ab-angle->ABCF C

Average Error: 20.4 → 20.4

Time: 19.4s

Precision: binary64

Cost: 84608

Math

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

↓

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

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 (fma (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 * fma(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(pi, Float64(angle * 0.005555555555555556), 1.0)
	return Float64((Float64(a * fma(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[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(N[Power[N[(a * N[(N[Cos[t$95$0], $MachinePrecision] * N[Cos[1.0], $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 20.4

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

2. Applied egg-rr 24.6

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

3. Applied egg-rr 20.4

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

4. Simplified 20.4

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

   [Start] 20.4	\[ {\left(a \cdot \left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right) + 1\right) \cdot \cos 1 + \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right) + 1\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   fma-def [=>] 20.4	\[ {\left(a \cdot \color{blue}{\mathsf{fma}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right) + 1\right), \cos 1, \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right) + 1\right) \cdot \sin 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   fma-def [=>] 20.4	\[ {\left(a \cdot \mathsf{fma}\left(\cos \color{blue}{\left(\mathsf{fma}\left(\pi, angle \cdot 0.005555555555555556, 1\right)\right)}, \cos 1, \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right) + 1\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   fma-def [=>] 20.4	\[ {\left(a \cdot \mathsf{fma}\left(\cos \left(\mathsf{fma}\left(\pi, angle \cdot 0.005555555555555556, 1\right)\right), \cos 1, \sin \color{blue}{\left(\mathsf{fma}\left(\pi, angle \cdot 0.005555555555555556, 1\right)\right)} \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

5. Final simplification 20.4

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

Alternatives

Alternative 1
Error	20.4
Cost	65792

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

Alternative 2
Error	20.4
Cost	65088

\[{\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{\frac{\frac{180}{angle}}{{\left(\sqrt[3]{\pi}\right)}^{2}}}\right)\right)}^{2} \]

Alternative 3
Error	20.4
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 4
Error	20.4
Cost	39360

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

Alternative 5
Error	20.4
Cost	26240

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

Alternative 6
Error	20.6
Cost	20425

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.0046 \lor \neg \left(angle \leq 0.004\right):\\ \;\;\;\;{a}^{2} + \frac{b}{\frac{2}{b}} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(angle \cdot \left(\pi \cdot b\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]

Alternative 7
Error	25.9
Cost	20360

\[\begin{array}{l} t_0 := \pi \cdot \left(angle \cdot b\right)\\ \mathbf{if}\;b \leq -1 \cdot 10^{+97}:\\ \;\;\;\;{a}^{2} + {\left(angle \cdot \left(\pi \cdot b\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+105}:\\ \;\;\;\;{a}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(t_0 \cdot \left(\pi \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot b\right) \cdot \left(angle \cdot t_0\right)\right)\\ \end{array} \]

Alternative 8
Error	26.2
Cost	19840

\[{a}^{2} + {\left(angle \cdot \left(\pi \cdot b\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} \]

Reproduce

herbie shell --seed 2023016 
(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)))