) * 2.0)) * 0.5) + 0.5) * b) * b));
	else
		tmp = fma(Float64(fma(-0.5, cos(Float64(0.011111111111111112 * Float64(angle * pi))), 0.5) * a), a, Float64(Float64(1.0 * b) * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e+14], N[(N[(N[(Pi * Pi), $MachinePrecision] * N[(N[(N[(angle * a), $MachinePrecision] * angle), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] * a + N[(N[(N[(N[(N[Cos[N[(N[(N[(angle * 0.005555555555555556), $MachinePrecision] * Pi), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * a), $MachinePrecision] * a + N[(N[(1.0 * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5e14

   1. Initial program 85.5%

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

   4. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\log \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot a\right)}, a, \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right)} \]

   5. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{32400} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}, a, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{32400} \cdot \color{blue}{\left(\left(a \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, a, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{32400} \cdot \left(a \cdot {angle}^{2}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{2}}, a, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{32400} \cdot \left(a \cdot {angle}^{2}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{2}}, a, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{32400} \cdot \left(a \cdot {angle}^{2}\right)\right)} \cdot {\mathsf{PI}\left(\right)}^{2}, a, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(a \cdot \color{blue}{\left(angle \cdot angle\right)}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{2}, a, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \color{blue}{\left(\left(a \cdot angle\right) \cdot angle\right)}\right) \cdot {\mathsf{PI}\left(\right)}^{2}, a, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \color{blue}{\left(\left(a \cdot angle\right) \cdot angle\right)}\right) \cdot {\mathsf{PI}\left(\right)}^{2}, a, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(\color{blue}{\left(a \cdot angle\right)} \cdot angle\right)\right) \cdot {\mathsf{PI}\left(\right)}^{2}, a, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(\left(a \cdot angle\right) \cdot angle\right)\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, a, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(\left(a \cdot angle\right) \cdot angle\right)\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, a, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

      11. lower-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{32400} \cdot \left(\left(a \cdot angle\right) \cdot angle\right)\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), a, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

      12. lower-PI.f64 80.5

          \[\leadsto \mathsf{fma}\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(a \cdot angle\right) \cdot angle\right)\right) \cdot \left(\pi \cdot \color{blue}{\pi}\right), a, \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

   7. Applied rewrites 80.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(a \cdot angle\right) \cdot angle\right)\right) \cdot \left(\pi \cdot \pi\right)}, a, \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

   if 5e14 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 64.0%

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

   4. Applied rewrites 33.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\log \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot a\right)}, a, \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right)} \]
   5. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. rem-exp-log 64.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot a}, a, \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

   6. Applied rewrites 63.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.5, \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5\right) \cdot a}, a, \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

   7. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), \frac{1}{2}\right) \cdot a, a, \left(\color{blue}{1} \cdot b\right) \cdot b\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 64.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5\right) \cdot a, a, \left(\color{blue}{1} \cdot b\right) \cdot b\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \left(\left(\left(angle \cdot a\right) \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right), a, \left(\left(\cos \left(\left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right) \cdot 2\right) \cdot 0.5 + 0.5\right) \cdot b\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5\right) \cdot a, a, \left(1 \cdot b\right) \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5.6 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5\right) \cdot a, a, \left(1 \cdot b\right) \cdot b\right)\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+139}:\\ \;\;\;\;\left(\left(\left(a \cdot a\right) \cdot \pi\right) \cdot \pi\right) \cdot \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) + {\left(1 \cdot b\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot angle\right) \cdot \left(angle \cdot a\right)\right) \cdot \left(\pi \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 5.6e-15)
   (fma
    (* (fma -0.5 (cos (* 0.011111111111111112 (* angle PI))) 0.5) a)
    a
    (* (* 1.0 b) b))
   (if (<= a 5e+139)
     (+
      (* (* (* (* a a) PI) PI) (* (* angle angle) 3.08641975308642e-5))
      (pow (* 1.0 b) 2.0))
     (* (* (* (* 3.08641975308642e-5 a) angle) (* angle a)) (* PI PI)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5.6e-15) {
		tmp = fma((fma(-0.5, cos((0.011111111111111112 * (angle * ((double) M_PI)))), 0.5) * a), a, ((1.0 * b) * b));
	} else if (a <= 5e+139) {
		tmp = ((((a * a) * ((double) M_PI)) * ((double) M_PI)) * ((angle * angle) * 3.08641975308642e-5)) + pow((1.0 * b), 2.0);
	} else {
		tmp = (((3.08641975308642e-5 * a) * angle) * (angle * a)) * (((double) M_PI) * ((double) M_PI));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 5.6e-15)
		tmp = fma(Float64(fma(-0.5, cos(Float64(0.011111111111111112 * Float64(angle * pi))), 0.5) * a), a, Float64(Float64(1.0 * b) * b));
	elseif (a <= 5e+139)
		tmp = Float64(Float64(Float64(Float64(Float64(a * a) * pi) * pi) * Float64(Float64(angle * angle) * 3.08641975308642e-5)) + (Float64(1.0 * b) ^ 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(3.08641975308642e-5 * a) * angle) * Float64(angle * a)) * Float64(pi * pi));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 5.6e-15], N[(N[(N[(-0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * a), $MachinePrecision] * a + N[(N[(1.0 * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5e+139], N[(N[(N[(N[(N[(a * a), $MachinePrecision] * Pi), $MachinePrecision] * Pi), $MachinePrecision] * N[(N[(angle * angle), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] + N[Power[N[(1.0 * b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(3.08641975308642e-5 * a), $MachinePrecision] * angle), $MachinePrecision] * N[(angle * a), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 5.60000000000000028e-15

   1. Initial program 78.3%

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

   4. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\log \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot a\right)}, a, \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right)} \]
   5. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. rem-exp-log 70.6

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot a}, a, \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

   6. Applied rewrites 70.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.5, \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5\right) \cdot a}, a, \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

   7. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), \frac{1}{2}\right) \cdot a, a, \left(\color{blue}{1} \cdot b\right) \cdot b\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 70.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5\right) \cdot a, a, \left(\color{blue}{1} \cdot b\right) \cdot b\right) \]

   if 5.60000000000000028e-15 < a < 5.0000000000000003e139

   1. Initial program 70.5%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-PI.f64 N/A

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

      18. lower-PI.f64 66.9

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

   8. Applied rewrites 66.9%

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

   if 5.0000000000000003e139 < a

   1. Initial program 95.6%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 46.9%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 78.5%

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

   13. Applied rewrites 92.9%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.6 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5\right) \cdot a, a, \left(1 \cdot b\right) \cdot b\right)\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+139}:\\ \;\;\;\;\left(\left(\left(a \cdot a\right) \cdot \pi\right) \cdot \pi\right) \cdot \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) + {\left(1 \cdot b\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot angle\right) \cdot \left(angle \cdot a\right)\right) \cdot \left(\pi \cdot \pi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5.6 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5\right) \cdot a, a, \left(1 \cdot b\right) \cdot b\right)\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right) \cdot \pi\right) \cdot \pi, angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot angle\right) \cdot \left(angle \cdot a\right)\right) \cdot \left(\pi \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 5.6e-15)
   (fma
    (* (fma -0.5 (cos (* 0.011111111111111112 (* angle PI))) 0.5) a)
    a
    (* (* 1.0 b) b))
   (if (<= a 5e+139)
     (fma
      (* (* (* 3.08641975308642e-5 (* a a)) PI) PI)
      (* angle angle)
      (* b b))
     (* (* (* (* 3.08641975308642e-5 a) angle) (* angle a)) (* PI PI)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5.6e-15) {
		tmp = fma((fma(-0.5, cos((0.011111111111111112 * (angle * ((double) M_PI)))), 0.5) * a), a, ((1.0 * b) * b));
	} else if (a <= 5e+139) {
		tmp = fma((((3.08641975308642e-5 * (a * a)) * ((double) M_PI)) * ((double) M_PI)), (angle * angle), (b * b));
	} else {
		tmp = (((3.08641975308642e-5 * a) * angle) * (angle * a)) * (((double) M_PI) * ((double) M_PI));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 5.6e-15)
		tmp = fma(Float64(fma(-0.5, cos(Float64(0.011111111111111112 * Float64(angle * pi))), 0.5) * a), a, Float64(Float64(1.0 * b) * b));
	elseif (a <= 5e+139)
		tmp = fma(Float64(Float64(Float64(3.08641975308642e-5 * Float64(a * a)) * pi) * pi), Float64(angle * angle), Float64(b * b));
	else
		tmp = Float64(Float64(Float64(Float64(3.08641975308642e-5 * a) * angle) * Float64(angle * a)) * Float64(pi * pi));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 5.6e-15], N[(N[(N[(-0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * a), $MachinePrecision] * a + N[(N[(1.0 * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5e+139], N[(N[(N[(N[(3.08641975308642e-5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] * Pi), $MachinePrecision] * N[(angle * angle), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(3.08641975308642e-5 * a), $MachinePrecision] * angle), $MachinePrecision] * N[(angle * a), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 5.60000000000000028e-15

   1. Initial program 78.3%

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

   4. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\log \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot a\right)}, a, \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right)} \]
   5. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. rem-exp-log 70.6

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot a}, a, \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

   6. Applied rewrites 70.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.5, \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5\right) \cdot a}, a, \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right) \]

   7. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), \frac{1}{2}\right) \cdot a, a, \left(\color{blue}{1} \cdot b\right) \cdot b\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 70.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5\right) \cdot a, a, \left(\color{blue}{1} \cdot b\right) \cdot b\right) \]

   if 5.60000000000000028e-15 < a < 5.0000000000000003e139

   1. Initial program 70.5%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 31.7%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 66.8%

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

   if 5.0000000000000003e139 < a

   1. Initial program 95.6%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 46.9%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 78.5%

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

   13. Applied rewrites 92.9%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.6 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5\right) \cdot a, a, \left(1 \cdot b\right) \cdot b\right)\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right) \cdot \pi\right) \cdot \pi, angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot angle\right) \cdot \left(angle \cdot a\right)\right) \cdot \left(\pi \cdot \pi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.75 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right) \cdot \pi\right) \cdot \pi, angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot angle\right) \cdot \left(angle \cdot a\right)\right) \cdot \left(\pi \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.75e-65)
   (* (fma (cos (* 0.011111111111111112 (* angle PI))) 0.5 0.5) (* b b))
   (if (<= a 5e+139)
     (fma
      (* (* (* 3.08641975308642e-5 (* a a)) PI) PI)
      (* angle angle)
      (* b b))
     (* (* (* (* 3.08641975308642e-5 a) angle) (* angle a)) (* PI PI)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.75e-65) {
		tmp = fma(cos((0.011111111111111112 * (angle * ((double) M_PI)))), 0.5, 0.5) * (b * b);
	} else if (a <= 5e+139) {
		tmp = fma((((3.08641975308642e-5 * (a * a)) * ((double) M_PI)) * ((double) M_PI)), (angle * angle), (b * b));
	} else {
		tmp = (((3.08641975308642e-5 * a) * angle) * (angle * a)) * (((double) M_PI) * ((double) M_PI));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 3.75e-65)
		tmp = Float64(fma(cos(Float64(0.011111111111111112 * Float64(angle * pi))), 0.5, 0.5) * Float64(b * b));
	elseif (a <= 5e+139)
		tmp = fma(Float64(Float64(Float64(3.08641975308642e-5 * Float64(a * a)) * pi) * pi), Float64(angle * angle), Float64(b * b));
	else
		tmp = Float64(Float64(Float64(Float64(3.08641975308642e-5 * a) * angle) * Float64(angle * a)) * Float64(pi * pi));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 3.75e-65], N[(N[(N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5e+139], N[(N[(N[(N[(3.08641975308642e-5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] * Pi), $MachinePrecision] * N[(angle * angle), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(3.08641975308642e-5 * a), $MachinePrecision] * angle), $MachinePrecision] * N[(angle * a), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 3.7500000000000001e-65

   1. Initial program 79.4%

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

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\log \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot a\right)}, a, \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot b\right) \cdot b\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot {b}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot {b}^{2}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{2}\right)} \cdot {b}^{2} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{2}} + \frac{1}{2}\right) \cdot {b}^{2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{2}, \frac{1}{2}\right)} \cdot {b}^{2} \]

      6. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}, \frac{1}{2}, \frac{1}{2}\right) \cdot {b}^{2} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}, \frac{1}{2}, \frac{1}{2}\right) \cdot {b}^{2} \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right), \frac{1}{2}, \frac{1}{2}\right) \cdot {b}^{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right), \frac{1}{2}, \frac{1}{2}\right) \cdot {b}^{2} \]

      10. lower-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot {b}^{2} \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), \frac{1}{2}, \frac{1}{2}\right) \cdot \color{blue}{\left(b \cdot b\right)} \]

      12. lower-*.f64 64.6

          \[\leadsto \mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5, 0.5\right) \cdot \color{blue}{\left(b \cdot b\right)} \]

   7. Applied rewrites 64.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)} \]

   if 3.7500000000000001e-65 < a < 5.0000000000000003e139

   1. Initial program 68.9%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 69.6%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 35.4%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 64.3%

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

   if 5.0000000000000003e139 < a

   1. Initial program 95.6%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 46.9%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 78.5%

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

   13. Applied rewrites 92.9%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.75 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right) \cdot \pi\right) \cdot \pi, angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot angle\right) \cdot \left(angle \cdot a\right)\right) \cdot \left(\pi \cdot \pi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.2% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.75 \cdot 10^{-65}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right) \cdot \pi\right) \cdot \pi, angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot angle\right) \cdot \left(angle \cdot a\right)\right) \cdot \left(\pi \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.75e-65)
   (* b b)
   (if (<= a 5e+139)
     (fma
      (* (* (* 3.08641975308642e-5 (* a a)) PI) PI)
      (* angle angle)
      (* b b))
     (* (* (* (* 3.08641975308642e-5 a) angle) (* angle a)) (* PI PI)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.75e-65) {
		tmp = b * b;
	} else if (a <= 5e+139) {
		tmp = fma((((3.08641975308642e-5 * (a * a)) * ((double) M_PI)) * ((double) M_PI)), (angle * angle), (b * b));
	} else {
		tmp = (((3.08641975308642e-5 * a) * angle) * (angle * a)) * (((double) M_PI) * ((double) M_PI));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 3.75e-65)
		tmp = Float64(b * b);
	elseif (a <= 5e+139)
		tmp = fma(Float64(Float64(Float64(3.08641975308642e-5 * Float64(a * a)) * pi) * pi), Float64(angle * angle), Float64(b * b));
	else
		tmp = Float64(Float64(Float64(Float64(3.08641975308642e-5 * a) * angle) * Float64(angle * a)) * Float64(pi * pi));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 3.75e-65], N[(b * b), $MachinePrecision], If[LessEqual[a, 5e+139], N[(N[(N[(N[(3.08641975308642e-5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] * Pi), $MachinePrecision] * N[(angle * angle), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(3.08641975308642e-5 * a), $MachinePrecision] * angle), $MachinePrecision] * N[(angle * a), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 3.7500000000000001e-65

   1. Initial program 79.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 64.6

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

   5. Applied rewrites 64.6%

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

   if 3.7500000000000001e-65 < a < 5.0000000000000003e139

   1. Initial program 68.9%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 69.6%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 35.4%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 64.3%

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

   if 5.0000000000000003e139 < a

   1. Initial program 95.6%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 46.9%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 78.5%

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

   13. Applied rewrites 92.9%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.75 \cdot 10^{-65}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right) \cdot \pi\right) \cdot \pi, angle \cdot angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot angle\right) \cdot \left(angle \cdot a\right)\right) \cdot \left(\pi \cdot \pi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.8% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.7 \cdot 10^{+139}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot angle\right) \cdot \left(angle \cdot a\right)\right) \cdot \left(\pi \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 2.7e+139)
   (* b b)
   (* (* (* (* 3.08641975308642e-5 a) angle) (* angle a)) (* PI PI))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.7e+139) {
		tmp = b * b;
	} else {
		tmp = (((3.08641975308642e-5 * a) * angle) * (angle * a)) * (((double) M_PI) * ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.7e+139) {
		tmp = b * b;
	} else {
		tmp = (((3.08641975308642e-5 * a) * angle) * (angle * a)) * (Math.PI * Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 2.7e+139:
		tmp = b * b
	else:
		tmp = (((3.08641975308642e-5 * a) * angle) * (angle * a)) * (math.pi * math.pi)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 2.7e+139)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(Float64(Float64(3.08641975308642e-5 * a) * angle) * Float64(angle * a)) * Float64(pi * pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 2.7e+139)
		tmp = b * b;
	else
		tmp = (((3.08641975308642e-5 * a) * angle) * (angle * a)) * (pi * pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 2.7e+139], N[(b * b), $MachinePrecision], N[(N[(N[(N[(3.08641975308642e-5 * a), $MachinePrecision] * angle), $MachinePrecision] * N[(angle * a), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.6999999999999998e139

   1. Initial program 77.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 62.6

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

   5. Applied rewrites 62.6%

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

   if 2.6999999999999998e139 < a

   1. Initial program 95.6%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 46.9%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 78.5%

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

   13. Applied rewrites 92.9%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.7 \cdot 10^{+139}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot angle\right) \cdot \left(angle \cdot a\right)\right) \cdot \left(\pi \cdot \pi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.4% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{+139}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(angle \cdot a\right) \cdot angle\right) \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\pi \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 5e+139)
   (* b b)
   (* (* (* (* (* angle a) angle) a) 3.08641975308642e-5) (* PI PI))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5e+139) {
		tmp = b * b;
	} else {
		tmp = ((((angle * a) * angle) * a) * 3.08641975308642e-5) * (((double) M_PI) * ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5e+139) {
		tmp = b * b;
	} else {
		tmp = ((((angle * a) * angle) * a) * 3.08641975308642e-5) * (Math.PI * Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 5e+139:
		tmp = b * b
	else:
		tmp = ((((angle * a) * angle) * a) * 3.08641975308642e-5) * (math.pi * math.pi)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 5e+139)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(angle * a) * angle) * a) * 3.08641975308642e-5) * Float64(pi * pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 5e+139)
		tmp = b * b;
	else
		tmp = ((((angle * a) * angle) * a) * 3.08641975308642e-5) * (pi * pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 5e+139], N[(b * b), $MachinePrecision], N[(N[(N[(N[(N[(angle * a), $MachinePrecision] * angle), $MachinePrecision] * a), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.0000000000000003e139

   1. Initial program 77.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 62.6

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

   5. Applied rewrites 62.6%

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

   if 5.0000000000000003e139 < a

   1. Initial program 95.6%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 46.9%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 78.5%

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

   13. Applied rewrites 88.2%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{+139}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(angle \cdot a\right) \cdot angle\right) \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\pi \cdot \pi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.0% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5.2 \cdot 10^{+139}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(angle \cdot angle\right) \cdot a\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 5.2e+139)
   (* b b)
   (* (* (* (* angle angle) a) (* PI PI)) (* 3.08641975308642e-5 a))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5.2e+139) {
		tmp = b * b;
	} else {
		tmp = (((angle * angle) * a) * (((double) M_PI) * ((double) M_PI))) * (3.08641975308642e-5 * a);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5.2e+139) {
		tmp = b * b;
	} else {
		tmp = (((angle * angle) * a) * (Math.PI * Math.PI)) * (3.08641975308642e-5 * a);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 5.2e+139:
		tmp = b * b
	else:
		tmp = (((angle * angle) * a) * (math.pi * math.pi)) * (3.08641975308642e-5 * a)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 5.2e+139)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(Float64(Float64(angle * angle) * a) * Float64(pi * pi)) * Float64(3.08641975308642e-5 * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 5.2e+139)
		tmp = b * b;
	else
		tmp = (((angle * angle) * a) * (pi * pi)) * (3.08641975308642e-5 * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 5.2e+139], N[(b * b), $MachinePrecision], N[(N[(N[(N[(angle * angle), $MachinePrecision] * a), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * N[(3.08641975308642e-5 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.20000000000000044e139

   1. Initial program 77.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 62.6

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

   5. Applied rewrites 62.6%

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

   if 5.20000000000000044e139 < a

   1. Initial program 95.6%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 46.9%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 78.5%

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

   13. Applied rewrites 78.6%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.2 \cdot 10^{+139}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(angle \cdot angle\right) \cdot a\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.8% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.0%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 56.7

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

5. Applied rewrites 56.7%

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

Reproduce

herbie shell --seed 2024235 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))