ab-angle->ABCF A

Percentage Accurate: 79.6% → 78.9%

Time: 14.8s

Alternatives: 12

Speedup: N/A×

• 37.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 78.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{-18}:\\ \;\;\;\;b \cdot b + {\left(angle\_m \cdot \left(a \cdot \left(\pi \cdot \mathsf{fma}\left(\left(angle\_m \cdot angle\_m\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \pi \cdot \pi, 0.005555555555555556\right)\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \left(0.5 + -0.5 \cdot \cos \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right), b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= (/ angle_m 180.0) 5e-18)
   (+
    (* b b)
    (pow
     (*
      angle_m
      (*
       a
       (*
        PI
        (fma
         (* (* angle_m angle_m) -2.8577960676726107e-8)
         (* PI PI)
         0.005555555555555556))))
     2.0))
   (fma
    a
    (* a (+ 0.5 (* -0.5 (cos (* PI (* angle_m 0.011111111111111112))))))
    (* b b))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e-18) {
		tmp = (b * b) + pow((angle_m * (a * (((double) M_PI) * fma(((angle_m * angle_m) * -2.8577960676726107e-8), (((double) M_PI) * ((double) M_PI)), 0.005555555555555556)))), 2.0);
	} else {
		tmp = fma(a, (a * (0.5 + (-0.5 * cos((((double) M_PI) * (angle_m * 0.011111111111111112)))))), (b * b));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e-18)
		tmp = Float64(Float64(b * b) + (Float64(angle_m * Float64(a * Float64(pi * fma(Float64(Float64(angle_m * angle_m) * -2.8577960676726107e-8), Float64(pi * pi), 0.005555555555555556)))) ^ 2.0));
	else
		tmp = fma(a, Float64(a * Float64(0.5 + Float64(-0.5 * cos(Float64(pi * Float64(angle_m * 0.011111111111111112)))))), Float64(b * b));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e-18], N[(N[(b * b), $MachinePrecision] + N[Power[N[(angle$95$m * N[(a * N[(Pi * N[(N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * -2.8577960676726107e-8), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(a * N[(a * N[(0.5 + N[(-0.5 * N[Cos[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000036e-18

   1. Initial program 85.9%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 86.0%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. PI-lowering-PI.f64 N/A

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

      6. /-lowering-/.f64 86.1

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

   7. Applied egg-rr 86.1%

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

      1. *-rgt-identity N/A

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

      2. pow2 N/A

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

      3. *-lowering-*.f64 86.1

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

   9. Applied egg-rr 86.1%

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

   10. Taylor expanded in angle around 0

       \[\leadsto {\color{blue}{\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + \frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + b \cdot b \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto {\left(angle \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \frac{-1}{34992000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)}\right)}^{2} + b \cdot b \]

       2. *-commutative N/A

          \[\leadsto {\left(angle \cdot \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \frac{-1}{34992000} \cdot \left(a \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot {angle}^{2}\right)}\right)\right)\right)}^{2} + b \cdot b \]

       3. associate-*r* N/A

          \[\leadsto {\left(angle \cdot \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \frac{-1}{34992000} \cdot \color{blue}{\left(\left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {angle}^{2}\right)}\right)\right)}^{2} + b \cdot b \]

       4. associate-*l* N/A

          \[\leadsto {\left(angle \cdot \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{\left(\frac{-1}{34992000} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot {angle}^{2}}\right)\right)}^{2} + b \cdot b \]

       5. *-lowering-*.f64 N/A

          \[\leadsto {\color{blue}{\left(angle \cdot \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \left(\frac{-1}{34992000} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot {angle}^{2}\right)\right)}}^{2} + b \cdot b \]

       6. *-commutative N/A

          \[\leadsto {\left(angle \cdot \left(\frac{1}{180} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot a\right)} + \left(\frac{-1}{34992000} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot {angle}^{2}\right)\right)}^{2} + b \cdot b \]

       7. associate-*r* N/A

          \[\leadsto {\left(angle \cdot \left(\color{blue}{\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot a} + \left(\frac{-1}{34992000} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot {angle}^{2}\right)\right)}^{2} + b \cdot b \]

       8. associate-*l* N/A

          \[\leadsto {\left(angle \cdot \left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot a + \color{blue}{\frac{-1}{34992000} \cdot \left(\left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {angle}^{2}\right)}\right)\right)}^{2} + b \cdot b \]

       9. associate-*r* N/A

          \[\leadsto {\left(angle \cdot \left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot a + \frac{-1}{34992000} \cdot \color{blue}{\left(a \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot {angle}^{2}\right)\right)}\right)\right)}^{2} + b \cdot b \]

       10. *-commutative N/A

           \[\leadsto {\left(angle \cdot \left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot a + \frac{-1}{34992000} \cdot \left(a \cdot \color{blue}{\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right)\right)\right)}^{2} + b \cdot b \]

       11. *-commutative N/A

           \[\leadsto {\left(angle \cdot \left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot a + \frac{-1}{34992000} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot a\right)}\right)\right)}^{2} + b \cdot b \]

   12. Simplified 83.4%

       \[\leadsto {\color{blue}{\left(angle \cdot \left(a \cdot \left(\pi \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \pi \cdot \pi, 0.005555555555555556\right)\right)\right)\right)}}^{2} + b \cdot b \]

   if 5.00000000000000036e-18 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 69.6%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 69.9%

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

      1. div-inv N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-down N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

   7. Applied egg-rr 70.0%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-18}:\\ \;\;\;\;b \cdot b + {\left(angle \cdot \left(a \cdot \left(\pi \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \pi \cdot \pi, 0.005555555555555556\right)\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \left(0.5 + -0.5 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right), b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.5% accurate, 1.9× speedup

Math

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

FPCore

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

Python

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

Julia

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

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((a * sin((pi / (180.0 / angle_m)))) ^ 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[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.2%

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

3. Taylor expanded in angle around 0

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

5. Simplified 82.4%

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

   1. *-commutative N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

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

   5. PI-lowering-PI.f64 N/A

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

   6. /-lowering-/.f64 82.4

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

7. Applied egg-rr 82.4%

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

   1. *-rgt-identity N/A

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

   2. pow2 N/A

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

   3. *-lowering-*.f64 82.4

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

9. Applied egg-rr 82.4%

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

Alternative 3: 79.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.2%

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

3. Taylor expanded in angle around 0

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

5. Simplified 82.4%

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

   1. *-commutative N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

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

   5. PI-lowering-PI.f64 N/A

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

   6. /-lowering-/.f64 82.4

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

7. Applied egg-rr 82.4%

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

   1. *-rgt-identity N/A

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

   2. pow2 N/A

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

   3. *-lowering-*.f64 82.4

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

9. Applied egg-rr 82.4%

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

    1. *-commutative N/A

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

    2. associate-/r/ N/A

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

    3. div-inv N/A

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

    4. metadata-eval N/A

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

    5. *-commutative N/A

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

    6. *-commutative N/A

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

    7. associate-*l* N/A

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

    8. *-lowering-*.f64 N/A

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

    9. associate-*l* N/A

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

    10. *-commutative N/A

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

    11. sin-lowering-sin.f64 N/A

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

    12. associate-*r* N/A

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

    13. *-commutative N/A

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

    14. *-commutative N/A

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

    15. *-lowering-*.f64 N/A

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

    16. *-lowering-*.f64 N/A

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

    17. PI-lowering-PI.f64 82.4

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

11. Applied egg-rr 82.4%

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

12. Final simplification 82.4%

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

Alternative 4: 78.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{-18}:\\ \;\;\;\;b \cdot b + {\left(0.005555555555555556 \cdot \left(angle\_m \cdot \left(a \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \left(0.5 + -0.5 \cdot \cos \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right), b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= (/ angle_m 180.0) 5e-18)
   (+ (* b b) (pow (* 0.005555555555555556 (* angle_m (* a PI))) 2.0))
   (fma
    a
    (* a (+ 0.5 (* -0.5 (cos (* PI (* angle_m 0.011111111111111112))))))
    (* b b))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e-18) {
		tmp = (b * b) + pow((0.005555555555555556 * (angle_m * (a * ((double) M_PI)))), 2.0);
	} else {
		tmp = fma(a, (a * (0.5 + (-0.5 * cos((((double) M_PI) * (angle_m * 0.011111111111111112)))))), (b * b));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e-18)
		tmp = Float64(Float64(b * b) + (Float64(0.005555555555555556 * Float64(angle_m * Float64(a * pi))) ^ 2.0));
	else
		tmp = fma(a, Float64(a * Float64(0.5 + Float64(-0.5 * cos(Float64(pi * Float64(angle_m * 0.011111111111111112)))))), Float64(b * b));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e-18], N[(N[(b * b), $MachinePrecision] + N[Power[N[(0.005555555555555556 * N[(angle$95$m * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(a * N[(a * N[(0.5 + N[(-0.5 * N[Cos[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000036e-18

   1. Initial program 85.9%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 86.0%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. PI-lowering-PI.f64 N/A

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

      6. /-lowering-/.f64 86.1

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

   7. Applied egg-rr 86.1%

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

      1. *-rgt-identity N/A

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

      2. pow2 N/A

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

      3. *-lowering-*.f64 86.1

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

   9. Applied egg-rr 86.1%

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

   10. Taylor expanded in angle around 0

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

       1. *-lowering-*.f64 N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 N/A

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

       8. PI-lowering-PI.f64 84.0

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

   12. Simplified 84.0%

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

   if 5.00000000000000036e-18 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 69.6%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 69.9%

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

      1. div-inv N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-down N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

   7. Applied egg-rr 70.0%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-18}:\\ \;\;\;\;b \cdot b + {\left(0.005555555555555556 \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \left(0.5 + -0.5 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right), b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{-18}:\\ \;\;\;\;b \cdot b + {\left(0.005555555555555556 \cdot \left(angle\_m \cdot \left(a \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\_m\right) \cdot 0.011111111111111112\right), -0.5, 0.5\right), a, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= (/ angle_m 180.0) 5e-18)
   (+ (* b b) (pow (* 0.005555555555555556 (* angle_m (* a PI))) 2.0))
   (fma
    (* a (fma (cos (* (* PI angle_m) 0.011111111111111112)) -0.5 0.5))
    a
    (* b b))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e-18) {
		tmp = (b * b) + pow((0.005555555555555556 * (angle_m * (a * ((double) M_PI)))), 2.0);
	} else {
		tmp = fma((a * fma(cos(((((double) M_PI) * angle_m) * 0.011111111111111112)), -0.5, 0.5)), a, (b * b));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e-18)
		tmp = Float64(Float64(b * b) + (Float64(0.005555555555555556 * Float64(angle_m * Float64(a * pi))) ^ 2.0));
	else
		tmp = fma(Float64(a * fma(cos(Float64(Float64(pi * angle_m) * 0.011111111111111112)), -0.5, 0.5)), a, Float64(b * b));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e-18], N[(N[(b * b), $MachinePrecision] + N[Power[N[(0.005555555555555556 * N[(angle$95$m * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[Cos[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000036e-18

   1. Initial program 85.9%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 86.0%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. PI-lowering-PI.f64 N/A

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

      6. /-lowering-/.f64 86.1

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

   7. Applied egg-rr 86.1%

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

      1. *-rgt-identity N/A

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

      2. pow2 N/A

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

      3. *-lowering-*.f64 86.1

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

   9. Applied egg-rr 86.1%

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

   10. Taylor expanded in angle around 0

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

       1. *-lowering-*.f64 N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 N/A

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

       8. PI-lowering-PI.f64 84.0

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

   12. Simplified 84.0%

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

   if 5.00000000000000036e-18 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 69.6%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 69.9%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. PI-lowering-PI.f64 N/A

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

      6. /-lowering-/.f64 70.1

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

   7. Applied egg-rr 70.1%

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

      1. *-rgt-identity N/A

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

      2. pow2 N/A

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

      3. *-lowering-*.f64 70.1

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

   9. Applied egg-rr 70.1%

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

   11. Applied egg-rr 70.1%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-18}:\\ \;\;\;\;b \cdot b + {\left(0.005555555555555556 \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), -0.5, 0.5\right), a, b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 9.2 \cdot 10^{-90}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + {\left(0.005555555555555556 \cdot \left(angle\_m \cdot \left(a \cdot \pi\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 9.2e-90)
   (* b b)
   (+ (* b b) (pow (* 0.005555555555555556 (* angle_m (* a PI))) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 9.2e-90) {
		tmp = b * b;
	} else {
		tmp = (b * b) + pow((0.005555555555555556 * (angle_m * (a * ((double) M_PI)))), 2.0);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 9.2e-90) {
		tmp = b * b;
	} else {
		tmp = (b * b) + Math.pow((0.005555555555555556 * (angle_m * (a * Math.PI))), 2.0);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 9.2e-90:
		tmp = b * b
	else:
		tmp = (b * b) + math.pow((0.005555555555555556 * (angle_m * (a * math.pi))), 2.0)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 9.2e-90)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(b * b) + (Float64(0.005555555555555556 * Float64(angle_m * Float64(a * pi))) ^ 2.0));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 9.2e-90)
		tmp = b * b;
	else
		tmp = (b * b) + ((0.005555555555555556 * (angle_m * (a * pi))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 9.2e-90], N[(b * b), $MachinePrecision], N[(N[(b * b), $MachinePrecision] + N[Power[N[(0.005555555555555556 * N[(angle$95$m * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 9.1999999999999992e-90

   1. Initial program 81.3%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 63.2

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

   5. Simplified 63.2%

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

   if 9.1999999999999992e-90 < a

   1. Initial program 84.0%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 84.0%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. PI-lowering-PI.f64 N/A

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

      6. /-lowering-/.f64 84.1

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

   7. Applied egg-rr 84.1%

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

      1. *-rgt-identity N/A

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

      2. pow2 N/A

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

      3. *-lowering-*.f64 84.1

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

   9. Applied egg-rr 84.1%

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

   10. Taylor expanded in angle around 0

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

       1. *-lowering-*.f64 N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 N/A

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

       8. PI-lowering-PI.f64 81.2

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

   12. Simplified 81.2%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 9.2 \cdot 10^{-90}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + {\left(0.005555555555555556 \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.6% accurate, 10.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 7.8 \cdot 10^{-90}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 3.08641975308642 \cdot 10^{-5}, \left(a \cdot angle\_m\right) \cdot \left(\pi \cdot \left(\pi \cdot angle\_m\right)\right), b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 7.8e-90)
   (* b b)
   (fma
    (* a 3.08641975308642e-5)
    (* (* a angle_m) (* PI (* PI angle_m)))
    (* b b))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 7.8e-90) {
		tmp = b * b;
	} else {
		tmp = fma((a * 3.08641975308642e-5), ((a * angle_m) * (((double) M_PI) * (((double) M_PI) * angle_m))), (b * b));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 7.8e-90)
		tmp = Float64(b * b);
	else
		tmp = fma(Float64(a * 3.08641975308642e-5), Float64(Float64(a * angle_m) * Float64(pi * Float64(pi * angle_m))), Float64(b * b));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 7.8e-90], N[(b * b), $MachinePrecision], N[(N[(a * 3.08641975308642e-5), $MachinePrecision] * N[(N[(a * angle$95$m), $MachinePrecision] * N[(Pi * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7.80000000000000009e-90

   1. Initial program 81.3%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 63.2

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

   5. Simplified 63.2%

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

   if 7.80000000000000009e-90 < a

   1. Initial program 84.0%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 84.0%

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

   6. Taylor expanded in angle around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. PI-lowering-PI.f64 N/A

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

      12. PI-lowering-PI.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 70.0

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

   8. Simplified 70.0%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. PI-lowering-PI.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. PI-lowering-PI.f64 N/A

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

      15. *-lowering-*.f64 81.2

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

   10. Applied egg-rr 81.2%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7.8 \cdot 10^{-90}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 3.08641975308642 \cdot 10^{-5}, \left(a \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot angle\right)\right), b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.5% accurate, 10.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.02 \cdot 10^{-89}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, a \cdot \left(\left(a \cdot angle\_m\right) \cdot \left(\pi \cdot \left(\pi \cdot angle\_m\right)\right)\right), b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 1.02e-89)
   (* b b)
   (fma
    3.08641975308642e-5
    (* a (* (* a angle_m) (* PI (* PI angle_m))))
    (* b b))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.02e-89) {
		tmp = b * b;
	} else {
		tmp = fma(3.08641975308642e-5, (a * ((a * angle_m) * (((double) M_PI) * (((double) M_PI) * angle_m)))), (b * b));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 1.02e-89)
		tmp = Float64(b * b);
	else
		tmp = fma(3.08641975308642e-5, Float64(a * Float64(Float64(a * angle_m) * Float64(pi * Float64(pi * angle_m)))), Float64(b * b));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 1.02e-89], N[(b * b), $MachinePrecision], N[(3.08641975308642e-5 * N[(a * N[(N[(a * angle$95$m), $MachinePrecision] * N[(Pi * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.0199999999999999e-89

   1. Initial program 81.3%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 63.2

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

   5. Simplified 63.2%

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

   if 1.0199999999999999e-89 < a

   1. Initial program 84.0%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 84.0%

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

   6. Taylor expanded in angle around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. PI-lowering-PI.f64 N/A

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

      12. PI-lowering-PI.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 70.0

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

   8. Simplified 70.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. PI-lowering-PI.f64 N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 N/A

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

      13. PI-lowering-PI.f64 81.2

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

   10. Applied egg-rr 81.2%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.02 \cdot 10^{-89}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, a \cdot \left(\left(a \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot angle\right)\right)\right), b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.4% accurate, 10.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot \left(\left(angle\_m \cdot angle\_m\right) \cdot \left(\pi \cdot \pi\right)\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 4.4e+111)
   (fma
    3.08641975308642e-5
    (* (* a a) (* (* angle_m angle_m) (* PI PI)))
    (* b b))
   (* b b)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 4.4e+111) {
		tmp = fma(3.08641975308642e-5, ((a * a) * ((angle_m * angle_m) * (((double) M_PI) * ((double) M_PI)))), (b * b));
	} else {
		tmp = b * b;
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 4.4e+111)
		tmp = fma(3.08641975308642e-5, Float64(Float64(a * a) * Float64(Float64(angle_m * angle_m) * Float64(pi * pi))), Float64(b * b));
	else
		tmp = Float64(b * b);
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 4.4e+111], N[(3.08641975308642e-5 * N[(N[(a * a), $MachinePrecision] * N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(b * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.39999999999999997e111

   1. Initial program 80.0%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 80.2%

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

   6. Taylor expanded in angle around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. PI-lowering-PI.f64 N/A

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

      12. PI-lowering-PI.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 68.4

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

   8. Simplified 68.4%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. PI-lowering-PI.f64 N/A

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

      6. PI-lowering-PI.f64 N/A

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

      7. *-lowering-*.f64 68.5

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

   10. Applied egg-rr 68.5%

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

   if 4.39999999999999997e111 < b

   1. Initial program 92.8%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 92.7

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

   5. Simplified 92.7%

      \[\leadsto \color{blue}{b \cdot b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.4% accurate, 10.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 6.9 \cdot 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot \left(angle\_m \cdot \left(angle\_m \cdot \left(\pi \cdot \pi\right)\right)\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 6.9e+112)
   (fma
    3.08641975308642e-5
    (* (* a a) (* angle_m (* angle_m (* PI PI))))
    (* b b))
   (* b b)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 6.9e+112) {
		tmp = fma(3.08641975308642e-5, ((a * a) * (angle_m * (angle_m * (((double) M_PI) * ((double) M_PI))))), (b * b));
	} else {
		tmp = b * b;
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 6.9e+112)
		tmp = fma(3.08641975308642e-5, Float64(Float64(a * a) * Float64(angle_m * Float64(angle_m * Float64(pi * pi)))), Float64(b * b));
	else
		tmp = Float64(b * b);
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 6.9e+112], N[(3.08641975308642e-5 * N[(N[(a * a), $MachinePrecision] * N[(angle$95$m * N[(angle$95$m * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(b * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.8999999999999999e112

   1. Initial program 80.0%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 80.2%

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

   6. Taylor expanded in angle around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. PI-lowering-PI.f64 N/A

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

      12. PI-lowering-PI.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 68.4

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

   8. Simplified 68.4%

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

   if 6.8999999999999999e112 < b

   1. Initial program 92.8%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 92.7

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

   5. Simplified 92.7%

      \[\leadsto \color{blue}{b \cdot b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 61.1% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 3.3 \cdot 10^{+143}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(angle\_m \cdot \left(angle\_m \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 3.3e+143)
   (* b b)
   (* (* angle_m (* angle_m (* PI PI))) (* 3.08641975308642e-5 (* a a)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 3.3e+143) {
		tmp = b * b;
	} else {
		tmp = (angle_m * (angle_m * (((double) M_PI) * ((double) M_PI)))) * (3.08641975308642e-5 * (a * a));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 3.3e+143) {
		tmp = b * b;
	} else {
		tmp = (angle_m * (angle_m * (Math.PI * Math.PI))) * (3.08641975308642e-5 * (a * a));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 3.3e+143:
		tmp = b * b
	else:
		tmp = (angle_m * (angle_m * (math.pi * math.pi))) * (3.08641975308642e-5 * (a * a))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 3.3e+143)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(angle_m * Float64(angle_m * Float64(pi * pi))) * Float64(3.08641975308642e-5 * Float64(a * a)));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 3.3e+143)
		tmp = b * b;
	else
		tmp = (angle_m * (angle_m * (pi * pi))) * (3.08641975308642e-5 * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 3.3e+143], N[(b * b), $MachinePrecision], N[(N[(angle$95$m * N[(angle$95$m * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(3.08641975308642e-5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.3e143

   1. Initial program 78.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 61.8

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

   5. Simplified 61.8%

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

   if 3.3e143 < a

   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. Add Preprocessing

   3. Taylor expanded in angle around 0

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

   5. Simplified 99.7%

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

   6. Taylor expanded in angle around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. PI-lowering-PI.f64 N/A

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

      12. PI-lowering-PI.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 76.2

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

   8. Simplified 76.2%

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

   9. Taylor expanded in a around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. PI-lowering-PI.f64 N/A

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

       11. PI-lowering-PI.f64 N/A

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

       12. *-commutative N/A

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

       13. *-lowering-*.f64 N/A

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

       14. unpow2 N/A

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

       15. *-lowering-*.f64 76.2

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

   11. Simplified 76.2%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.3 \cdot 10^{+143}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.3% accurate, 74.7× 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 = abs(angle)
real(8) function code(a, b, angle_m)
    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 82.2%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 57.0

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

5. Simplified 57.0%

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

Reproduce

herbie shell --seed 2024198 
(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)))