ab-angle->ABCF A

Average Error: 20.3 → 20.3

Time: 18.8s

Precision: binary64

Cost: 26240

Math

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

↓

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

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 (* angle (/ PI 180.0)))) 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((angle * (((double) M_PI) / 180.0)))), 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((angle * (Math.PI / 180.0)))), 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((angle * (math.pi / 180.0)))), 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(angle * Float64(pi / 180.0)))) ^ 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((angle * (pi / 180.0)))) ^ 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[(angle * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.3

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

2. Simplified 20.3

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

   [Start] 20.3	\[ {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   associate-*l/ [=>] 20.3	\[ {\left(a \cdot \sin \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   associate-*r/ [<=] 20.3	\[ {\left(a \cdot \sin \color{blue}{\left(angle \cdot \frac{\pi}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   associate-*l/ [=>] 20.3	\[ {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)}\right)}^{2} \]
   associate-*r/ [<=] 20.3	\[ {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(angle \cdot \frac{\pi}{180}\right)}\right)}^{2} \]

3. Taylor expanded in angle around 0 20.3

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

4. Final simplification 20.3

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

Alternatives

Alternative 1
Error	20.6
Cost	20425

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.0044 \lor \neg \left(angle \leq 1.32 \cdot 10^{-12}\right):\\ \;\;\;\;{b}^{2} + a \cdot \left(\left(1 - \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \frac{a}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 2
Error	20.6
Cost	20424

\[\begin{array}{l} t_0 := \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;angle \leq -0.0044:\\ \;\;\;\;{b}^{2} + \left(a \cdot a\right) \cdot \left(0.5 - \frac{t_0}{2}\right)\\ \mathbf{elif}\;angle \leq 1.32 \cdot 10^{-12}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + a \cdot \left(\left(1 - t_0\right) \cdot \frac{a}{2}\right)\\ \end{array} \]

Alternative 3
Error	20.5
Cost	20424

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.0044:\\ \;\;\;\;{b}^{2} + \frac{a}{\frac{2}{a}} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;angle \leq 1.32 \cdot 10^{-12}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + a \cdot \left(\left(1 - \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \frac{a}{2}\right)\\ \end{array} \]

Alternative 4
Error	20.5
Cost	20424

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.0044:\\ \;\;\;\;{b}^{2} - a \cdot \frac{a}{\frac{2}{-1 + \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}}\\ \mathbf{elif}\;angle \leq 1.32 \cdot 10^{-12}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + a \cdot \left(\left(1 - \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \frac{a}{2}\right)\\ \end{array} \]

Alternative 5
Error	20.5
Cost	20424

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.0044:\\ \;\;\;\;{b}^{2} + \frac{-1}{\frac{\frac{2}{-1 + \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}}{a \cdot a}}\\ \mathbf{elif}\;angle \leq 1.32 \cdot 10^{-12}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + a \cdot \left(\left(1 - \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \frac{a}{2}\right)\\ \end{array} \]

Alternative 6
Error	22.0
Cost	20104

\[\begin{array}{l} \mathbf{if}\;angle \leq -2.45 \cdot 10^{+30}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;angle \leq 6.4 \cdot 10^{+47}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \]

Alternative 7
Error	22.0
Cost	19977

\[\begin{array}{l} \mathbf{if}\;angle \leq -9.4 \cdot 10^{+32} \lor \neg \left(angle \leq 5.8 \cdot 10^{+47}\right):\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(b, \pi \cdot \left(0.005555555555555556 \cdot \left(a \cdot angle\right)\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 8
Error	32.7
Cost	192

\[b \cdot b \]

Reproduce

herbie shell --seed 2023187 
(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)))