ab-angle->ABCF C

Percentage Accurate: 80.1% → 80.1%

Time: 5.7s

Alternatives: 12

Speedup: 1.0×

• 37.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.1% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (*
   (pow
    (sin (+ (fma -0.005555555555555556 (* (fabs angle) PI) (* 0.5 PI)) PI))
    2.0)
   (* a a))
  (pow (* b (sin (* PI (/ (fabs angle) 180.0)))) 2.0)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.1%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. sqr-neg-rev N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. distribute-lft-neg-in N/A

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

   8. unpow-prod-down N/A

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

   9. lower-unsound-pow.f32 N/A

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

   10. lower-pow.f32 N/A

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

   11. pow2 N/A

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

   12. lower-unsound-*.f64 N/A

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

3. Applied rewrites 80.1%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. mult-flip N/A

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

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

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

   12. sin-neg-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-neg.f64 N/A

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

5. Applied rewrites 80.0%

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

   1. lift-sin.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. sin-neg N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. mult-flip N/A

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

   8. lift-PI.f64 N/A

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

   9. lower-fma.f64 N/A

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

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

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

   11. cos-neg-rev N/A

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

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

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

7. Applied rewrites 80.0%

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

Alternative 2: 80.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (*
   (-
    0.5
    (* 0.5 (cos (* 2.0 (fma (* PI 0.005555555555555556) angle (* 0.5 PI))))))
   (* a a))
  (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.1%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. sqr-neg-rev N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. distribute-lft-neg-in N/A

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

   8. unpow-prod-down N/A

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

   9. lower-unsound-pow.f32 N/A

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

   10. lower-pow.f32 N/A

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

   11. pow2 N/A

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

   12. lower-unsound-*.f64 N/A

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

3. Applied rewrites 80.1%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. mult-flip N/A

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

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

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

   12. sin-neg-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-neg.f64 N/A

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

5. Applied rewrites 80.0%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. sin-neg N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. sin-neg N/A

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

   9. sqr-neg N/A

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

   10. sqr-sin-a N/A

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

   11. lower--.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

7. Applied rewrites 80.1%

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

Alternative 3: 80.1% accurate, 1.0× speedup

Math

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

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]]

Derivation

1. Initial program 80.1%

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

Alternative 4: 80.0% accurate, 1.0× speedup

Math

\[\left(1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (*
   (- 1.0 (- 0.5 (* 0.5 (cos (* 2.0 (* (* angle 0.005555555555555556) PI))))))
   (* a a))
  (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))

C

double code(double a, double b, double angle) {
	return ((1.0 - (0.5 - (0.5 * cos((2.0 * ((angle * 0.005555555555555556) * ((double) M_PI))))))) * (a * a)) + pow((b * sin((((double) M_PI) * (angle / 180.0)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return ((1.0 - (0.5 - (0.5 * Math.cos((2.0 * ((angle * 0.005555555555555556) * Math.PI)))))) * (a * a)) + Math.pow((b * Math.sin((Math.PI * (angle / 180.0)))), 2.0);
}

Python

def code(a, b, angle):
	return ((1.0 - (0.5 - (0.5 * math.cos((2.0 * ((angle * 0.005555555555555556) * math.pi)))))) * (a * a)) + math.pow((b * math.sin((math.pi * (angle / 180.0)))), 2.0)

Julia

function code(a, b, angle)
	return Float64(Float64(Float64(1.0 - Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(angle * 0.005555555555555556) * pi)))))) * Float64(a * a)) + (Float64(b * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((1.0 - (0.5 - (0.5 * cos((2.0 * ((angle * 0.005555555555555556) * pi)))))) * (a * a)) + ((b * sin((pi * (angle / 180.0)))) ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 80.1%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. sqr-neg-rev N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. distribute-lft-neg-in N/A

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

   8. unpow-prod-down N/A

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

   9. lower-unsound-pow.f32 N/A

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

   10. lower-pow.f32 N/A

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

   11. pow2 N/A

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

   12. lower-unsound-*.f64 N/A

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

3. Applied rewrites 80.1%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. mult-flip N/A

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

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

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

   12. sin-neg-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-neg.f64 N/A

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

5. Applied rewrites 80.0%

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

7. Applied rewrites 80.1%

   \[\leadsto \color{blue}{\left(1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)} \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
8. Add Preprocessing

Alternative 5: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left|angle\right|\\ \mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \mathsf{fma}\left(0.005555555555555556, t\_0, \pi\right)\right), a \cdot a, {\left(\sin \left(0.005555555555555556 \cdot t\_0\right) \cdot b\right)}^{2}\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (fabs angle))))
   (fma
    (+ 0.5 (* 0.5 (cos (* 2.0 (fma 0.005555555555555556 t_0 PI)))))
    (* a a)
    (pow (* (sin (* 0.005555555555555556 t_0)) b) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * fabs(angle);
	return fma((0.5 + (0.5 * cos((2.0 * fma(0.005555555555555556, t_0, ((double) M_PI)))))), (a * a), pow((sin((0.005555555555555556 * t_0)) * b), 2.0));
}

Julia

function code(a, b, angle)
	t_0 = Float64(pi * abs(angle))
	return fma(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * fma(0.005555555555555556, t_0, pi))))), Float64(a * a), (Float64(sin(Float64(0.005555555555555556 * t_0)) * b) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[Abs[angle], $MachinePrecision]), $MachinePrecision]}, N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(0.005555555555555556 * t$95$0 + Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[Power[N[(N[Sin[N[(0.005555555555555556 * t$95$0), $MachinePrecision]], $MachinePrecision] * b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 80.1%

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

   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-fabs-rev N/A

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

   3. cos-neg-rev N/A

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

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

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

   5. lower-sin.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. associate-*r/ N/A

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

   10. fabs-div N/A

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

   11. metadata-eval N/A

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

   12. distribute-neg-frac2 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-fabs.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. metadata-eval N/A

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

   18. lift-PI.f64 N/A

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

   19. mult-flip N/A

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

   20. *-commutative N/A

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

   21. lower-*.f64 N/A

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

   22. metadata-eval 80.0

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

3. Applied rewrites 80.0%

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

5. Applied rewrites 80.0%

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

Alternative 6: 79.9% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+ (pow (* a 1.0) 2.0) (pow (* b (sin (* PI (/ 1.0 (/ 180.0 angle))))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * 1.0), 2.0) + pow((b * sin((((double) M_PI) * (1.0 / (180.0 / angle))))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * 1.0), 2.0) + Math.pow((b * Math.sin((Math.PI * (1.0 / (180.0 / angle))))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((a * 1.0), 2.0) + math.pow((b * math.sin((math.pi * (1.0 / (180.0 / angle))))), 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(a * 1.0) ^ 2.0) + (Float64(b * sin(Float64(pi * Float64(1.0 / Float64(180.0 / angle))))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * 1.0) ^ 2.0) + ((b * sin((pi * (1.0 / (180.0 / angle))))) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * 1.0), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(1.0 / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.1%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

   3. lower-unsound-/.f64 N/A

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

   4. lower-unsound-/.f64 80.1

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

3. Applied rewrites 80.1%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

   3. lower-unsound-/.f64 N/A

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

   4. lower-unsound-/.f64 80.0

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

5. Applied rewrites 80.0%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 79.8%

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

Alternative 7: 79.8% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+ (pow (* a 1.0) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * 1.0), 2.0) + pow((b * sin((((double) M_PI) * (angle / 180.0)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * 1.0), 2.0) + Math.pow((b * Math.sin((Math.PI * (angle / 180.0)))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((a * 1.0), 2.0) + math.pow((b * math.sin((math.pi * (angle / 180.0)))), 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(a * 1.0) ^ 2.0) + (Float64(b * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * 1.0) ^ 2.0) + ((b * sin((pi * (angle / 180.0)))) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * 1.0), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.1%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 79.9%

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

Alternative 8: 79.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := 0.5 \cdot \cos \left(\left(\left|angle\right| \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \mathbf{if}\;\left|angle\right| \leq 175000:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 + t\_0\right) \cdot a, a, \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(\left(\pi \cdot \pi\right) \cdot b\right) \cdot \left|angle\right|\right) \cdot \left|angle\right|\right)\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 \cdot a, a, \left(\left(0.5 - t\_0\right) \cdot b\right) \cdot b\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos (* (* (fabs angle) PI) 0.011111111111111112)))))
   (if (<= (fabs angle) 175000.0)
     (fma
      (* (+ 0.5 t_0) a)
      a
      (*
       (*
        3.08641975308642e-5
        (* (* (* (* PI PI) b) (fabs angle)) (fabs angle)))
       b))
     (fma (* 1.0 a) a (* (* (- 0.5 t_0) b) b)))))

C

double code(double a, double b, double angle) {
	double t_0 = 0.5 * cos(((fabs(angle) * ((double) M_PI)) * 0.011111111111111112));
	double tmp;
	if (fabs(angle) <= 175000.0) {
		tmp = fma(((0.5 + t_0) * a), a, ((3.08641975308642e-5 * ((((((double) M_PI) * ((double) M_PI)) * b) * fabs(angle)) * fabs(angle))) * b));
	} else {
		tmp = fma((1.0 * a), a, (((0.5 - t_0) * b) * b));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(0.5 * cos(Float64(Float64(abs(angle) * pi) * 0.011111111111111112)))
	tmp = 0.0
	if (abs(angle) <= 175000.0)
		tmp = fma(Float64(Float64(0.5 + t_0) * a), a, Float64(Float64(3.08641975308642e-5 * Float64(Float64(Float64(Float64(pi * pi) * b) * abs(angle)) * abs(angle))) * b));
	else
		tmp = fma(Float64(1.0 * a), a, Float64(Float64(Float64(0.5 - t_0) * b) * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(0.5 * N[Cos[N[(N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[angle], $MachinePrecision], 175000.0], N[(N[(N[(0.5 + t$95$0), $MachinePrecision] * a), $MachinePrecision] * a + N[(N[(3.08641975308642e-5 * N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * b), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * a), $MachinePrecision] * a + N[(N[(N[(0.5 - t$95$0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < 175000

   1. Initial program 80.1%

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

   3. Applied rewrites 68.3%

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

   4. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)} \cdot b\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right) \cdot b\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right) \cdot b\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left(\color{blue}{b} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \cdot b\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left(b \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right)\right) \cdot b\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{2}}\right)\right)\right) \cdot b\right) \]

      6. lower-PI.f64 71.1

         \[\leadsto \mathsf{fma}\left(\left(0.5 + 0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \cdot a, a, \left(3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left(b \cdot {\pi}^{2}\right)\right)\right) \cdot b\right) \]

   6. Applied rewrites 71.1%

      \[\leadsto \mathsf{fma}\left(\left(0.5 + 0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \cdot a, a, \color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left(b \cdot {\pi}^{2}\right)\right)\right)} \cdot b\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \color{blue}{\left(b \cdot {\pi}^{2}\right)}\right)\right) \cdot b\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \left(\left(b \cdot {\pi}^{2}\right) \cdot \color{blue}{{angle}^{2}}\right)\right) \cdot b\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \left(\left(b \cdot {\pi}^{2}\right) \cdot {angle}^{\color{blue}{2}}\right)\right) \cdot b\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \left(\left(b \cdot {\pi}^{2}\right) \cdot \left(angle \cdot \color{blue}{angle}\right)\right)\right) \cdot b\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \left(\left(\left(b \cdot {\pi}^{2}\right) \cdot angle\right) \cdot \color{blue}{angle}\right)\right) \cdot b\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \left(\left(\left(b \cdot {\pi}^{2}\right) \cdot angle\right) \cdot \color{blue}{angle}\right)\right) \cdot b\right) \]

      7. lower-*.f64 74.2

         \[\leadsto \mathsf{fma}\left(\left(0.5 + 0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \cdot a, a, \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(b \cdot {\pi}^{2}\right) \cdot angle\right) \cdot angle\right)\right) \cdot b\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \left(\left(\left(b \cdot {\pi}^{2}\right) \cdot angle\right) \cdot angle\right)\right) \cdot b\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \left(\left(\left({\pi}^{2} \cdot b\right) \cdot angle\right) \cdot angle\right)\right) \cdot b\right) \]

      10. lower-*.f64 74.2

          \[\leadsto \mathsf{fma}\left(\left(0.5 + 0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \cdot a, a, \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left({\pi}^{2} \cdot b\right) \cdot angle\right) \cdot angle\right)\right) \cdot b\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \left(\left(\left({\pi}^{2} \cdot b\right) \cdot angle\right) \cdot angle\right)\right) \cdot b\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \left(\left(\left(\left(\pi \cdot \pi\right) \cdot b\right) \cdot angle\right) \cdot angle\right)\right) \cdot b\right) \]

      13. lower-*.f64 74.2

          \[\leadsto \mathsf{fma}\left(\left(0.5 + 0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \cdot a, a, \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(\left(\pi \cdot \pi\right) \cdot b\right) \cdot angle\right) \cdot angle\right)\right) \cdot b\right) \]

   8. Applied rewrites 74.2%

      \[\leadsto \mathsf{fma}\left(\left(0.5 + 0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \cdot a, a, \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(\left(\pi \cdot \pi\right) \cdot b\right) \cdot angle\right) \cdot \color{blue}{angle}\right)\right) \cdot b\right) \]

   if 175000 < angle

   1. Initial program 80.1%

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

   3. Applied rewrites 68.3%

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

   4. Taylor expanded in angle around 0

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

   6. Applied rewrites 68.1%

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

Alternative 9: 76.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \left|angle\right| \cdot \pi\\ \mathbf{if}\;\left|angle\right| \leq 175000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(-0.011111111111111112 \cdot t\_0\right), 0.5, 0.5\right) \cdot a, a, \left(\left(\left(\left(\left|angle\right| \cdot \left|angle\right|\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 \cdot a, a, \left(\left(0.5 - 0.5 \cdot \cos \left(t\_0 \cdot 0.011111111111111112\right)\right) \cdot b\right) \cdot b\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (fabs angle) PI)))
   (if (<= (fabs angle) 175000.0)
     (fma
      (* (fma (cos (* -0.011111111111111112 t_0)) 0.5 0.5) a)
      a
      (*
       (*
        (* (* (* (fabs angle) (fabs angle)) 3.08641975308642e-5) b)
        (* PI PI))
       b))
     (fma
      (* 1.0 a)
      a
      (* (* (- 0.5 (* 0.5 (cos (* t_0 0.011111111111111112)))) b) b)))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(angle) * ((double) M_PI);
	double tmp;
	if (fabs(angle) <= 175000.0) {
		tmp = fma((fma(cos((-0.011111111111111112 * t_0)), 0.5, 0.5) * a), a, (((((fabs(angle) * fabs(angle)) * 3.08641975308642e-5) * b) * (((double) M_PI) * ((double) M_PI))) * b));
	} else {
		tmp = fma((1.0 * a), a, (((0.5 - (0.5 * cos((t_0 * 0.011111111111111112)))) * b) * b));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(abs(angle) * pi)
	tmp = 0.0
	if (abs(angle) <= 175000.0)
		tmp = fma(Float64(fma(cos(Float64(-0.011111111111111112 * t_0)), 0.5, 0.5) * a), a, Float64(Float64(Float64(Float64(Float64(abs(angle) * abs(angle)) * 3.08641975308642e-5) * b) * Float64(pi * pi)) * b));
	else
		tmp = fma(Float64(1.0 * a), a, Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(t_0 * 0.011111111111111112)))) * b) * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision]}, If[LessEqual[N[Abs[angle], $MachinePrecision], 175000.0], N[(N[(N[(N[Cos[N[(-0.011111111111111112 * t$95$0), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * a), $MachinePrecision] * a + N[(N[(N[(N[(N[(N[Abs[angle], $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] * b), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * a), $MachinePrecision] * a + N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(t$95$0 * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < 175000

   1. Initial program 80.1%

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

   3. Applied rewrites 68.3%

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

   4. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)} \cdot b\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right) \cdot b\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right) \cdot b\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left(\color{blue}{b} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \cdot b\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left(b \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right)\right) \cdot b\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{2}}\right)\right)\right) \cdot b\right) \]

      6. lower-PI.f64 71.1

         \[\leadsto \mathsf{fma}\left(\left(0.5 + 0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \cdot a, a, \left(3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left(b \cdot {\pi}^{2}\right)\right)\right) \cdot b\right) \]

   6. Applied rewrites 71.1%

      \[\leadsto \mathsf{fma}\left(\left(0.5 + 0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \cdot a, a, \color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left(b \cdot {\pi}^{2}\right)\right)\right)} \cdot b\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 71.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5, 0.5\right) \cdot a, a, \left(\left(\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot b\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot b\right)} \]

   if 175000 < angle

   1. Initial program 80.1%

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

   3. Applied rewrites 68.3%

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

   4. Taylor expanded in angle around 0

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

   6. Applied rewrites 68.1%

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

Alternative 10: 76.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 0.0028:\\ \;\;\;\;\mathsf{fma}\left(1 \cdot a, a, \left(3.08641975308642 \cdot 10^{-5} \cdot \left({\left(\left|angle\right|\right)}^{2} \cdot \left(b \cdot {\pi}^{2}\right)\right)\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 \cdot a, a, \left(\left(0.5 - 0.5 \cdot \cos \left(\left(\left|angle\right| \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \cdot b\right) \cdot b\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (fabs angle) 0.0028)
   (fma
    (* 1.0 a)
    a
    (*
     (* 3.08641975308642e-5 (* (pow (fabs angle) 2.0) (* b (pow PI 2.0))))
     b))
   (fma
    (* 1.0 a)
    a
    (*
     (* (- 0.5 (* 0.5 (cos (* (* (fabs angle) PI) 0.011111111111111112)))) b)
     b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(angle) <= 0.0028) {
		tmp = fma((1.0 * a), a, ((3.08641975308642e-5 * (pow(fabs(angle), 2.0) * (b * pow(((double) M_PI), 2.0)))) * b));
	} else {
		tmp = fma((1.0 * a), a, (((0.5 - (0.5 * cos(((fabs(angle) * ((double) M_PI)) * 0.011111111111111112)))) * b) * b));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(angle) <= 0.0028)
		tmp = fma(Float64(1.0 * a), a, Float64(Float64(3.08641975308642e-5 * Float64((abs(angle) ^ 2.0) * Float64(b * (pi ^ 2.0)))) * b));
	else
		tmp = fma(Float64(1.0 * a), a, Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(Float64(abs(angle) * pi) * 0.011111111111111112)))) * b) * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[Abs[angle], $MachinePrecision], 0.0028], N[(N[(1.0 * a), $MachinePrecision] * a + N[(N[(3.08641975308642e-5 * N[(N[Power[N[Abs[angle], $MachinePrecision], 2.0], $MachinePrecision] * N[(b * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * a), $MachinePrecision] * a + N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 0.00279999999999999997

   1. Initial program 80.1%

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

   3. Applied rewrites 68.3%

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

   4. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)} \cdot b\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right) \cdot b\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right) \cdot b\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left(\color{blue}{b} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \cdot b\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left(b \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right)\right) \cdot b\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{2}}\right)\right)\right) \cdot b\right) \]

      6. lower-PI.f64 71.1

         \[\leadsto \mathsf{fma}\left(\left(0.5 + 0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \cdot a, a, \left(3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left(b \cdot {\pi}^{2}\right)\right)\right) \cdot b\right) \]

   6. Applied rewrites 71.1%

      \[\leadsto \mathsf{fma}\left(\left(0.5 + 0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \cdot a, a, \color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left(b \cdot {\pi}^{2}\right)\right)\right)} \cdot b\right) \]

   7. Taylor expanded in angle around 0

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

   9. Applied rewrites 70.8%

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

   if 0.00279999999999999997 < angle

   1. Initial program 80.1%

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

   3. Applied rewrites 68.3%

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

   4. Taylor expanded in angle around 0

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

   6. Applied rewrites 68.1%

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

Alternative 11: 70.8% accurate, 2.2× speedup

Math

\[\mathsf{fma}\left(1 \cdot a, a, \left(3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left(b \cdot {\pi}^{2}\right)\right)\right) \cdot b\right) \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (fma
  (* 1.0 a)
  a
  (* (* 3.08641975308642e-5 (* (pow angle 2.0) (* b (pow PI 2.0)))) b)))

C

double code(double a, double b, double angle) {
	return fma((1.0 * a), a, ((3.08641975308642e-5 * (pow(angle, 2.0) * (b * pow(((double) M_PI), 2.0)))) * b));
}

Julia

function code(a, b, angle)
	return fma(Float64(1.0 * a), a, Float64(Float64(3.08641975308642e-5 * Float64((angle ^ 2.0) * Float64(b * (pi ^ 2.0)))) * b))
end

Wolfram

code[a_, b_, angle_] := N[(N[(1.0 * a), $MachinePrecision] * a + N[(N[(3.08641975308642e-5 * N[(N[Power[angle, 2.0], $MachinePrecision] * N[(b * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.1%

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

3. Applied rewrites 68.3%

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

4. Taylor expanded in angle around 0

   \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)} \cdot b\right) \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right) \cdot b\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right) \cdot b\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left(\color{blue}{b} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \cdot b\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left(b \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right)\right) \cdot b\right) \]

   5. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a, a, \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{2}}\right)\right)\right) \cdot b\right) \]

   6. lower-PI.f64 71.1

      \[\leadsto \mathsf{fma}\left(\left(0.5 + 0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \cdot a, a, \left(3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left(b \cdot {\pi}^{2}\right)\right)\right) \cdot b\right) \]

6. Applied rewrites 71.1%

   \[\leadsto \mathsf{fma}\left(\left(0.5 + 0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \cdot a, a, \color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left(b \cdot {\pi}^{2}\right)\right)\right)} \cdot b\right) \]

7. Taylor expanded in angle around 0

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

9. Applied rewrites 70.8%

   \[\leadsto \mathsf{fma}\left(\color{blue}{1} \cdot a, a, \left(3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left(b \cdot {\pi}^{2}\right)\right)\right) \cdot b\right) \]
10. Add Preprocessing

Alternative 12: 57.1% accurate, 29.7× speedup

Math

\[a \cdot a \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.1%

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

2. Taylor expanded in angle around 0

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

   1. lower-pow.f64 57.1

      \[\leadsto {a}^{\color{blue}{2}} \]

4. Applied rewrites 57.1%

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

   1. lift-pow.f64 N/A

      \[\leadsto {a}^{\color{blue}{2}} \]

   2. unpow2 N/A

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

   3. lower-*.f64 57.1

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

6. Applied rewrites 57.1%

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

Reproduce

herbie shell --seed 2025171 
(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)))