ab-angle->ABCF C

Percentage Accurate: 80.4% → 80.4%

Time: 35.7s

Precision: binary64

Cost: 78336

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, 1\right)\\ {\left(a \cdot \left(\cos t_0 \cdot \cos 1 + \sin t_0 \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \end{array} \]

FPCore

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

↓

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (fma (* PI angle) 0.005555555555555556 1.0)))
   (+
    (pow (* a (+ (* (cos t_0) (cos 1.0)) (* (sin t_0) (sin 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 * cos((((double) M_PI) * (angle / 180.0)))), 2.0) + pow((b * sin((((double) M_PI) * (angle / 180.0)))), 2.0);
}

↓

double code(double a, double b, double angle) {
	double t_0 = fma((((double) M_PI) * angle), 0.005555555555555556, 1.0);
	return pow((a * ((cos(t_0) * cos(1.0)) + (sin(t_0) * sin(1.0)))), 2.0) + pow((b * sin((((double) M_PI) * (angle / 180.0)))), 2.0);
}

Julia

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

↓

function code(a, b, angle)
	t_0 = fma(Float64(pi * angle), 0.005555555555555556, 1.0)
	return Float64((Float64(a * Float64(Float64(cos(t_0) * cos(1.0)) + Float64(sin(t_0) * sin(1.0)))) ^ 2.0) + (Float64(b * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0))
end

Wolfram

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

↓

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556 + 1.0), $MachinePrecision]}, N[(N[Power[N[(a * N[(N[(N[Cos[t$95$0], $MachinePrecision] * N[Cos[1.0], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[t$95$0], $MachinePrecision] * N[Sin[1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 79.5%

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

2. Applied egg-rr 62.9%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   Step-by-step derivation

   [Start] 79.5%	\[ {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   expm1-log1p-u [=>] 62.9%	\[ {\left(a \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   div-inv [=>] 62.9%	\[ {\left(a \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   metadata-eval [=>] 62.9%	\[ {\left(a \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

3. Applied egg-rr 79.7%

   \[\leadsto {\left(a \cdot \color{blue}{\left(\cos \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, 1\right)\right) \cdot \cos 1 + \sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, 1\right)\right) \cdot \sin 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   Step-by-step derivation

   [Start] 62.9%	\[ {\left(a \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   expm1-udef [=>] 62.9%	\[ {\left(a \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)} - 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   cos-diff [=>] 62.9%	\[ {\left(a \cdot \color{blue}{\left(\cos \left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \sin 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   log1p-udef [=>] 62.9%	\[ {\left(a \cdot \left(\cos \left(e^{\color{blue}{\log \left(1 + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   add-exp-log [<=] 62.8%	\[ {\left(a \cdot \left(\cos \color{blue}{\left(1 + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)} \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   +-commutative [=>] 62.8%	\[ {\left(a \cdot \left(\cos \color{blue}{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right) + 1\right)} \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   associate-*r* [=>] 62.8%	\[ {\left(a \cdot \left(\cos \left(\color{blue}{\left(\pi \cdot angle\right) \cdot 0.005555555555555556} + 1\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   fma-def [=>] 62.8%	\[ {\left(a \cdot \left(\cos \color{blue}{\left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, 1\right)\right)} \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   log1p-udef [=>] 62.8%	\[ {\left(a \cdot \left(\cos \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, 1\right)\right) \cdot \cos 1 + \sin \left(e^{\color{blue}{\log \left(1 + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   add-exp-log [<=] 79.5%	\[ {\left(a \cdot \left(\cos \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, 1\right)\right) \cdot \cos 1 + \sin \color{blue}{\left(1 + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)} \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   +-commutative [=>] 79.5%	\[ {\left(a \cdot \left(\cos \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, 1\right)\right) \cdot \cos 1 + \sin \color{blue}{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right) + 1\right)} \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   associate-*r* [=>] 79.7%	\[ {\left(a \cdot \left(\cos \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, 1\right)\right) \cdot \cos 1 + \sin \left(\color{blue}{\left(\pi \cdot angle\right) \cdot 0.005555555555555556} + 1\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   fma-def [=>] 79.7%	\[ {\left(a \cdot \left(\cos \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, 1\right)\right) \cdot \cos 1 + \sin \color{blue}{\left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, 1\right)\right)} \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

4. Final simplification 79.7%

   \[\leadsto {\left(a \cdot \left(\cos \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, 1\right)\right) \cdot \cos 1 + \sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, 1\right)\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

Alternatives

Alternative 1
Accuracy	80.4%
Cost	78336

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, 1\right)\\ {\left(a \cdot \left(\cos t_0 \cdot \cos 1 + \sin t_0 \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \end{array} \]

Alternative 2
Accuracy	80.4%
Cost	39488

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

Alternative 3
Accuracy	80.4%
Cost	39360

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

Alternative 4
Accuracy	80.4%
Cost	39360

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

Alternative 5
Accuracy	80.5%
Cost	39360

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

Alternative 6
Accuracy	80.4%
Cost	26368

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

Alternative 7
Accuracy	80.5%
Cost	26240

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

Alternative 8
Accuracy	75.5%
Cost	19840

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

Alternative 9
Accuracy	75.5%
Cost	19840

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

Alternative 10
Accuracy	75.5%
Cost	19840

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

Alternative 11
Accuracy	57.8%
Cost	13248

\[{a}^{2} + {\left(b \cdot 0\right)}^{2} \]

Reproduce

herbie shell --seed 2023272 
(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)))