ab-angle->ABCF A

Average Error: 32.09% → 32.08%

Time: 18.0s

Precision: binary64

Cost: 26240

Math

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

↓

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

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 (sin (/ PI (/ 180.0 angle)))) 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 * sin((((double) M_PI) / (180.0 / angle)))), 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.sin((Math.PI / (180.0 / angle)))), 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.sin((math.pi / (180.0 / angle)))), 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 * sin(Float64(pi / Float64(180.0 / angle)))) ^ 2.0) + (b ^ 2.0))
end

MATLAB

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

↓

function tmp = code(a, b, angle)
	tmp = ((a * sin((pi / (180.0 / angle)))) ^ 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[Sin[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.09

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

2. Simplified 32

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

   [Start] 32.09	\[ {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   associate-*l/ [=>] 32.09	\[ {\left(a \cdot \sin \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   associate-*r/ [<=] 32.02	\[ {\left(a \cdot \sin \color{blue}{\left(angle \cdot \frac{\pi}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   associate-*l/ [=>] 32	\[ {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)}\right)}^{2} \]
   associate-*r/ [<=] 32	\[ {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(angle \cdot \frac{\pi}{180}\right)}\right)}^{2} \]

3. Taylor expanded in angle around 0 32.01

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

4. Taylor expanded in angle around 0 32.06

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

5. Simplified 32.08

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

   [Start] 32.06	\[ {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
   metadata-eval [<=] 32.06	\[ {\left(a \cdot \sin \left(\color{blue}{\frac{-1}{-180}} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
   metadata-eval [<=] 32.06	\[ {\left(a \cdot \sin \left(\frac{\color{blue}{-1}}{-180} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
   associate-/r/ [<=] 32.11	\[ {\left(a \cdot \sin \color{blue}{\left(\frac{-1}{\frac{-180}{angle \cdot \pi}}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
   associate-/l* [<=] 32.08	\[ {\left(a \cdot \sin \color{blue}{\left(\frac{\left(-1\right) \cdot \left(angle \cdot \pi\right)}{-180}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
   *-commutative [=>] 32.08	\[ {\left(a \cdot \sin \left(\frac{\color{blue}{\left(angle \cdot \pi\right) \cdot \left(-1\right)}}{-180}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
   associate-/l* [=>] 32.08	\[ {\left(a \cdot \sin \color{blue}{\left(\frac{angle \cdot \pi}{\frac{-180}{-1}}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
   *-commutative [=>] 32.08	\[ {\left(a \cdot \sin \left(\frac{\color{blue}{\pi \cdot angle}}{\frac{-180}{-1}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
   metadata-eval [=>] 32.08	\[ {\left(a \cdot \sin \left(\frac{\pi \cdot angle}{\frac{-180}{\color{blue}{-1}}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
   metadata-eval [=>] 32.08	\[ {\left(a \cdot \sin \left(\frac{\pi \cdot angle}{\color{blue}{180}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
   associate-/l* [=>] 32.08	\[ {\left(a \cdot \sin \color{blue}{\left(\frac{\pi}{\frac{180}{angle}}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

6. Final simplification 32.08

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

Alternatives

Alternative 1
Error	32.06%
Cost	26240

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

Alternative 2
Error	32.01%
Cost	26240

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

Alternative 3
Error	32.25%
Cost	20425

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.0045 \lor \neg \left(angle \leq 0.005\right):\\ \;\;\;\;{b}^{2} + \frac{a}{\frac{2}{a}} \cdot \left(1 - \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(\pi \cdot \left(a \cdot angle\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]

Alternative 4
Error	32.25%
Cost	20424

\[\begin{array}{l} t_0 := 1 - \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\\ \mathbf{if}\;angle \leq -0.0045:\\ \;\;\;\;{b}^{2} + \frac{a}{\frac{2}{a}} \cdot t_0\\ \mathbf{elif}\;angle \leq 0.005:\\ \;\;\;\;{b}^{2} + {\left(\pi \cdot \left(a \cdot angle\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \frac{a \cdot a}{\frac{2}{t_0}}\\ \end{array} \]

Alternative 5
Error	34.7%
Cost	20105

\[\begin{array}{l} \mathbf{if}\;angle \leq -9 \cdot 10^{+45} \lor \neg \left(angle \leq 5.7 \cdot 10^{+20}\right):\\ \;\;\;\;{b}^{2} + \frac{{\left(a \cdot \sqrt{0}\right)}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(\pi \cdot \left(a \cdot angle\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]

Alternative 6
Error	40.86%
Cost	19840

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

Alternative 7
Error	40.87%
Cost	19840

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

Reproduce

herbie shell --seed 2023121 
(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)))