ab-angle->ABCF A

Percentage Accurate: 79.8% → 79.8%

Time: 10.3s

Alternatives: 9

Speedup: 1.0×

• 36.6% 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: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\pi, 0.005555555555555556 \cdot angle, \frac{\pi}{2}\right)\right)\right)}^{2} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-PI.f64 79.8

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

3. Applied rewrites 79.8%

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

4. Taylor expanded in angle around 0

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

   1. lower-*.f64 79.8

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

6. Applied rewrites 79.8%

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

7. Taylor expanded in angle around 0

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

   1. lower-*.f64 79.8

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

9. Applied rewrites 79.8%

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

Alternative 2: 79.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 angle) 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 = (0.005555555555555556 * angle) * ((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 = (0.005555555555555556 * angle) * 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 = (0.005555555555555556 * angle) * 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(0.005555555555555556 * angle) * 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 = (0.005555555555555556 * angle) * 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[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[Power[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.8%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 79.8

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

4. Applied rewrites 79.8%

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

5. Taylor expanded in angle around 0

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

   1. lower-*.f64 79.8

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

7. Applied rewrites 79.8%

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

Alternative 3: 79.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

2. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 79.7

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

4. Applied rewrites 79.7%

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

5. Taylor expanded in angle around 0

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

   1. lower-*.f64 79.8

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

7. Applied rewrites 79.8%

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

Alternative 4: 76.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 0.165:\\ \;\;\;\;{\left(a \cdot \left(\mathsf{fma}\left(0.005555555555555556, \pi, \left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot angle\right)\right)}^{2} + b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 - \cos \left(\left(\pi + \pi\right) \cdot \frac{angle}{180}\right) \cdot 0.5\right) \cdot a, a, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 0.165)
   (+
    (pow
     (*
      a
      (*
       (fma
        0.005555555555555556
        PI
        (* (* -2.8577960676726107e-8 (* angle angle)) (* (* PI PI) PI)))
       angle))
     2.0)
    (* b b))
   (fma (* (- 0.5 (* (cos (* (+ PI PI) (/ angle 180.0))) 0.5)) a) a (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 0.165) {
		tmp = pow((a * (fma(0.005555555555555556, ((double) M_PI), ((-2.8577960676726107e-8 * (angle * angle)) * ((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)))) * angle)), 2.0) + (b * b);
	} else {
		tmp = fma(((0.5 - (cos(((((double) M_PI) + ((double) M_PI)) * (angle / 180.0))) * 0.5)) * a), a, (b * b));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 0.165)
		tmp = Float64((Float64(a * Float64(fma(0.005555555555555556, pi, Float64(Float64(-2.8577960676726107e-8 * Float64(angle * angle)) * Float64(Float64(pi * pi) * pi))) * angle)) ^ 2.0) + Float64(b * b));
	else
		tmp = fma(Float64(Float64(0.5 - Float64(cos(Float64(Float64(pi + pi) * Float64(angle / 180.0))) * 0.5)) * a), a, Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 0.165], N[(N[Power[N[(a * N[(N[(0.005555555555555556 * Pi + N[(N[(-2.8577960676726107e-8 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 - N[(N[Cos[N[(N[(Pi + Pi), $MachinePrecision] * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 0.165000000000000008

   1. Initial program 86.5%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 86.4

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

   4. Applied rewrites 86.4%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 82.1%

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

   if 0.165000000000000008 < angle

   1. Initial program 60.9%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.0

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

   4. Applied rewrites 61.0%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 60.9%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. cos-neg-rev N/A

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

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

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

      8. lower-sin.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-PI.f64 61.0

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

   8. Applied rewrites 61.0%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   10. Applied rewrites 60.9%

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

Alternative 5: 76.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 0.165:\\ \;\;\;\;{\left(a \cdot \left(\mathsf{fma}\left(0.005555555555555556, \pi, \left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot angle\right)\right)}^{2} + b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 - \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot 0.5\right) \cdot a, a, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 0.165)
   (+
    (pow
     (*
      a
      (*
       (fma
        0.005555555555555556
        PI
        (* (* -2.8577960676726107e-8 (* angle angle)) (* (* PI PI) PI)))
       angle))
     2.0)
    (* b b))
   (fma
    (* (- 0.5 (* (cos (* 0.011111111111111112 (* PI angle))) 0.5)) a)
    a
    (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 0.165) {
		tmp = pow((a * (fma(0.005555555555555556, ((double) M_PI), ((-2.8577960676726107e-8 * (angle * angle)) * ((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)))) * angle)), 2.0) + (b * b);
	} else {
		tmp = fma(((0.5 - (cos((0.011111111111111112 * (((double) M_PI) * angle))) * 0.5)) * a), a, (b * b));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 0.165)
		tmp = Float64((Float64(a * Float64(fma(0.005555555555555556, pi, Float64(Float64(-2.8577960676726107e-8 * Float64(angle * angle)) * Float64(Float64(pi * pi) * pi))) * angle)) ^ 2.0) + Float64(b * b));
	else
		tmp = fma(Float64(Float64(0.5 - Float64(cos(Float64(0.011111111111111112 * Float64(pi * angle))) * 0.5)) * a), a, Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 0.165], N[(N[Power[N[(a * N[(N[(0.005555555555555556 * Pi + N[(N[(-2.8577960676726107e-8 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 - N[(N[Cos[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 0.165000000000000008

   1. Initial program 86.5%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 86.4

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

   4. Applied rewrites 86.4%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 82.1%

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

   if 0.165000000000000008 < angle

   1. Initial program 60.9%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.0

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

   4. Applied rewrites 61.0%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 60.9%

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

   7. Taylor expanded in angle around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 60.9

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

   9. Applied rewrites 60.9%

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

Alternative 6: 76.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 3.6:\\ \;\;\;\;{\left(a \cdot \left(\mathsf{fma}\left(0.005555555555555556, \pi, \left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot angle\right)\right)}^{2} + b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \sin \left(-0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right) \cdot a, a, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 3.6)
   (+
    (pow
     (*
      a
      (*
       (fma
        0.005555555555555556
        PI
        (* (* -2.8577960676726107e-8 (* angle angle)) (* (* PI PI) PI)))
       angle))
     2.0)
    (* b b))
   (fma
    (* (- 0.5 (* 0.5 (sin (* -0.011111111111111112 (* PI angle))))) a)
    a
    (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 3.6) {
		tmp = pow((a * (fma(0.005555555555555556, ((double) M_PI), ((-2.8577960676726107e-8 * (angle * angle)) * ((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)))) * angle)), 2.0) + (b * b);
	} else {
		tmp = fma(((0.5 - (0.5 * sin((-0.011111111111111112 * (((double) M_PI) * angle))))) * a), a, (b * b));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 3.6)
		tmp = Float64((Float64(a * Float64(fma(0.005555555555555556, pi, Float64(Float64(-2.8577960676726107e-8 * Float64(angle * angle)) * Float64(Float64(pi * pi) * pi))) * angle)) ^ 2.0) + Float64(b * b));
	else
		tmp = fma(Float64(Float64(0.5 - Float64(0.5 * sin(Float64(-0.011111111111111112 * Float64(pi * angle))))) * a), a, Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 3.6], N[(N[Power[N[(a * N[(N[(0.005555555555555556 * Pi + N[(N[(-2.8577960676726107e-8 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 - N[(0.5 * N[Sin[N[(-0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 3.60000000000000009

   1. Initial program 86.5%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 86.4

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

   4. Applied rewrites 86.4%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 82.1%

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

   if 3.60000000000000009 < angle

   1. Initial program 60.7%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 60.8

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

   4. Applied rewrites 60.8%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 60.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. cos-neg-rev N/A

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

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

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

      8. lower-sin.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-PI.f64 60.9

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

   8. Applied rewrites 60.9%

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

   9. Taylor expanded in angle around inf

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

       1. lower-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-PI.f64 60.1

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

   11. Applied rewrites 60.1%

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

Alternative 7: 66.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.2 \cdot 10^{-138}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot 0.005555555555555556\right)}^{2} + b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 2.2e-138)
   (* b b)
   (+ (pow (* (* (* PI angle) a) 0.005555555555555556) 2.0) (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.2e-138) {
		tmp = b * b;
	} else {
		tmp = pow((((((double) M_PI) * angle) * a) * 0.005555555555555556), 2.0) + (b * b);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.2e-138) {
		tmp = b * b;
	} else {
		tmp = Math.pow((((Math.PI * angle) * a) * 0.005555555555555556), 2.0) + (b * b);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 2.2e-138:
		tmp = b * b
	else:
		tmp = math.pow((((math.pi * angle) * a) * 0.005555555555555556), 2.0) + (b * b)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 2.2e-138)
		tmp = Float64(b * b);
	else
		tmp = Float64((Float64(Float64(Float64(pi * angle) * a) * 0.005555555555555556) ^ 2.0) + Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 2.2e-138)
		tmp = b * b;
	else
		tmp = ((((pi * angle) * a) * 0.005555555555555556) ^ 2.0) + (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 2.2e-138], N[(b * b), $MachinePrecision], N[(N[Power[N[(N[(N[(Pi * angle), $MachinePrecision] * a), $MachinePrecision] * 0.005555555555555556), $MachinePrecision], 2.0], $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.1999999999999999e-138

   1. Initial program 79.4%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.0

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

   4. Applied rewrites 61.0%

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

   if 2.1999999999999999e-138 < a

   1. Initial program 80.4%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 80.4

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

   4. Applied rewrites 80.4%

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

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 77.0

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

   7. Applied rewrites 77.0%

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

Alternative 8: 64.9% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.2 \cdot 10^{-138}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right), a, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 2.2e-138)
   (* b b)
   (fma
    (* (* 3.08641975308642e-5 a) (* (* angle angle) (* PI PI)))
    a
    (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.2e-138) {
		tmp = b * b;
	} else {
		tmp = fma(((3.08641975308642e-5 * a) * ((angle * angle) * (((double) M_PI) * ((double) M_PI)))), a, (b * b));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 2.2e-138)
		tmp = Float64(b * b);
	else
		tmp = fma(Float64(Float64(3.08641975308642e-5 * a) * Float64(Float64(angle * angle) * Float64(pi * pi))), a, Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 2.2e-138], N[(b * b), $MachinePrecision], N[(N[(N[(3.08641975308642e-5 * a), $MachinePrecision] * N[(N[(angle * angle), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.1999999999999999e-138

   1. Initial program 79.4%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.0

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

   4. Applied rewrites 61.0%

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

   if 2.1999999999999999e-138 < a

   1. Initial program 80.4%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 80.4

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

   4. Applied rewrites 80.4%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 63.6%

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

   7. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-PI.f64 72.0

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

   9. Applied rewrites 72.0%

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

Alternative 9: 56.3% accurate, 29.7× speedup

Math

\[\begin{array}{l} \\ b \cdot b \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

2. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 56.3

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

4. Applied rewrites 56.3%

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

Reproduce

herbie shell --seed 2025124 
(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)))