ab-angle->ABCF C

Percentage Accurate: 79.6% → 79.5%

Time: 5.1s

Alternatives: 8

Speedup: 1.2×

• 36.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \mathsf{fma}\left({\sin \left(\mathsf{fma}\left(\frac{\pi}{angle\_m}, 0.5, -0.005555555555555556 \cdot \pi\right) \cdot angle\_m\right)}^{2}, a \cdot a, {\left(\sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot b\right)}^{2}\right) \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (fma
  (pow
   (sin (* (fma (/ PI angle_m) 0.5 (* -0.005555555555555556 PI)) angle_m))
   2.0)
  (* a a)
  (pow (* (sin (* (* angle_m PI) 0.005555555555555556)) b) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return fma(pow(sin((fma((((double) M_PI) / angle_m), 0.5, (-0.005555555555555556 * ((double) M_PI))) * angle_m)), 2.0), (a * a), pow((sin(((angle_m * ((double) M_PI)) * 0.005555555555555556)) * b), 2.0));
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return fma((sin(Float64(fma(Float64(pi / angle_m), 0.5, Float64(-0.005555555555555556 * pi)) * angle_m)) ^ 2.0), Float64(a * a), (Float64(sin(Float64(Float64(angle_m * pi) * 0.005555555555555556)) * b) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[Sin[N[(N[(N[(Pi / angle$95$m), $MachinePrecision] * 0.5 + N[(-0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(a * a), $MachinePrecision] + N[Power[N[(N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.2%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-PI.f64 78.2

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

4. Applied rewrites 78.2%

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

5. Taylor expanded in angle around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. fp-cancel-sub-sign-inv N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-PI.f64 78.3

       \[\leadsto {\left(a \cdot \sin \left(\mathsf{fma}\left(\frac{\pi}{angle}, 0.5, -0.005555555555555556 \cdot \pi\right) \cdot angle\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

7. Applied rewrites 78.3%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{\pi}{angle}, 0.5, -0.005555555555555556 \cdot \pi\right) \cdot angle\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. unpow-prod-down N/A

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

   6. lift-pow.f64 N/A

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

9. Applied rewrites 78.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\sin \left(\mathsf{fma}\left(\frac{\pi}{angle}, 0.5, -0.005555555555555556 \cdot \pi\right) \cdot angle\right)}^{2}, a \cdot a, {\left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot b\right)}^{2}\right)} \]

10. Taylor expanded in angle around inf

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

    1. lift-/.f64 N/A

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

    2. *-commutative N/A

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

    3. lift-/.f64 N/A

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

    4. lower-sin.f64 N/A

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

    5. *-commutative N/A

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

    6. lower-*.f64 N/A

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

    7. lower-*.f64 N/A

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

    8. lift-PI.f64 78.4

       \[\leadsto \mathsf{fma}\left({\sin \left(\mathsf{fma}\left(\frac{\pi}{angle}, 0.5, -0.005555555555555556 \cdot \pi\right) \cdot angle\right)}^{2}, a \cdot a, {\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot b\right)}^{2}\right) \]

12. Applied rewrites 78.4%

    \[\leadsto \mathsf{fma}\left({\sin \left(\mathsf{fma}\left(\frac{\pi}{angle}, 0.5, -0.005555555555555556 \cdot \pi\right) \cdot angle\right)}^{2}, a \cdot a, {\left(\color{blue}{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)} \cdot b\right)}^{2}\right) \]
13. Add Preprocessing

Alternative 2: 79.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{fma}\left(\frac{\pi}{angle\_m}, 0.5, -0.005555555555555556 \cdot \pi\right) \cdot angle\_m\right)\right), a \cdot a, {\left(\sin \left(\frac{angle\_m}{180} \cdot \pi\right) \cdot b\right)}^{2}\right) \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (fma
  (-
   0.5
   (*
    0.5
    (cos
     (*
      2.0
      (* (fma (/ PI angle_m) 0.5 (* -0.005555555555555556 PI)) angle_m)))))
  (* a a)
  (pow (* (sin (* (/ angle_m 180.0) PI)) b) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return fma((0.5 - (0.5 * cos((2.0 * (fma((((double) M_PI) / angle_m), 0.5, (-0.005555555555555556 * ((double) M_PI))) * angle_m))))), (a * a), pow((sin(((angle_m / 180.0) * ((double) M_PI))) * b), 2.0));
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(fma(Float64(pi / angle_m), 0.5, Float64(-0.005555555555555556 * pi)) * angle_m))))), Float64(a * a), (Float64(sin(Float64(Float64(angle_m / 180.0) * pi)) * b) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(N[(Pi / angle$95$m), $MachinePrecision] * 0.5 + N[(-0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[Power[N[(N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.2%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-PI.f64 78.2

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

4. Applied rewrites 78.2%

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

5. Taylor expanded in angle around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. fp-cancel-sub-sign-inv N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-PI.f64 78.3

       \[\leadsto {\left(a \cdot \sin \left(\mathsf{fma}\left(\frac{\pi}{angle}, 0.5, -0.005555555555555556 \cdot \pi\right) \cdot angle\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

7. Applied rewrites 78.3%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{\pi}{angle}, 0.5, -0.005555555555555556 \cdot \pi\right) \cdot angle\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. unpow-prod-down N/A

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

   6. lift-pow.f64 N/A

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

9. Applied rewrites 78.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\sin \left(\mathsf{fma}\left(\frac{\pi}{angle}, 0.5, -0.005555555555555556 \cdot \pi\right) \cdot angle\right)}^{2}, a \cdot a, {\left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot b\right)}^{2}\right)} \]
10. Step-by-step derivation

    1. lift-pow.f64 N/A

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

    2. unpow2 N/A

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

    3. lift-sin.f64 N/A

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

    4. lift-sin.f64 N/A

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

    5. sqr-sin-a N/A

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

    6. lower--.f64 N/A

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

11. Applied rewrites 78.3%

    \[\leadsto \mathsf{fma}\left(\color{blue}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{fma}\left(\frac{\pi}{angle}, 0.5, -0.005555555555555556 \cdot \pi\right) \cdot angle\right)\right)}, a \cdot a, {\left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot b\right)}^{2}\right) \]
12. Add Preprocessing

Alternative 3: 79.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.2%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 77.7

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

5. Applied rewrites 77.7%

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

Alternative 4: 66.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \left(b \cdot \pi\right) \cdot angle\_m\\ \mathbf{if}\;b \leq 6.5 \cdot 10^{-51}:\\ \;\;\;\;{\left(\sin \left(\mathsf{fma}\left(-0.005555555555555556, \pi \cdot angle\_m, 0.5 \cdot \pi\right)\right) \cdot a\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + t\_0 \cdot \left(t\_0 \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* b PI) angle_m)))
   (if (<= b 6.5e-51)
     (pow
      (* (sin (fma -0.005555555555555556 (* PI angle_m) (* 0.5 PI))) a)
      2.0)
     (+ (* a a) (* t_0 (* t_0 3.08641975308642e-5))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = (b * ((double) M_PI)) * angle_m;
	double tmp;
	if (b <= 6.5e-51) {
		tmp = pow((sin(fma(-0.005555555555555556, (((double) M_PI) * angle_m), (0.5 * ((double) M_PI)))) * a), 2.0);
	} else {
		tmp = (a * a) + (t_0 * (t_0 * 3.08641975308642e-5));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(Float64(b * pi) * angle_m)
	tmp = 0.0
	if (b <= 6.5e-51)
		tmp = Float64(sin(fma(-0.005555555555555556, Float64(pi * angle_m), Float64(0.5 * pi))) * a) ^ 2.0;
	else
		tmp = Float64(Float64(a * a) + Float64(t_0 * Float64(t_0 * 3.08641975308642e-5)));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(b * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision]}, If[LessEqual[b, 6.5e-51], N[Power[N[(N[Sin[N[(-0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(t$95$0 * N[(t$95$0 * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 6.5000000000000003e-51

   1. Initial program 76.2%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

      3. sin-+PI/2-rev N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-PI.f64 76.3

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

   4. Applied rewrites 76.3%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 57.0%

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

   if 6.5000000000000003e-51 < b

   1. Initial program 82.5%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 79.9

          \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(\left(\pi \cdot b\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} \]

   5. Applied rewrites 79.9%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \color{blue}{{\left(\left(\pi \cdot b\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}} \]

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 79.9

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

   8. Applied rewrites 79.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-PI.f64 80.0

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

   10. Applied rewrites 80.0%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-PI.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lift-PI.f64 N/A

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

       13. lower-*.f64 80.0

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

       14. lift-*.f64 N/A

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

       15. lift-PI.f64 N/A

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

       16. lift-*.f64 N/A

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

       17. associate-*r* N/A

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

       18. *-commutative N/A

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

       19. lower-*.f64 N/A

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

       20. lower-*.f64 N/A

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

       21. lift-PI.f64 79.9

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

   12. Applied rewrites 79.9%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{-51}:\\ \;\;\;\;{\left(\sin \left(\mathsf{fma}\left(-0.005555555555555556, \pi \cdot angle, 0.5 \cdot \pi\right)\right) \cdot a\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \left(b \cdot \pi\right) \cdot angle\_m\\ \mathbf{if}\;b \leq 2.6 \cdot 10^{-51}:\\ \;\;\;\;{\left(\cos \left(\left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\right) \cdot a\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + t\_0 \cdot \left(t\_0 \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* b PI) angle_m)))
   (if (<= b 2.6e-51)
     (pow (* (cos (* (* PI angle_m) 0.005555555555555556)) a) 2.0)
     (+ (* a a) (* t_0 (* t_0 3.08641975308642e-5))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = (b * ((double) M_PI)) * angle_m;
	double tmp;
	if (b <= 2.6e-51) {
		tmp = pow((cos(((((double) M_PI) * angle_m) * 0.005555555555555556)) * a), 2.0);
	} else {
		tmp = (a * a) + (t_0 * (t_0 * 3.08641975308642e-5));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = (b * Math.PI) * angle_m;
	double tmp;
	if (b <= 2.6e-51) {
		tmp = Math.pow((Math.cos(((Math.PI * angle_m) * 0.005555555555555556)) * a), 2.0);
	} else {
		tmp = (a * a) + (t_0 * (t_0 * 3.08641975308642e-5));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = (b * math.pi) * angle_m
	tmp = 0
	if b <= 2.6e-51:
		tmp = math.pow((math.cos(((math.pi * angle_m) * 0.005555555555555556)) * a), 2.0)
	else:
		tmp = (a * a) + (t_0 * (t_0 * 3.08641975308642e-5))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(Float64(b * pi) * angle_m)
	tmp = 0.0
	if (b <= 2.6e-51)
		tmp = Float64(cos(Float64(Float64(pi * angle_m) * 0.005555555555555556)) * a) ^ 2.0;
	else
		tmp = Float64(Float64(a * a) + Float64(t_0 * Float64(t_0 * 3.08641975308642e-5)));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	t_0 = (b * pi) * angle_m;
	tmp = 0.0;
	if (b <= 2.6e-51)
		tmp = (cos(((pi * angle_m) * 0.005555555555555556)) * a) ^ 2.0;
	else
		tmp = (a * a) + (t_0 * (t_0 * 3.08641975308642e-5));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(b * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision]}, If[LessEqual[b, 2.6e-51], N[Power[N[(N[Cos[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(t$95$0 * N[(t$95$0 * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.6e-51

   1. Initial program 76.2%

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

   3. Taylor expanded in a around inf

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

      1. pow-prod-down N/A

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

      2. lower-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 56.9

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

   5. Applied rewrites 56.9%

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

   if 2.6e-51 < b

   1. Initial program 82.5%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 79.9

          \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(\left(\pi \cdot b\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} \]

   5. Applied rewrites 79.9%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \color{blue}{{\left(\left(\pi \cdot b\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}} \]

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 79.9

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

   8. Applied rewrites 79.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-PI.f64 80.0

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

   10. Applied rewrites 80.0%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-PI.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lift-PI.f64 N/A

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

       13. lower-*.f64 80.0

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

       14. lift-*.f64 N/A

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

       15. lift-PI.f64 N/A

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

       16. lift-*.f64 N/A

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

       17. associate-*r* N/A

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

       18. *-commutative N/A

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

       19. lower-*.f64 N/A

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

       20. lower-*.f64 N/A

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

       21. lift-PI.f64 79.9

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

   12. Applied rewrites 79.9%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.6 \cdot 10^{-51}:\\ \;\;\;\;{\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot a\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.7% accurate, 10.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \left(b \cdot \pi\right) \cdot angle\_m\\ \mathbf{if}\;b \leq 2.6 \cdot 10^{-51}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + t\_0 \cdot \left(t\_0 \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* b PI) angle_m)))
   (if (<= b 2.6e-51)
     (* a a)
     (+ (* a a) (* t_0 (* t_0 3.08641975308642e-5))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = (b * ((double) M_PI)) * angle_m;
	double tmp;
	if (b <= 2.6e-51) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (t_0 * (t_0 * 3.08641975308642e-5));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = (b * Math.PI) * angle_m;
	double tmp;
	if (b <= 2.6e-51) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (t_0 * (t_0 * 3.08641975308642e-5));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = (b * math.pi) * angle_m
	tmp = 0
	if b <= 2.6e-51:
		tmp = a * a
	else:
		tmp = (a * a) + (t_0 * (t_0 * 3.08641975308642e-5))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(Float64(b * pi) * angle_m)
	tmp = 0.0
	if (b <= 2.6e-51)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + Float64(t_0 * Float64(t_0 * 3.08641975308642e-5)));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	t_0 = (b * pi) * angle_m;
	tmp = 0.0;
	if (b <= 2.6e-51)
		tmp = a * a;
	else
		tmp = (a * a) + (t_0 * (t_0 * 3.08641975308642e-5));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(b * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision]}, If[LessEqual[b, 2.6e-51], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(t$95$0 * N[(t$95$0 * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.6e-51

   1. Initial program 76.2%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 56.1

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

   5. Applied rewrites 56.1%

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

   if 2.6e-51 < b

   1. Initial program 82.5%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 79.9

          \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(\left(\pi \cdot b\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} \]

   5. Applied rewrites 79.9%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \color{blue}{{\left(\left(\pi \cdot b\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}} \]

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 79.9

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

   8. Applied rewrites 79.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-PI.f64 80.0

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

   10. Applied rewrites 80.0%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-PI.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lift-PI.f64 N/A

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

       13. lower-*.f64 80.0

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

       14. lift-*.f64 N/A

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

       15. lift-PI.f64 N/A

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

       16. lift-*.f64 N/A

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

       17. associate-*r* N/A

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

       18. *-commutative N/A

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

       19. lower-*.f64 N/A

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

       20. lower-*.f64 N/A

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

       21. lift-PI.f64 79.9

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

   12. Applied rewrites 79.9%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.6 \cdot 10^{-51}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.7% accurate, 10.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 2.6 \cdot 10^{-51}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(\left(b \cdot angle\_m\right) \cdot \left(\pi \cdot \left(\left(b \cdot \pi\right) \cdot angle\_m\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 2.6e-51)
   (* a a)
   (+
    (* a a)
    (* (* (* b angle_m) (* PI (* (* b PI) angle_m))) 3.08641975308642e-5))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 2.6e-51) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (((b * angle_m) * (((double) M_PI) * ((b * ((double) M_PI)) * angle_m))) * 3.08641975308642e-5);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 2.6e-51) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (((b * angle_m) * (Math.PI * ((b * Math.PI) * angle_m))) * 3.08641975308642e-5);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if b <= 2.6e-51:
		tmp = a * a
	else:
		tmp = (a * a) + (((b * angle_m) * (math.pi * ((b * math.pi) * angle_m))) * 3.08641975308642e-5)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 2.6e-51)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + Float64(Float64(Float64(b * angle_m) * Float64(pi * Float64(Float64(b * pi) * angle_m))) * 3.08641975308642e-5));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (b <= 2.6e-51)
		tmp = a * a;
	else
		tmp = (a * a) + (((b * angle_m) * (pi * ((b * pi) * angle_m))) * 3.08641975308642e-5);
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 2.6e-51], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(N[(N[(b * angle$95$m), $MachinePrecision] * N[(Pi * N[(N[(b * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.6e-51

   1. Initial program 76.2%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 56.1

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

   5. Applied rewrites 56.1%

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

   if 2.6e-51 < b

   1. Initial program 82.5%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 79.9

          \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(\left(\pi \cdot b\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} \]

   5. Applied rewrites 79.9%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \color{blue}{{\left(\left(\pi \cdot b\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}} \]

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 79.9

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

   8. Applied rewrites 79.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-PI.f64 80.0

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

   10. Applied rewrites 80.0%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lift-PI.f64 80.0

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

       11. lift-*.f64 N/A

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

       12. lift-PI.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. associate-*r* N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 N/A

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

       17. lower-*.f64 N/A

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

       18. lift-PI.f64 80.1

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

   12. Applied rewrites 80.1%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.6 \cdot 10^{-51}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(\left(b \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b \cdot \pi\right) \cdot angle\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.7% accurate, 74.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ a \cdot a \end{array} \]

FPCore

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

C

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

Fortran

angle_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle_m
    code = a * a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(a * a), $MachinePrecision]

Derivation

1. Initial program 78.2%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 52.7

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

5. Applied rewrites 52.7%

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

Reproduce

herbie shell --seed 2025084 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))