ab-angle->ABCF A

Average Accuracy: 67.4% → 67.3%

Time: 18.1s

Precision: binary64

Cost: 39040

Math

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

↓

\[{\left(a \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)}^{2} + {b}^{2} \]

FPCore

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

↓

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (expm1 (log1p (sin (* PI (* angle 0.005555555555555556)))))) 2.0)
  (pow b 2.0)))

C

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

↓

double code(double a, double b, double angle) {
	return pow((a * expm1(log1p(sin((((double) M_PI) * (angle * 0.005555555555555556)))))), 2.0) + pow(b, 2.0);
}

Java

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

↓

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.expm1(Math.log1p(Math.sin((Math.PI * (angle * 0.005555555555555556)))))), 2.0) + Math.pow(b, 2.0);
}

Python

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

↓

def code(a, b, angle):
	return math.pow((a * math.expm1(math.log1p(math.sin((math.pi * (angle * 0.005555555555555556)))))), 2.0) + math.pow(b, 2.0)

Julia

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

↓

function code(a, b, angle)
	return Float64((Float64(a * expm1(log1p(sin(Float64(pi * Float64(angle * 0.005555555555555556)))))) ^ 2.0) + (b ^ 2.0))
end

Wolfram

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

↓

code[a_, b_, angle_] := N[(N[Power[N[(a * N[(Exp[N[Log[1 + N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.4%

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

2. Simplified 67.5%

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

   [Start] 67.4	\[ {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   associate-*l/ [=>] 67.4	\[ {\left(a \cdot \sin \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   associate-*r/ [<=] 67.5	\[ {\left(a \cdot \sin \color{blue}{\left(angle \cdot \frac{\pi}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   associate-*l/ [=>] 67.5	\[ {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)}\right)}^{2} \]
   associate-*r/ [<=] 67.5	\[ {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(angle \cdot \frac{\pi}{180}\right)}\right)}^{2} \]

3. Taylor expanded in angle around 0 67.3%

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

4. Taylor expanded in angle around inf 67.3%

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

5. Simplified 67.3%

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

   [Start] 67.3	\[ {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
   *-commutative [=>] 67.3	\[ {\left(a \cdot \sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
   *-commutative [=>] 67.3	\[ {\left(a \cdot \sin \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
   associate-*l* [=>] 67.3	\[ {\left(a \cdot \sin \color{blue}{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

6. Applied egg-rr 67.3%

   \[\leadsto {\left(a \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
   Proof

   [Start] 67.3	\[ {\left(a \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
   expm1-log1p-u [=>] 67.3	\[ {\left(a \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

7. Final simplification 67.3%

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

Alternatives

Alternative 1
Accuracy	65.8%
Cost	26372

\[\begin{array}{l} \mathbf{if}\;angle \leq 3300:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \mathsf{expm1}\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot 0\right)}^{2}\\ \end{array} \]

Alternative 2
Accuracy	67.3%
Cost	26240

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

Alternative 3
Accuracy	67.3%
Cost	26240

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

Alternative 4
Accuracy	67.3%
Cost	26240

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

Alternative 5
Accuracy	64.6%
Cost	20105

\[\begin{array}{l} \mathbf{if}\;angle \leq -1.9 \cdot 10^{+19} \lor \neg \left(angle \leq 8 \cdot 10^{+44}\right):\\ \;\;\;\;{b}^{2} + {\left(a \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(angle \cdot \left(a \cdot \pi\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]

Alternative 6
Accuracy	49.4%
Cost	13248

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

Reproduce

herbie shell --seed 2023151 
(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)))