ab-angle->ABCF A

Percentage Accurate: 80.5% → 80.5%

Time: 5.4s

Alternatives: 14

Speedup: 1.0×

• 63.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.5%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-PI.f64 N/A

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

   7. mult-flip N/A

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

   8. distribute-lft-out N/A

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

   9. lower-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. mult-flip N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval 80.4%

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

3. Applied rewrites 80.4%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-in N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

   8. mult-flip N/A

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

   9. lift-PI.f64 N/A

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

   10. lower-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. mult-flip N/A

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

   15. metadata-eval N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 80.4%

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

5. Applied rewrites 80.4%

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

Alternative 2: 80.5% accurate, 1.0× speedup

Math

\[{\left(a \cdot \sin \left(\frac{\left|angle\right|}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{90 + \left|angle\right|}{180}\right)\right)}^{2} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (+
 (pow (* a (sin (* (/ (fabs angle) 180.0) PI))) 2.0)
 (pow (* b (sin (* PI (/ (+ 90.0 (fabs angle)) 180.0)))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * sin(((fabs(angle) / 180.0) * ((double) M_PI)))), 2.0) + pow((b * sin((((double) M_PI) * ((90.0 + fabs(angle)) / 180.0)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin(((Math.abs(angle) / 180.0) * Math.PI))), 2.0) + Math.pow((b * Math.sin((Math.PI * ((90.0 + Math.abs(angle)) / 180.0)))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((a * math.sin(((math.fabs(angle) / 180.0) * math.pi))), 2.0) + math.pow((b * math.sin((math.pi * ((90.0 + math.fabs(angle)) / 180.0)))), 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(Float64(abs(angle) / 180.0) * pi))) ^ 2.0) + (Float64(b * sin(Float64(pi * Float64(Float64(90.0 + abs(angle)) / 180.0)))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * sin(((abs(angle) / 180.0) * pi))) ^ 2.0) + ((b * sin((pi * ((90.0 + abs(angle)) / 180.0)))) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(N[(N[Abs[angle], $MachinePrecision] / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(N[(90.0 + N[Abs[angle], $MachinePrecision]), $MachinePrecision] / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.5%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-PI.f64 N/A

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

   7. mult-flip N/A

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

   8. distribute-lft-out N/A

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

   9. lower-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. mult-flip N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval 80.4%

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

3. Applied rewrites 80.4%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. associate-/r/ N/A

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

   7. div-flip-rev N/A

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

   8. add-to-fraction N/A

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

   9. lower-/.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. metadata-eval 80.4%

       \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{\color{blue}{90} + angle}{180}\right)\right)}^{2} \]

5. Applied rewrites 80.4%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \color{blue}{\frac{90 + angle}{180}}\right)\right)}^{2} \]
6. Add Preprocessing

Alternative 3: 80.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double a, double b, double angle) {
	double t_0 = (angle * 0.005555555555555556) * Math.PI;
	return Math.pow((a * Math.sin(t_0)), 2.0) + Math.pow((b * Math.cos(t_0)), 2.0);
}

Python

def code(a, b, angle):
	t_0 = (angle * 0.005555555555555556) * math.pi
	return math.pow((a * math.sin(t_0)), 2.0) + math.pow((b * math.cos(t_0)), 2.0)

Julia

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

MATLAB

function tmp = code(a, b, angle)
	t_0 = (angle * 0.005555555555555556) * pi;
	tmp = ((a * sin(t_0)) ^ 2.0) + ((b * cos(t_0)) ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 80.5%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval 80.5%

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

3. Applied rewrites 80.5%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval 80.5%

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

5. Applied rewrites 80.5%

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

Alternative 4: 80.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double a, double b, double angle) {
	double t_0 = (0.005555555555555556 * Math.PI) * angle;
	return Math.pow((a * Math.sin(t_0)), 2.0) + Math.pow((b * Math.cos(t_0)), 2.0);
}

Python

def code(a, b, angle):
	t_0 = (0.005555555555555556 * math.pi) * angle
	return math.pow((a * math.sin(t_0)), 2.0) + math.pow((b * math.cos(t_0)), 2.0)

Julia

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

MATLAB

function tmp = code(a, b, angle)
	t_0 = (0.005555555555555556 * pi) * angle;
	tmp = ((a * sin(t_0)) ^ 2.0) + ((b * cos(t_0)) ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 80.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 80.5%

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

3. Applied rewrites 80.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 80.5%

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

5. Applied rewrites 80.5%

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

Alternative 5: 80.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.5%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

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

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

   4. lower-unsound-/.f64 80.4%

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

3. Applied rewrites 80.4%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

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

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

   4. lower-unsound-/.f64 80.4%

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

5. Applied rewrites 80.4%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 80.3%

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

Alternative 6: 80.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.5%

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

2. Taylor expanded in angle around 0

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

   1. lower-pow.f64 80.4%

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

4. Applied rewrites 80.4%

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

Alternative 7: 69.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| \cdot \left|b\right|\\ \mathbf{if}\;\left|b\right| \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(\left|b\right|, \left|b\right|, \left(\mathsf{fma}\left(t\_0, -3.08641975308642 \cdot 10^{-5}, 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left|angle\right|\right)\right) \cdot \left|angle\right|\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\mathsf{fma}\left(0.005555555555555556, \left|angle\right|, 0.5\right) \cdot \pi\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* (fabs b) (fabs b))))
  (if (<= (fabs b) 1.4e+77)
    (fma
     (fabs b)
     (fabs b)
     (*
      (*
       (fma t_0 -3.08641975308642e-5 (* 3.08641975308642e-5 (* a a)))
       (* (* PI PI) (fabs angle)))
      (fabs angle)))
    (*
     t_0
     (-
      0.5
      (*
       0.5
       (cos
        (*
         2.0
         (* (fma 0.005555555555555556 (fabs angle) 0.5) PI)))))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) * fabs(b);
	double tmp;
	if (fabs(b) <= 1.4e+77) {
		tmp = fma(fabs(b), fabs(b), ((fma(t_0, -3.08641975308642e-5, (3.08641975308642e-5 * (a * a))) * ((((double) M_PI) * ((double) M_PI)) * fabs(angle))) * fabs(angle)));
	} else {
		tmp = t_0 * (0.5 - (0.5 * cos((2.0 * (fma(0.005555555555555556, fabs(angle), 0.5) * ((double) M_PI))))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) * abs(b))
	tmp = 0.0
	if (abs(b) <= 1.4e+77)
		tmp = fma(abs(b), abs(b), Float64(Float64(fma(t_0, -3.08641975308642e-5, Float64(3.08641975308642e-5 * Float64(a * a))) * Float64(Float64(pi * pi) * abs(angle))) * abs(angle)));
	else
		tmp = Float64(t_0 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(fma(0.005555555555555556, abs(angle), 0.5) * pi))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 1.4e+77], N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision] + N[(N[(N[(t$95$0 * -3.08641975308642e-5 + N[(3.08641975308642e-5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(0.005555555555555556 * N[Abs[angle], $MachinePrecision] + 0.5), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.4e77

   1. Initial program 80.5%

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

   2. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   4. Applied rewrites 41.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 41.5%

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

   6. Applied rewrites 41.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 43.9%

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

   8. Applied rewrites 43.9%

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

   if 1.4e77 < b

   1. Initial program 80.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 57.6%

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

   4. Applied rewrites 57.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 57.6%

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

   6. Applied rewrites 57.6%

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

   8. Applied rewrites 57.6%

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

Alternative 8: 69.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(\left|b\right|, \left|b\right|, \left(\mathsf{fma}\left(\left|b\right| \cdot \left|b\right|, -3.08641975308642 \cdot 10^{-5}, 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left|angle\right|\right)\right) \cdot \left|angle\right|\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left|b\right|\right)}^{2} \cdot {\sin \left(\pi \cdot \left(0.5 + 0.005555555555555556 \cdot \left|angle\right|\right)\right)}^{2}\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (if (<= (fabs b) 1.4e+77)
  (fma
   (fabs b)
   (fabs b)
   (*
    (*
     (fma
      (* (fabs b) (fabs b))
      -3.08641975308642e-5
      (* 3.08641975308642e-5 (* a a)))
     (* (* PI PI) (fabs angle)))
    (fabs angle)))
  (*
   (pow (fabs b) 2.0)
   (pow
    (sin (* PI (+ 0.5 (* 0.005555555555555556 (fabs angle)))))
    2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(b) <= 1.4e+77) {
		tmp = fma(fabs(b), fabs(b), ((fma((fabs(b) * fabs(b)), -3.08641975308642e-5, (3.08641975308642e-5 * (a * a))) * ((((double) M_PI) * ((double) M_PI)) * fabs(angle))) * fabs(angle)));
	} else {
		tmp = pow(fabs(b), 2.0) * pow(sin((((double) M_PI) * (0.5 + (0.005555555555555556 * fabs(angle))))), 2.0);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(b) <= 1.4e+77)
		tmp = fma(abs(b), abs(b), Float64(Float64(fma(Float64(abs(b) * abs(b)), -3.08641975308642e-5, Float64(3.08641975308642e-5 * Float64(a * a))) * Float64(Float64(pi * pi) * abs(angle))) * abs(angle)));
	else
		tmp = Float64((abs(b) ^ 2.0) * (sin(Float64(pi * Float64(0.5 + Float64(0.005555555555555556 * abs(angle))))) ^ 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[Abs[b], $MachinePrecision], 1.4e+77], N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision] + N[(N[(N[(N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * -3.08641975308642e-5 + N[(3.08641975308642e-5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Sin[N[(Pi * N[(0.5 + N[(0.005555555555555556 * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.4e77

   1. Initial program 80.5%

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

   2. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   4. Applied rewrites 41.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 41.5%

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

   6. Applied rewrites 41.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 43.9%

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

   8. Applied rewrites 43.9%

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

   if 1.4e77 < b

   1. Initial program 80.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-PI.f64 N/A

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

      7. mult-flip N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval 80.4%

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

   3. Applied rewrites 80.4%

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

   4. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 57.6%

         \[\leadsto {b}^{2} \cdot {\sin \left(\pi \cdot \left(0.5 + 0.005555555555555556 \cdot angle\right)\right)}^{2} \]

   6. Applied rewrites 57.6%

      \[\leadsto \color{blue}{{b}^{2} \cdot {\sin \left(\pi \cdot \left(0.5 + 0.005555555555555556 \cdot angle\right)\right)}^{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 69.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| \cdot \left|b\right|\\ \mathbf{if}\;\left|b\right| \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(\left|b\right|, \left|b\right|, \left(\mathsf{fma}\left(t\_0, -3.08641975308642 \cdot 10^{-5}, 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot angle\right)\right) \cdot angle\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* (fabs b) (fabs b))))
  (if (<= (fabs b) 1.4e+77)
    (fma
     (fabs b)
     (fabs b)
     (*
      (*
       (fma t_0 -3.08641975308642e-5 (* 3.08641975308642e-5 (* a a)))
       (* (* PI PI) angle))
      angle))
    (*
     t_0
     (+
      0.5
      (* 0.5 (cos (* 2.0 (* (* PI angle) 0.005555555555555556)))))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) * fabs(b);
	double tmp;
	if (fabs(b) <= 1.4e+77) {
		tmp = fma(fabs(b), fabs(b), ((fma(t_0, -3.08641975308642e-5, (3.08641975308642e-5 * (a * a))) * ((((double) M_PI) * ((double) M_PI)) * angle)) * angle));
	} else {
		tmp = t_0 * (0.5 + (0.5 * cos((2.0 * ((((double) M_PI) * angle) * 0.005555555555555556)))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) * abs(b))
	tmp = 0.0
	if (abs(b) <= 1.4e+77)
		tmp = fma(abs(b), abs(b), Float64(Float64(fma(t_0, -3.08641975308642e-5, Float64(3.08641975308642e-5 * Float64(a * a))) * Float64(Float64(pi * pi) * angle)) * angle));
	else
		tmp = Float64(t_0 * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(Float64(pi * angle) * 0.005555555555555556))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 1.4e+77], N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision] + N[(N[(N[(t$95$0 * -3.08641975308642e-5 + N[(3.08641975308642e-5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.4e77

   1. Initial program 80.5%

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

   2. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   4. Applied rewrites 41.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 41.5%

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

   6. Applied rewrites 41.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 43.9%

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

   8. Applied rewrites 43.9%

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

   if 1.4e77 < b

   1. Initial program 80.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 57.6%

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

   4. Applied rewrites 57.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 57.6%

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

   6. Applied rewrites 57.6%

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

   8. Applied rewrites 57.6%

      \[\leadsto \left(b \cdot b\right) \cdot \left(0.5 + \color{blue}{0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 68.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| \cdot \left|b\right|\\ \mathbf{if}\;\left|b\right| \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(\left|b\right|, \left|b\right|, \left(\mathsf{fma}\left(t\_0, -3.08641975308642 \cdot 10^{-5}, 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot angle\right)\right) \cdot angle\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {1}^{2}\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* (fabs b) (fabs b))))
  (if (<= (fabs b) 1.4e+77)
    (fma
     (fabs b)
     (fabs b)
     (*
      (*
       (fma t_0 -3.08641975308642e-5 (* 3.08641975308642e-5 (* a a)))
       (* (* PI PI) angle))
      angle))
    (* t_0 (pow 1.0 2.0)))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) * fabs(b);
	double tmp;
	if (fabs(b) <= 1.4e+77) {
		tmp = fma(fabs(b), fabs(b), ((fma(t_0, -3.08641975308642e-5, (3.08641975308642e-5 * (a * a))) * ((((double) M_PI) * ((double) M_PI)) * angle)) * angle));
	} else {
		tmp = t_0 * pow(1.0, 2.0);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) * abs(b))
	tmp = 0.0
	if (abs(b) <= 1.4e+77)
		tmp = fma(abs(b), abs(b), Float64(Float64(fma(t_0, -3.08641975308642e-5, Float64(3.08641975308642e-5 * Float64(a * a))) * Float64(Float64(pi * pi) * angle)) * angle));
	else
		tmp = Float64(t_0 * (1.0 ^ 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 1.4e+77], N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision] + N[(N[(N[(t$95$0 * -3.08641975308642e-5 + N[(3.08641975308642e-5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[1.0, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.4e77

   1. Initial program 80.5%

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

   2. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   4. Applied rewrites 41.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 41.5%

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

   6. Applied rewrites 41.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 43.9%

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

   8. Applied rewrites 43.9%

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

   if 1.4e77 < b

   1. Initial program 80.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 57.6%

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

   4. Applied rewrites 57.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 57.6%

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

   6. Applied rewrites 57.6%

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

   7. Taylor expanded in angle around 0

      \[\leadsto \left(b \cdot b\right) \cdot {1}^{2} \]
   8. Step-by-step derivation

   9. Applied rewrites 57.7%

      \[\leadsto \left(b \cdot b\right) \cdot {1}^{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 68.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| \cdot \left|b\right|\\ \mathbf{if}\;\left|b\right| \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(t\_0, -3.08641975308642 \cdot 10^{-5}, 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right) \cdot angle\right) \cdot angle, 9.869604401089358, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {1}^{2}\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* (fabs b) (fabs b))))
  (if (<= (fabs b) 1.4e+77)
    (fma
     (*
      (*
       (fma t_0 -3.08641975308642e-5 (* 3.08641975308642e-5 (* a a)))
       angle)
      angle)
     9.869604401089358
     t_0)
    (* t_0 (pow 1.0 2.0)))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) * fabs(b);
	double tmp;
	if (fabs(b) <= 1.4e+77) {
		tmp = fma(((fma(t_0, -3.08641975308642e-5, (3.08641975308642e-5 * (a * a))) * angle) * angle), 9.869604401089358, t_0);
	} else {
		tmp = t_0 * pow(1.0, 2.0);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) * abs(b))
	tmp = 0.0
	if (abs(b) <= 1.4e+77)
		tmp = fma(Float64(Float64(fma(t_0, -3.08641975308642e-5, Float64(3.08641975308642e-5 * Float64(a * a))) * angle) * angle), 9.869604401089358, t_0);
	else
		tmp = Float64(t_0 * (1.0 ^ 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 1.4e+77], N[(N[(N[(N[(t$95$0 * -3.08641975308642e-5 + N[(3.08641975308642e-5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * angle), $MachinePrecision] * 9.869604401089358 + t$95$0), $MachinePrecision], N[(t$95$0 * N[Power[1.0, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.4e77

   1. Initial program 80.5%

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

   2. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   4. Applied rewrites 41.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 41.5%

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

   6. Applied rewrites 41.5%

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

   7. Evaluated real constant 41.5%

      \[\leadsto \mathsf{fma}\left(b, b, \left(9.869604401089358 \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, b \cdot b, \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(angle \cdot angle\right)\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto b \cdot b + \color{blue}{\left(\frac{2778046668940015}{281474976710656} \cdot \mathsf{fma}\left(\frac{-1}{32400}, b \cdot b, \left(a \cdot a\right) \cdot \frac{1}{32400}\right)\right) \cdot \left(angle \cdot angle\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto b \cdot b + \color{blue}{\left(\frac{2778046668940015}{281474976710656} \cdot \mathsf{fma}\left(\frac{-1}{32400}, b \cdot b, \left(a \cdot a\right) \cdot \frac{1}{32400}\right)\right)} \cdot \left(angle \cdot angle\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{2778046668940015}{281474976710656} \cdot \mathsf{fma}\left(\frac{-1}{32400}, b \cdot b, \left(a \cdot a\right) \cdot \frac{1}{32400}\right)\right) \cdot \left(angle \cdot angle\right) + \color{blue}{b \cdot b} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\frac{2778046668940015}{281474976710656} \cdot \mathsf{fma}\left(\frac{-1}{32400}, b \cdot b, \left(a \cdot a\right) \cdot \frac{1}{32400}\right)\right) \cdot \left(angle \cdot angle\right) + \color{blue}{b} \cdot b \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{2778046668940015}{281474976710656} \cdot \mathsf{fma}\left(\frac{-1}{32400}, b \cdot b, \left(a \cdot a\right) \cdot \frac{1}{32400}\right)\right) \cdot \left(angle \cdot angle\right) + b \cdot b \]

      6. associate-*l* N/A

         \[\leadsto \frac{2778046668940015}{281474976710656} \cdot \left(\mathsf{fma}\left(\frac{-1}{32400}, b \cdot b, \left(a \cdot a\right) \cdot \frac{1}{32400}\right) \cdot \left(angle \cdot angle\right)\right) + \color{blue}{b} \cdot b \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{32400}, b \cdot b, \left(a \cdot a\right) \cdot \frac{1}{32400}\right) \cdot \left(angle \cdot angle\right)\right) \cdot \frac{2778046668940015}{281474976710656} + \color{blue}{b} \cdot b \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{32400}, b \cdot b, \left(a \cdot a\right) \cdot \frac{1}{32400}\right) \cdot \left(angle \cdot angle\right), \color{blue}{\frac{2778046668940015}{281474976710656}}, b \cdot b\right) \]

   9. Applied rewrites 43.9%

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(b \cdot b, -3.08641975308642 \cdot 10^{-5}, 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right) \cdot angle\right) \cdot angle, \color{blue}{9.869604401089358}, b \cdot b\right) \]

   if 1.4e77 < b

   1. Initial program 80.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 57.6%

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

   4. Applied rewrites 57.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 57.6%

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

   6. Applied rewrites 57.6%

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

   7. Taylor expanded in angle around 0

      \[\leadsto \left(b \cdot b\right) \cdot {1}^{2} \]
   8. Step-by-step derivation

   9. Applied rewrites 57.7%

      \[\leadsto \left(b \cdot b\right) \cdot {1}^{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 68.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| \cdot \left|b\right|\\ \mathbf{if}\;\left|b\right| \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(\left|b\right|, \left|b\right|, \left(\mathsf{fma}\left(0.00030461741978670857, a \cdot a, -0.00030461741978670857 \cdot t\_0\right) \cdot angle\right) \cdot angle\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {1}^{2}\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* (fabs b) (fabs b))))
  (if (<= (fabs b) 1.4e+77)
    (fma
     (fabs b)
     (fabs b)
     (*
      (*
       (fma
        0.00030461741978670857
        (* a a)
        (* -0.00030461741978670857 t_0))
       angle)
      angle))
    (* t_0 (pow 1.0 2.0)))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) * fabs(b);
	double tmp;
	if (fabs(b) <= 1.4e+77) {
		tmp = fma(fabs(b), fabs(b), ((fma(0.00030461741978670857, (a * a), (-0.00030461741978670857 * t_0)) * angle) * angle));
	} else {
		tmp = t_0 * pow(1.0, 2.0);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) * abs(b))
	tmp = 0.0
	if (abs(b) <= 1.4e+77)
		tmp = fma(abs(b), abs(b), Float64(Float64(fma(0.00030461741978670857, Float64(a * a), Float64(-0.00030461741978670857 * t_0)) * angle) * angle));
	else
		tmp = Float64(t_0 * (1.0 ^ 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 1.4e+77], N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision] + N[(N[(N[(0.00030461741978670857 * N[(a * a), $MachinePrecision] + N[(-0.00030461741978670857 * t$95$0), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[1.0, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.4e77

   1. Initial program 80.5%

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

   2. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   4. Applied rewrites 41.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 41.5%

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

   6. Applied rewrites 41.5%

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

   7. Evaluated real constant 41.5%

      \[\leadsto \mathsf{fma}\left(b, b, \left(9.869604401089358 \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, b \cdot b, \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(angle \cdot angle\right)\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, b, \left(\frac{2778046668940015}{281474976710656} \cdot \mathsf{fma}\left(\frac{-1}{32400}, b \cdot b, \left(a \cdot a\right) \cdot \frac{1}{32400}\right)\right) \cdot \left(angle \cdot angle\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, b, \left(\frac{2778046668940015}{281474976710656} \cdot \mathsf{fma}\left(\frac{-1}{32400}, b \cdot b, \left(a \cdot a\right) \cdot \frac{1}{32400}\right)\right) \cdot \left(angle \cdot angle\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(b, b, \left(\left(\frac{2778046668940015}{281474976710656} \cdot \mathsf{fma}\left(\frac{-1}{32400}, b \cdot b, \left(a \cdot a\right) \cdot \frac{1}{32400}\right)\right) \cdot angle\right) \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, b, \left(\left(\frac{2778046668940015}{281474976710656} \cdot \mathsf{fma}\left(\frac{-1}{32400}, b \cdot b, \left(a \cdot a\right) \cdot \frac{1}{32400}\right)\right) \cdot angle\right) \cdot angle\right) \]

   9. Applied rewrites 43.9%

      \[\leadsto \mathsf{fma}\left(b, b, \left(\mathsf{fma}\left(0.00030461741978670857, a \cdot a, -0.00030461741978670857 \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot angle\right) \]

   if 1.4e77 < b

   1. Initial program 80.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 57.6%

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

   4. Applied rewrites 57.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 57.6%

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

   6. Applied rewrites 57.6%

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

   7. Taylor expanded in angle around 0

      \[\leadsto \left(b \cdot b\right) \cdot {1}^{2} \]
   8. Step-by-step derivation

   9. Applied rewrites 57.7%

      \[\leadsto \left(b \cdot b\right) \cdot {1}^{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 57.7% accurate, 5.2× speedup

Math

\[\left(b \cdot b\right) \cdot {1}^{2} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (* (* b b) (pow 1.0 2.0)))

C

double code(double a, double b, double angle) {
	return (b * b) * pow(1.0, 2.0);
}

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 = (b * b) * (1.0d0 ** 2.0d0)
end function

Java

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

Python

def code(a, b, angle):
	return (b * b) * math.pow(1.0, 2.0)

Julia

function code(a, b, angle)
	return Float64(Float64(b * b) * (1.0 ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (b * b) * (1.0 ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[(b * b), $MachinePrecision] * N[Power[1.0, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.5%

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

2. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-PI.f64 57.6%

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

4. Applied rewrites 57.6%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 57.6%

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

6. Applied rewrites 57.6%

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

7. Taylor expanded in angle around 0

   \[\leadsto \left(b \cdot b\right) \cdot {1}^{2} \]
8. Step-by-step derivation

9. Applied rewrites 57.7%

   \[\leadsto \left(b \cdot b\right) \cdot {1}^{2} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))