ab-angle->ABCF A

Average Error: 68.2% → 68.1%

Time: 31.4s

Precision: binary64

Cost: 26240.00

Math

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

↓

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

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

Derivation

1. Initial program 68.2

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

2. Simplified 68.2

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

   [Start] 68.2	\[ {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   associate-/r/ [<=] 68.2	\[ {\left(a \cdot \sin \color{blue}{\left(\frac{angle}{\frac{180}{\pi}}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   associate-/r/ [<=] 68.2	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle}{\frac{180}{\pi}}\right)}\right)}^{2} \]

3. Taylor expanded in angle around 0 68.1

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

4. Final simplification 68.1

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

Alternatives

Alternative 1
Error	67.4%
Cost	26500.00

\[\begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \mathsf{expm1}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \frac{a \cdot a}{2} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	68.0%
Cost	26240.00

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

Alternative 3
Error	68.1%
Cost	26240.00

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

Alternative 4
Error	67.7%
Cost	20681.00

\[\begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -0.002 \lor \neg \left(\frac{angle}{180} \leq 5 \cdot 10^{-21}\right):\\ \;\;\;\;{b}^{2} + \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right) \cdot \frac{a}{\frac{2}{a}}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(0.005555555555555556 \cdot \left(\pi \cdot \left(a \cdot angle\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 5
Error	67.7%
Cost	20680.00

\[\begin{array}{l} t_0 := \frac{a}{\frac{2}{a}}\\ \mathbf{if}\;\frac{angle}{180} \leq -0.002:\\ \;\;\;\;{b}^{2} + \left(1 - \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) \cdot t_0\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;{b}^{2} + {\left(0.005555555555555556 \cdot \left(\pi \cdot \left(a \cdot angle\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right) \cdot t_0\\ \end{array} \]

Alternative 6
Error	67.7%
Cost	20680.00

\[\begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -0.002:\\ \;\;\;\;{b}^{2} + \left(1 - \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) \cdot \frac{a}{\frac{2}{a}}\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;{b}^{2} + {\left(0.005555555555555556 \cdot \left(\pi \cdot \left(a \cdot angle\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \frac{a \cdot a}{2} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 7
Error	67.7%
Cost	20680.00

\[\begin{array}{l} t_0 := \frac{a \cdot a}{2}\\ \mathbf{if}\;\frac{angle}{180} \leq -0.002:\\ \;\;\;\;{b}^{2} + t_0 \cdot \left(1 - \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;{b}^{2} + {\left(0.005555555555555556 \cdot \left(\pi \cdot \left(a \cdot angle\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + t_0 \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 8
Error	67.6%
Cost	20680.00

\[\begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -0.002:\\ \;\;\;\;{b}^{2} + \frac{a \cdot a}{\frac{2}{1 - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}}\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;{b}^{2} + {\left(0.005555555555555556 \cdot \left(\pi \cdot \left(a \cdot angle\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \frac{a \cdot a}{2} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 9
Error	59.3%
Cost	19840.00

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

Alternative 10
Error	59.3%
Cost	19840.00

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

Alternative 11
Error	59.3%
Cost	19840.00

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

Alternative 12
Error	59.4%
Cost	19840.00

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

Alternative 13
Error	59.4%
Cost	19840.00

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

Reproduce

herbie shell --seed 2023093 
(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)))