ab-angle->ABCF A

Percentage Accurate: 79.7% → 79.6%

Time: 4.0s

Alternatives: 7

Speedup: N/A×

• 36.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;angle\_m \leq 7.5 \cdot 10^{-26}:\\ \;\;\;\;{\left(\left(\left(angle\_m \cdot \pi\right) \cdot a\right) \cdot 0.005555555555555556\right)}^{2} + b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \frac{angle\_m}{180}\right)\right)\right) \cdot a, a, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= angle_m 7.5e-26)
   (+ (pow (* (* (* angle_m PI) a) 0.005555555555555556) 2.0) (* b b))
   (fma
    (* (- 0.5 (* 0.5 (cos (* 2.0 (* PI (/ angle_m 180.0)))))) a)
    a
    (* b b))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 7.5e-26) {
		tmp = pow((((angle_m * ((double) M_PI)) * a) * 0.005555555555555556), 2.0) + (b * b);
	} else {
		tmp = fma(((0.5 - (0.5 * cos((2.0 * (((double) M_PI) * (angle_m / 180.0)))))) * a), a, (b * b));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 7.5e-26)
		tmp = Float64((Float64(Float64(Float64(angle_m * pi) * a) * 0.005555555555555556) ^ 2.0) + Float64(b * b));
	else
		tmp = fma(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(pi * Float64(angle_m / 180.0)))))) * a), a, Float64(b * b));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[angle$95$m, 7.5e-26], N[(N[Power[N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * a), $MachinePrecision] * 0.005555555555555556), $MachinePrecision], 2.0], $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 7.4999999999999994e-26

   1. Initial program 99.7%

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

   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. unpow2 N/A

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

      2. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in angle around inf

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

      1. lift-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-PI.f64 99.7

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

   10. Applied rewrites 99.7%

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

   if 7.4999999999999994e-26 < angle

   1. Initial program 63.0%

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

   2. 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. unpow2 N/A

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

      2. lower-*.f64 62.9

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

   4. Applied rewrites 62.9%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 60.3%

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

Alternative 2: 78.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;angle\_m \leq 7.5 \cdot 10^{-26}:\\ \;\;\;\;{\left(\left(\left(angle\_m \cdot \pi\right) \cdot a\right) \cdot 0.005555555555555556\right)}^{2} + b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 - \cos \left(\left(0.011111111111111112 \cdot angle\_m\right) \cdot \pi\right) \cdot 0.5\right) \cdot a, a, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= angle_m 7.5e-26)
   (+ (pow (* (* (* angle_m PI) a) 0.005555555555555556) 2.0) (* b b))
   (fma
    (* (- 0.5 (* (cos (* (* 0.011111111111111112 angle_m) PI)) 0.5)) a)
    a
    (* b b))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 7.5e-26) {
		tmp = pow((((angle_m * ((double) M_PI)) * a) * 0.005555555555555556), 2.0) + (b * b);
	} else {
		tmp = fma(((0.5 - (cos(((0.011111111111111112 * angle_m) * ((double) M_PI))) * 0.5)) * a), a, (b * b));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 7.5e-26)
		tmp = Float64((Float64(Float64(Float64(angle_m * pi) * a) * 0.005555555555555556) ^ 2.0) + Float64(b * b));
	else
		tmp = fma(Float64(Float64(0.5 - Float64(cos(Float64(Float64(0.011111111111111112 * angle_m) * pi)) * 0.5)) * a), a, Float64(b * b));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[angle$95$m, 7.5e-26], N[(N[Power[N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * a), $MachinePrecision] * 0.005555555555555556), $MachinePrecision], 2.0], $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 - N[(N[Cos[N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 7.4999999999999994e-26

   1. Initial program 99.7%

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

   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. unpow2 N/A

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

      2. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in angle around inf

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

      1. lift-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-PI.f64 99.7

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

   10. Applied rewrites 99.7%

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

   if 7.4999999999999994e-26 < angle

   1. Initial program 63.0%

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

   2. 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. unpow2 N/A

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

      2. lower-*.f64 62.9

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

   4. Applied rewrites 62.9%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 60.3%

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

   7. Taylor expanded in angle around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-PI.f64 60.2

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

   9. Applied rewrites 60.2%

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lift-PI.f64 60.3

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

   11. Applied rewrites 60.3%

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

Alternative 3: 78.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;angle\_m \leq 7.5 \cdot 10^{-26}:\\ \;\;\;\;{\left(\left(\left(angle\_m \cdot \pi\right) \cdot a\right) \cdot 0.005555555555555556\right)}^{2} + b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 - \cos \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot 0.5\right) \cdot a, a, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= angle_m 7.5e-26)
   (+ (pow (* (* (* angle_m PI) a) 0.005555555555555556) 2.0) (* b b))
   (fma
    (* (- 0.5 (* (cos (* 0.011111111111111112 (* angle_m PI))) 0.5)) a)
    a
    (* b b))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 7.5e-26) {
		tmp = pow((((angle_m * ((double) M_PI)) * a) * 0.005555555555555556), 2.0) + (b * b);
	} else {
		tmp = fma(((0.5 - (cos((0.011111111111111112 * (angle_m * ((double) M_PI)))) * 0.5)) * a), a, (b * b));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 7.5e-26)
		tmp = Float64((Float64(Float64(Float64(angle_m * pi) * a) * 0.005555555555555556) ^ 2.0) + Float64(b * b));
	else
		tmp = fma(Float64(Float64(0.5 - Float64(cos(Float64(0.011111111111111112 * Float64(angle_m * pi))) * 0.5)) * a), a, Float64(b * b));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[angle$95$m, 7.5e-26], N[(N[Power[N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * a), $MachinePrecision] * 0.005555555555555556), $MachinePrecision], 2.0], $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 - N[(N[Cos[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 7.4999999999999994e-26

   1. Initial program 99.7%

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

   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. unpow2 N/A

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

      2. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in angle around inf

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

      1. lift-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-PI.f64 99.7

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

   10. Applied rewrites 99.7%

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

   if 7.4999999999999994e-26 < angle

   1. Initial program 63.0%

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

   2. 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. unpow2 N/A

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

      2. lower-*.f64 62.9

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

   4. Applied rewrites 62.9%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 60.3%

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

   7. Taylor expanded in angle around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-PI.f64 60.2

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

   9. Applied rewrites 60.2%

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

Alternative 4: 78.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \sin \left(\left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\right)\right)}^{2} + b \cdot b \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

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. unpow2 N/A

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

   2. lower-*.f64 79.7

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

4. Applied rewrites 79.7%

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

5. Taylor expanded in angle around inf

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

   1. lift-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-*.f64 79.6

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

7. Applied rewrites 79.6%

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

Alternative 5: 66.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.15 \cdot 10^{-45}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(angle\_m \cdot \pi\right) \cdot a\right) \cdot 0.005555555555555556\right)}^{2} + b \cdot b\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 1.15e-45)
   (* b b)
   (+ (pow (* (* (* angle_m PI) a) 0.005555555555555556) 2.0) (* b b))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.15e-45) {
		tmp = b * b;
	} else {
		tmp = pow((((angle_m * ((double) M_PI)) * a) * 0.005555555555555556), 2.0) + (b * b);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.15e-45) {
		tmp = b * b;
	} else {
		tmp = Math.pow((((angle_m * Math.PI) * a) * 0.005555555555555556), 2.0) + (b * b);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 1.15e-45:
		tmp = b * b
	else:
		tmp = math.pow((((angle_m * math.pi) * a) * 0.005555555555555556), 2.0) + (b * b)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 1.15e-45)
		tmp = Float64(b * b);
	else
		tmp = Float64((Float64(Float64(Float64(angle_m * pi) * a) * 0.005555555555555556) ^ 2.0) + Float64(b * b));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 1.15e-45)
		tmp = b * b;
	else
		tmp = ((((angle_m * pi) * a) * 0.005555555555555556) ^ 2.0) + (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 1.15e-45], N[(b * b), $MachinePrecision], N[(N[Power[N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * a), $MachinePrecision] * 0.005555555555555556), $MachinePrecision], 2.0], $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.14999999999999996e-45

   1. Initial program 78.4%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.6

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

   4. Applied rewrites 61.6%

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

   if 1.14999999999999996e-45 < a

   1. Initial program 83.1%

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

   2. 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. unpow2 N/A

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

      2. lower-*.f64 83.1

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

   4. Applied rewrites 83.1%

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

   5. Taylor expanded in angle around inf

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

      1. lift-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 83.0

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

   7. Applied rewrites 83.0%

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

   8. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-PI.f64 79.8

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

   10. Applied rewrites 79.8%

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

Alternative 6: 64.8% accurate, 4.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.15 \cdot 10^{-45}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(angle\_m \cdot angle\_m\right)\right), a, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 1.15e-45)
   (* b b)
   (fma
    (* (* 3.08641975308642e-5 a) (* (* PI PI) (* angle_m angle_m)))
    a
    (* b b))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.15e-45) {
		tmp = b * b;
	} else {
		tmp = fma(((3.08641975308642e-5 * a) * ((((double) M_PI) * ((double) M_PI)) * (angle_m * angle_m))), a, (b * b));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 1.15e-45)
		tmp = Float64(b * b);
	else
		tmp = fma(Float64(Float64(3.08641975308642e-5 * a) * Float64(Float64(pi * pi) * Float64(angle_m * angle_m))), a, Float64(b * b));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 1.15e-45], N[(b * b), $MachinePrecision], N[(N[(N[(3.08641975308642e-5 * a), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.14999999999999996e-45

   1. Initial program 78.4%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.6

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

   4. Applied rewrites 61.6%

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

   if 1.14999999999999996e-45 < a

   1. Initial program 83.1%

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

   2. 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. unpow2 N/A

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

      2. lower-*.f64 83.1

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

   4. Applied rewrites 83.1%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 62.3%

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

   7. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 72.9

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

   9. Applied rewrites 72.9%

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

Alternative 7: 56.5% accurate, 29.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

2. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 56.5

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

4. Applied rewrites 56.5%

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

Reproduce

herbie shell --seed 2025115 
(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)))