ab-angle->ABCF A

Percentage Accurate: 79.9% → 79.7%

Time: 3.7s

Alternatives: 1

• 37.8% 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: 79.9% 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: 79.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1 \cdot b, 1 \cdot b, {\left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot a\right)}^{2}\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 80.4%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-pow.f64 N/A

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

   10. unpow2 N/A

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

7. Applied rewrites 80.4%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 80.4

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

9. Applied rewrites 80.4%

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

Reproduce

herbie shell --seed 2025059 
(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)))