ab-angle->ABCF C

Percentage Accurate: 79.6% → 79.5%

Time: 15.6s

Alternatives: 13

Speedup: N/A×

• Could not converge on a ground truth

  1. a = 4.2752679276015455e-181
  2. b = 9.600748853557302e+191
  3. angle = 5.11036906049915e+144

• 36.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 78.8

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

5. Applied rewrites 78.8%

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

   1. lift-PI.f64 N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 78.9

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

7. Applied rewrites 78.9%

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

Alternative 2: 79.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 78.8

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

5. Applied rewrites 78.8%

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

   1. lift-PI.f64 N/A

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

   2. associate-*r/ N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 78.9

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

7. Applied rewrites 78.9%

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

Alternative 3: 79.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 78.8

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

5. Applied rewrites 78.8%

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

7. Applied rewrites 63.0%

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

9. Applied rewrites 78.8%

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

10. Final simplification 78.8%

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

Alternative 4: 79.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 78.8

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

5. Applied rewrites 78.8%

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

   1. lift-PI.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 78.8

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-/.f64 N/A

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

   10. div-inv N/A

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

   11. metadata-eval N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 78.8

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

7. Applied rewrites 78.8%

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

8. Final simplification 78.8%

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

Alternative 5: 76.3% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 1e-5)
   (+
    (* a a)
    (pow
     (*
      b
      (*
       angle
       (*
        PI
        (fma
         (* (* angle angle) -2.8577960676726107e-8)
         (* PI PI)
         0.005555555555555556))))
     2.0))
   (fma
    (* b (fma -0.5 (cos (* PI (* angle 0.011111111111111112))) 0.5))
    b
    (* a a))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 1e-5) {
		tmp = (a * a) + pow((b * (angle * (((double) M_PI) * fma(((angle * angle) * -2.8577960676726107e-8), (((double) M_PI) * ((double) M_PI)), 0.005555555555555556)))), 2.0);
	} else {
		tmp = fma((b * fma(-0.5, cos((((double) M_PI) * (angle * 0.011111111111111112))), 0.5)), b, (a * a));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 1e-5)
		tmp = Float64(Float64(a * a) + (Float64(b * Float64(angle * Float64(pi * fma(Float64(Float64(angle * angle) * -2.8577960676726107e-8), Float64(pi * pi), 0.005555555555555556)))) ^ 2.0));
	else
		tmp = fma(Float64(b * fma(-0.5, cos(Float64(pi * Float64(angle * 0.011111111111111112))), 0.5)), b, Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e-5], N[(N[(a * a), $MachinePrecision] + N[Power[N[(b * N[(angle * N[(Pi * N[(N[(N[(angle * angle), $MachinePrecision] * -2.8577960676726107e-8), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(-0.5 * N[Cos[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * b + N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1.00000000000000008e-5

   1. Initial program 87.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow3 N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. distribute-rgt-out N/A

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

      13. lower-*.f64 N/A

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

   8. Applied rewrites 81.9%

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

   if 1.00000000000000008e-5 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 57.3%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.8

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

   5. Applied rewrites 59.8%

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

      1. lift-PI.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 59.9

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

   7. Applied rewrites 59.9%

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

   9. Applied rewrites 59.8%

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

Alternative 6: 76.3% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 1e-5)
   (+
    (* a a)
    (pow
     (*
      angle
      (*
       b
       (*
        PI
        (fma
         (* (* angle angle) -2.8577960676726107e-8)
         (* PI PI)
         0.005555555555555556))))
     2.0))
   (fma
    (* b (fma -0.5 (cos (* PI (* angle 0.011111111111111112))) 0.5))
    b
    (* a a))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 1e-5) {
		tmp = (a * a) + pow((angle * (b * (((double) M_PI) * fma(((angle * angle) * -2.8577960676726107e-8), (((double) M_PI) * ((double) M_PI)), 0.005555555555555556)))), 2.0);
	} else {
		tmp = fma((b * fma(-0.5, cos((((double) M_PI) * (angle * 0.011111111111111112))), 0.5)), b, (a * a));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 1e-5)
		tmp = Float64(Float64(a * a) + (Float64(angle * Float64(b * Float64(pi * fma(Float64(Float64(angle * angle) * -2.8577960676726107e-8), Float64(pi * pi), 0.005555555555555556)))) ^ 2.0));
	else
		tmp = fma(Float64(b * fma(-0.5, cos(Float64(pi * Float64(angle * 0.011111111111111112))), 0.5)), b, Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e-5], N[(N[(a * a), $MachinePrecision] + N[Power[N[(angle * N[(b * N[(Pi * N[(N[(N[(angle * angle), $MachinePrecision] * -2.8577960676726107e-8), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(-0.5 * N[Cos[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * b + N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1.00000000000000008e-5

   1. Initial program 87.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

   8. Applied rewrites 81.9%

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

   if 1.00000000000000008e-5 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 57.3%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.8

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

   5. Applied rewrites 59.8%

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

      1. lift-PI.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 59.9

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

   7. Applied rewrites 59.9%

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

   9. Applied rewrites 59.8%

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

Alternative 7: 76.9% accurate, 3.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 1e-5)
   (+ (* a a) (pow (* b (* angle (* PI 0.005555555555555556))) 2.0))
   (fma
    (* b (fma -0.5 (cos (* PI (* angle 0.011111111111111112))) 0.5))
    b
    (* a a))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 1e-5) {
		tmp = (a * a) + pow((b * (angle * (((double) M_PI) * 0.005555555555555556))), 2.0);
	} else {
		tmp = fma((b * fma(-0.5, cos((((double) M_PI) * (angle * 0.011111111111111112))), 0.5)), b, (a * a));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 1e-5)
		tmp = Float64(Float64(a * a) + (Float64(b * Float64(angle * Float64(pi * 0.005555555555555556))) ^ 2.0));
	else
		tmp = fma(Float64(b * fma(-0.5, cos(Float64(pi * Float64(angle * 0.011111111111111112))), 0.5)), b, Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e-5], N[(N[(a * a), $MachinePrecision] + N[Power[N[(b * N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(-0.5 * N[Cos[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * b + N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1.00000000000000008e-5

   1. Initial program 87.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 83.1

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

   8. Applied rewrites 83.1%

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

   if 1.00000000000000008e-5 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 57.3%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.8

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

   5. Applied rewrites 59.8%

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

      1. lift-PI.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 59.9

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

   7. Applied rewrites 59.9%

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

   9. Applied rewrites 59.8%

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

4. Final simplification 75.9%

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

Alternative 8: 66.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{-9}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(b \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3.2e-9)
   (* a a)
   (+ (* a a) (pow (* b (* angle (* PI 0.005555555555555556))) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.2e-9) {
		tmp = a * a;
	} else {
		tmp = (a * a) + pow((b * (angle * (((double) M_PI) * 0.005555555555555556))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.2e-9) {
		tmp = a * a;
	} else {
		tmp = (a * a) + Math.pow((b * (angle * (Math.PI * 0.005555555555555556))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 3.2e-9:
		tmp = a * a
	else:
		tmp = (a * a) + math.pow((b * (angle * (math.pi * 0.005555555555555556))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 3.2e-9)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + (Float64(b * Float64(angle * Float64(pi * 0.005555555555555556))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 3.2e-9)
		tmp = a * a;
	else
		tmp = (a * a) + ((b * (angle * (pi * 0.005555555555555556))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 3.2e-9], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[Power[N[(b * N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.20000000000000012e-9

   1. Initial program 76.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.0

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

   5. Applied rewrites 61.0%

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

   if 3.20000000000000012e-9 < b

   1. Initial program 83.2%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 83.2

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

   5. Applied rewrites 83.2%

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

   6. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 80.2

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

   8. Applied rewrites 80.2%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{-9}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + {\left(b \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.9% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{-9}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \left(b \cdot \left(angle \cdot \left(b \cdot angle\right)\right)\right), 3.08641975308642 \cdot 10^{-5}, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3.2e-9)
   (* a a)
   (fma
    (* (* PI PI) (* b (* angle (* b angle))))
    3.08641975308642e-5
    (* a a))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.2e-9) {
		tmp = a * a;
	} else {
		tmp = fma(((((double) M_PI) * ((double) M_PI)) * (b * (angle * (b * angle)))), 3.08641975308642e-5, (a * a));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 3.2e-9)
		tmp = Float64(a * a);
	else
		tmp = fma(Float64(Float64(pi * pi) * Float64(b * Float64(angle * Float64(b * angle)))), 3.08641975308642e-5, Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 3.2e-9], N[(a * a), $MachinePrecision], N[(N[(N[(Pi * Pi), $MachinePrecision] * N[(b * N[(angle * N[(b * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5 + N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.20000000000000012e-9

   1. Initial program 76.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.0

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

   5. Applied rewrites 61.0%

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

   if 3.20000000000000012e-9 < b

   1. Initial program 83.2%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 83.2

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

   5. Applied rewrites 83.2%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 61.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

   10. Applied rewrites 78.9%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{-9}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \left(b \cdot \left(angle \cdot \left(b \cdot angle\right)\right)\right), 3.08641975308642 \cdot 10^{-5}, a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.9% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{-9}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \left(angle \cdot \left(b \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right), a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3.2e-9)
   (* a a)
   (fma
    b
    (* (* angle (* b angle)) (* (* PI PI) 3.08641975308642e-5))
    (* a a))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.2e-9) {
		tmp = a * a;
	} else {
		tmp = fma(b, ((angle * (b * angle)) * ((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5)), (a * a));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 3.2e-9)
		tmp = Float64(a * a);
	else
		tmp = fma(b, Float64(Float64(angle * Float64(b * angle)) * Float64(Float64(pi * pi) * 3.08641975308642e-5)), Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 3.2e-9], N[(a * a), $MachinePrecision], N[(b * N[(N[(angle * N[(b * angle), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.20000000000000012e-9

   1. Initial program 76.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.0

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

   5. Applied rewrites 61.0%

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

   if 3.20000000000000012e-9 < b

   1. Initial program 83.2%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 83.2

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

   5. Applied rewrites 83.2%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 61.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

   10. Applied rewrites 78.8%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{-9}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \left(angle \cdot \left(b \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right), a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.2% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{-9}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot \left(angle \cdot \left(angle \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right), a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3.2e-9)
   (* a a)
   (fma
    b
    (* b (* angle (* angle (* (* PI PI) 3.08641975308642e-5))))
    (* a a))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.2e-9) {
		tmp = a * a;
	} else {
		tmp = fma(b, (b * (angle * (angle * ((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5)))), (a * a));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 3.2e-9)
		tmp = Float64(a * a);
	else
		tmp = fma(b, Float64(b * Float64(angle * Float64(angle * Float64(Float64(pi * pi) * 3.08641975308642e-5)))), Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 3.2e-9], N[(a * a), $MachinePrecision], N[(b * N[(b * N[(angle * N[(angle * N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.20000000000000012e-9

   1. Initial program 76.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.0

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

   5. Applied rewrites 61.0%

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

   if 3.20000000000000012e-9 < b

   1. Initial program 83.2%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 83.2

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

   5. Applied rewrites 83.2%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 61.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

   10. Applied rewrites 72.2%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{-9}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot \left(angle \cdot \left(angle \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right), a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.3% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(b \cdot \pi\right)\\ \mathbf{if}\;b \leq 7.2 \cdot 10^{+152}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;3.08641975308642 \cdot 10^{-5} \cdot \left(t\_0 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* b PI))))
   (if (<= b 7.2e+152) (* a a) (* 3.08641975308642e-5 (* t_0 t_0)))))

C

double code(double a, double b, double angle) {
	double t_0 = angle * (b * ((double) M_PI));
	double tmp;
	if (b <= 7.2e+152) {
		tmp = a * a;
	} else {
		tmp = 3.08641975308642e-5 * (t_0 * t_0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = angle * (b * Math.PI);
	double tmp;
	if (b <= 7.2e+152) {
		tmp = a * a;
	} else {
		tmp = 3.08641975308642e-5 * (t_0 * t_0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = angle * (b * math.pi)
	tmp = 0
	if b <= 7.2e+152:
		tmp = a * a
	else:
		tmp = 3.08641975308642e-5 * (t_0 * t_0)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(angle * Float64(b * pi))
	tmp = 0.0
	if (b <= 7.2e+152)
		tmp = Float64(a * a);
	else
		tmp = Float64(3.08641975308642e-5 * Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = angle * (b * pi);
	tmp = 0.0;
	if (b <= 7.2e+152)
		tmp = a * a;
	else
		tmp = 3.08641975308642e-5 * (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(b * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 7.2e+152], N[(a * a), $MachinePrecision], N[(3.08641975308642e-5 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.1999999999999998e152

   1. Initial program 74.5%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.8

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

   5. Applied rewrites 57.8%

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

   if 7.1999999999999998e152 < b

   1. Initial program 99.8%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 62.5%

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

   9. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. associate-*l* N/A

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

       13. lower-*.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. unpow2 N/A

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

       17. lower-*.f64 N/A

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

       18. lower-PI.f64 N/A

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

       19. lower-PI.f64 62.5

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

   11. Applied rewrites 62.5%

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

   12. Taylor expanded in b around 0

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. unpow2 N/A

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

       4. unpow2 N/A

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

       5. unswap-sqr N/A

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

       6. unswap-sqr N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lower-PI.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. lower-PI.f64 84.2

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

   14. Applied rewrites 84.2%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.2 \cdot 10^{+152}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(b \cdot \pi\right)\right) \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.9% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 54.1

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

5. Applied rewrites 54.1%

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

Reproduce

herbie shell --seed 2024216 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))