ab-angle->ABCF C

Percentage Accurate: 79.8% → 79.9%

Time: 6.2s

Alternatives: 8

Speedup: 1.0×

• 63.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 79.8% accurate, 1.0× speedup

Math

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

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: 79.9% accurate, 0.9× speedup

Math

\[\left(a \cdot a\right) \cdot {\sin \left(\pi \cdot \left(0.5 - 0.005555555555555556 \cdot \left|angle\right|\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{\left|angle\right|}{180}\right)\right)}^{2} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (*
   (* a a)
   (pow (sin (* PI (- 0.5 (* 0.005555555555555556 (fabs angle))))) 2.0))
  (pow (* b (sin (* PI (/ (fabs angle) 180.0)))) 2.0)))

C

double code(double a, double b, double angle) {
	return ((a * a) * pow(sin((((double) M_PI) * (0.5 - (0.005555555555555556 * fabs(angle))))), 2.0)) + pow((b * sin((((double) M_PI) * (fabs(angle) / 180.0)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return ((a * a) * Math.pow(Math.sin((Math.PI * (0.5 - (0.005555555555555556 * Math.abs(angle))))), 2.0)) + Math.pow((b * Math.sin((Math.PI * (Math.abs(angle) / 180.0)))), 2.0);
}

Python

def code(a, b, angle):
	return ((a * a) * math.pow(math.sin((math.pi * (0.5 - (0.005555555555555556 * math.fabs(angle))))), 2.0)) + math.pow((b * math.sin((math.pi * (math.fabs(angle) / 180.0)))), 2.0)

Julia

function code(a, b, angle)
	return Float64(Float64(Float64(a * a) * (sin(Float64(pi * Float64(0.5 - Float64(0.005555555555555556 * abs(angle))))) ^ 2.0)) + (Float64(b * sin(Float64(pi * Float64(abs(angle) / 180.0)))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * a) * (sin((pi * (0.5 - (0.005555555555555556 * abs(angle))))) ^ 2.0)) + ((b * sin((pi * (abs(angle) / 180.0)))) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[(N[(a * a), $MachinePrecision] * N[Power[N[Sin[N[(Pi * N[(0.5 - N[(0.005555555555555556 * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(N[Abs[angle], $MachinePrecision] / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.8%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. lift-/.f64 N/A

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

   8. distribute-neg-frac2 N/A

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

   9. distribute-neg-frac2 N/A

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

   10. distribute-neg-frac N/A

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

   11. lift-PI.f64 N/A

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

   12. mult-flip N/A

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

   13. distribute-lft-out N/A

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

   14. lower-*.f64 N/A

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

   15. distribute-neg-frac N/A

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

   16. mult-flip N/A

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

   17. *-commutative N/A

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

   18. distribute-lft-neg-in N/A

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

   19. distribute-frac-neg2 N/A

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

   20. lower-fma.f64 N/A

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

3. Applied rewrites 79.8%

   \[\leadsto {\left(a \cdot \color{blue}{\sin \left(\pi \cdot \mathsf{fma}\left(-0.005555555555555556, angle, 0.5\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

4. Taylor expanded in angle around inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-*.f64 79.8%

      \[\leadsto {a}^{2} \cdot {\sin \left(\pi \cdot \left(0.5 - 0.005555555555555556 \cdot angle\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

6. Applied rewrites 79.8%

   \[\leadsto \color{blue}{{a}^{2} \cdot {\sin \left(\pi \cdot \left(0.5 - 0.005555555555555556 \cdot angle\right)\right)}^{2}} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 79.8%

      \[\leadsto \left(a \cdot a\right) \cdot {\color{blue}{\sin \left(\pi \cdot \left(0.5 - 0.005555555555555556 \cdot angle\right)\right)}}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

8. Applied rewrites 79.8%

   \[\leadsto \left(a \cdot a\right) \cdot {\color{blue}{\sin \left(\pi \cdot \left(0.5 - 0.005555555555555556 \cdot angle\right)\right)}}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
9. Add Preprocessing

Alternative 2: 79.9% accurate, 1.0× speedup

Math

\[{\left(a \cdot \sin \left(\pi \cdot \mathsf{fma}\left(-0.005555555555555556, \left|angle\right|, 0.5\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{\left|angle\right|}{180}\right)\right)}^{2} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* PI (fma -0.005555555555555556 (fabs angle) 0.5)))) 2.0)
  (pow (* b (sin (* PI (/ (fabs angle) 180.0)))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * sin((((double) M_PI) * fma(-0.005555555555555556, fabs(angle), 0.5)))), 2.0) + pow((b * sin((((double) M_PI) * (fabs(angle) / 180.0)))), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(pi * fma(-0.005555555555555556, abs(angle), 0.5)))) ^ 2.0) + (Float64(b * sin(Float64(pi * Float64(abs(angle) / 180.0)))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(Pi * N[(-0.005555555555555556 * N[Abs[angle], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(N[Abs[angle], $MachinePrecision] / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.8%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. lift-/.f64 N/A

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

   8. distribute-neg-frac2 N/A

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

   9. distribute-neg-frac2 N/A

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

   10. distribute-neg-frac N/A

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

   11. lift-PI.f64 N/A

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

   12. mult-flip N/A

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

   13. distribute-lft-out N/A

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

   14. lower-*.f64 N/A

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

   15. distribute-neg-frac N/A

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

   16. mult-flip N/A

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

   17. *-commutative N/A

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

   18. distribute-lft-neg-in N/A

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

   19. distribute-frac-neg2 N/A

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

   20. lower-fma.f64 N/A

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

3. Applied rewrites 79.8%

   \[\leadsto {\left(a \cdot \color{blue}{\sin \left(\pi \cdot \mathsf{fma}\left(-0.005555555555555556, angle, 0.5\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
4. Add Preprocessing

Alternative 3: 79.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 PI) angle)))
   (+ (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 = (0.005555555555555556 * ((double) M_PI)) * angle;
	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 = (0.005555555555555556 * Math.PI) * angle;
	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 = (0.005555555555555556 * math.pi) * angle
	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(Float64(0.005555555555555556 * pi) * angle)
	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 = (0.005555555555555556 * pi) * angle;
	tmp = ((a * cos(t_0)) ^ 2.0) + ((b * sin(t_0)) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(0.005555555555555556 * Pi), $MachinePrecision] * angle), $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]]

Derivation

1. Initial program 79.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval 79.8%

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

3. Applied rewrites 79.8%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval 79.9%

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

5. Applied rewrites 79.9%

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

Alternative 4: 79.8% accurate, 1.0× speedup

Math

\[\left(a \cdot a\right) \cdot \left(1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot -0.005555555555555556\right)\right)\right)\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (*
   (* a a)
   (-
    1.0
    (- 0.5 (* 0.5 (cos (* 2.0 (* (* PI angle) -0.005555555555555556)))))))
  (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))

C

double code(double a, double b, double angle) {
	return ((a * a) * (1.0 - (0.5 - (0.5 * cos((2.0 * ((((double) M_PI) * angle) * -0.005555555555555556))))))) + 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) * (1.0 - (0.5 - (0.5 * Math.cos((2.0 * ((Math.PI * angle) * -0.005555555555555556))))))) + Math.pow((b * Math.sin((Math.PI * (angle / 180.0)))), 2.0);
}

Python

def code(a, b, angle):
	return ((a * a) * (1.0 - (0.5 - (0.5 * math.cos((2.0 * ((math.pi * angle) * -0.005555555555555556))))))) + math.pow((b * math.sin((math.pi * (angle / 180.0)))), 2.0)

Julia

function code(a, b, angle)
	return Float64(Float64(Float64(a * a) * Float64(1.0 - Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(pi * angle) * -0.005555555555555556))))))) + (Float64(b * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * a) * (1.0 - (0.5 - (0.5 * cos((2.0 * ((pi * angle) * -0.005555555555555556))))))) + ((b * sin((pi * (angle / 180.0)))) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(Pi * angle), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.8%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. lift-/.f64 N/A

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

   8. distribute-neg-frac2 N/A

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

   9. distribute-neg-frac2 N/A

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

   10. distribute-neg-frac N/A

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

   11. lift-PI.f64 N/A

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

   12. mult-flip N/A

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

   13. distribute-lft-out N/A

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

   14. lower-*.f64 N/A

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

   15. distribute-neg-frac N/A

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

   16. mult-flip N/A

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

   17. *-commutative N/A

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

   18. distribute-lft-neg-in N/A

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

   19. distribute-frac-neg2 N/A

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

   20. lower-fma.f64 N/A

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

3. Applied rewrites 79.8%

   \[\leadsto {\left(a \cdot \color{blue}{\sin \left(\pi \cdot \mathsf{fma}\left(-0.005555555555555556, angle, 0.5\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

4. Taylor expanded in angle around inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-*.f64 79.8%

      \[\leadsto {a}^{2} \cdot {\sin \left(\pi \cdot \left(0.5 - 0.005555555555555556 \cdot angle\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

6. Applied rewrites 79.8%

   \[\leadsto \color{blue}{{a}^{2} \cdot {\sin \left(\pi \cdot \left(0.5 - 0.005555555555555556 \cdot angle\right)\right)}^{2}} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 79.8%

      \[\leadsto \left(a \cdot a\right) \cdot {\color{blue}{\sin \left(\pi \cdot \left(0.5 - 0.005555555555555556 \cdot angle\right)\right)}}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

8. Applied rewrites 79.8%

   \[\leadsto \left(a \cdot a\right) \cdot {\color{blue}{\sin \left(\pi \cdot \left(0.5 - 0.005555555555555556 \cdot angle\right)\right)}}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

10. Applied rewrites 79.8%

    \[\leadsto \left(a \cdot a\right) \cdot \left(1 - \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot -0.005555555555555556\right)\right)\right)}\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
11. Add Preprocessing

Alternative 5: 79.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * 1.0), 2.0) + Math.pow((b * Math.sin((Math.PI * (angle / 180.0)))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((a * 1.0), 2.0) + math.pow((b * math.sin((math.pi * (angle / 180.0)))), 2.0)

Julia

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

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * 1.0) ^ 2.0) + ((b * sin((pi * (angle / 180.0)))) ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 79.8%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 79.9%

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

Alternative 6: 71.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|a\right| \leq 2.5 \cdot 10^{-145}:\\ \;\;\;\;{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \left|a\right| \cdot \left|a\right|, \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)\right) \cdot b\right) \cdot b\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (fabs a) 2.5e-145)
   (* (pow b 2.0) (pow (sin (* 0.005555555555555556 (* angle PI))) 2.0))
   (fma
    1.0
    (* (fabs a) (fabs a))
    (*
     (* (- 0.5 (* 0.5 (cos (* 2.0 (* (* 0.005555555555555556 angle) PI))))) b)
     b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(a) <= 2.5e-145) {
		tmp = pow(b, 2.0) * pow(sin((0.005555555555555556 * (angle * ((double) M_PI)))), 2.0);
	} else {
		tmp = fma(1.0, (fabs(a) * fabs(a)), (((0.5 - (0.5 * cos((2.0 * ((0.005555555555555556 * angle) * ((double) M_PI)))))) * b) * b));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(a) <= 2.5e-145)
		tmp = Float64((b ^ 2.0) * (sin(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 2.0));
	else
		tmp = fma(1.0, Float64(abs(a) * abs(a)), Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(0.005555555555555556 * angle) * pi))))) * b) * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[Abs[a], $MachinePrecision], 2.5e-145], N[(N[Power[b, 2.0], $MachinePrecision] * N[Power[N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.4999999999999999e-145

   1. Initial program 79.8%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 34.8%

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

   4. Applied rewrites 34.8%

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

   if 2.4999999999999999e-145 < a

   1. Initial program 79.8%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac2 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. distribute-neg-frac N/A

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

      11. lift-PI.f64 N/A

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

      12. mult-flip N/A

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

      13. distribute-lft-out N/A

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

      14. lower-*.f64 N/A

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

      15. distribute-neg-frac N/A

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

      16. mult-flip N/A

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

      17. *-commutative N/A

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

      18. distribute-lft-neg-in N/A

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

      19. distribute-frac-neg2 N/A

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

      20. lower-fma.f64 N/A

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

   3. Applied rewrites 79.8%

      \[\leadsto {\left(a \cdot \color{blue}{\sin \left(\pi \cdot \mathsf{fma}\left(-0.005555555555555556, angle, 0.5\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. Applied rewrites 68.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right), a \cdot a, \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)\right) \cdot b\right) \cdot b\right)} \]

   6. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, a \cdot a, \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)\right) \cdot b\right) \cdot b\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 68.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, a \cdot a, \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)\right) \cdot b\right) \cdot b\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 68.6% accurate, 1.9× speedup

Math

\[\mathsf{fma}\left(1, a \cdot a, \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)\right) \cdot b\right) \cdot b\right) \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (fma
  1.0
  (* a a)
  (*
   (* (- 0.5 (* 0.5 (cos (* 2.0 (* (* 0.005555555555555556 angle) PI))))) b)
   b)))

C

double code(double a, double b, double angle) {
	return fma(1.0, (a * a), (((0.5 - (0.5 * cos((2.0 * ((0.005555555555555556 * angle) * ((double) M_PI)))))) * b) * b));
}

Julia

function code(a, b, angle)
	return fma(1.0, Float64(a * a), Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(0.005555555555555556 * angle) * pi))))) * b) * b))
end

Wolfram

code[a_, b_, angle_] := N[(1.0 * N[(a * a), $MachinePrecision] + N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.8%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. lift-/.f64 N/A

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

   8. distribute-neg-frac2 N/A

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

   9. distribute-neg-frac2 N/A

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

   10. distribute-neg-frac N/A

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

   11. lift-PI.f64 N/A

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

   12. mult-flip N/A

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

   13. distribute-lft-out N/A

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

   14. lower-*.f64 N/A

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

   15. distribute-neg-frac N/A

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

   16. mult-flip N/A

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

   17. *-commutative N/A

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

   18. distribute-lft-neg-in N/A

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

   19. distribute-frac-neg2 N/A

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

   20. lower-fma.f64 N/A

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

3. Applied rewrites 79.8%

   \[\leadsto {\left(a \cdot \color{blue}{\sin \left(\pi \cdot \mathsf{fma}\left(-0.005555555555555556, angle, 0.5\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

5. Applied rewrites 68.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right), a \cdot a, \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)\right) \cdot b\right) \cdot b\right)} \]

6. Taylor expanded in angle around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{1}, a \cdot a, \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)\right) \cdot b\right) \cdot b\right) \]
7. Step-by-step derivation

8. Applied rewrites 68.6%

   \[\leadsto \mathsf{fma}\left(\color{blue}{1}, a \cdot a, \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)\right) \cdot b\right) \cdot b\right) \]
9. Add Preprocessing

Alternative 8: 57.6% accurate, 29.7× speedup

Math

\[a \cdot a \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    code = 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 79.8%

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

2. Taylor expanded in angle around 0

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

   1. lower-pow.f64 57.6%

      \[\leadsto {a}^{\color{blue}{2}} \]

4. Applied rewrites 57.6%

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

   1. lift-pow.f64 N/A

      \[\leadsto {a}^{\color{blue}{2}} \]

   2. unpow2 N/A

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

   3. lower-*.f64 57.6%

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

6. Applied rewrites 57.6%

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

Reproduce

herbie shell --seed 2025183 
(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)))