ab-angle->ABCF C

Average Error: 20.8 → 20.9

Time: 13.3s

Precision: binary64

Cost: 26240

Math

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

↓

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

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

Java

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

↓

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

Python

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

↓

def code(a, b, angle):
	return math.pow(a, 2.0) + math.pow((b * math.sin((0.005555555555555556 * (angle * math.pi)))), 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)
	return Float64((a ^ 2.0) + (Float64(b * sin(Float64(0.005555555555555556 * Float64(angle * pi)))) ^ 2.0))
end

MATLAB

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

↓

function tmp = code(a, b, angle)
	tmp = (a ^ 2.0) + ((b * sin((0.005555555555555556 * (angle * pi)))) ^ 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_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.8

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

2. Taylor expanded in angle around 0 20.9

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

3. Taylor expanded in b around 0 20.9

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

4. Final simplification 20.9

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

Alternatives

Alternative 1
Error	22.6
Cost	20104

\[\begin{array}{l} \mathbf{if}\;angle \leq -2.5 \cdot 10^{+56}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;angle \leq 1.12 \cdot 10^{+49}:\\ \;\;\;\;{a}^{2} + {\left(0.005555555555555556 \cdot \left(\pi \cdot \left(b \cdot angle\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]

Alternative 2
Error	22.5
Cost	20104

\[\begin{array}{l} \mathbf{if}\;angle \leq -3.2 \cdot 10^{+56}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;angle \leq 2.2 \cdot 10^{+33}:\\ \;\;\;\;{a}^{2} + {\left(angle \cdot \left(b \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]

Alternative 3
Error	22.5
Cost	20104

\[\begin{array}{l} \mathbf{if}\;angle \leq -2.3 \cdot 10^{+56}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;angle \leq 2.1 \cdot 10^{+49}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]

Alternative 4
Error	22.6
Cost	19976

\[\begin{array}{l} \mathbf{if}\;angle \leq -2.3 \cdot 10^{+56}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;angle \leq 4.6 \cdot 10^{+39}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, angle \cdot \left(0.005555555555555556 \cdot \left(b \cdot \pi\right)\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]

Alternative 5
Error	22.6
Cost	19976

\[\begin{array}{l} \mathbf{if}\;angle \leq -2.4 \cdot 10^{+56}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;angle \leq 1.02 \cdot 10^{+49}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, 0.005555555555555556 \cdot \left(\pi \cdot \left(b \cdot angle\right)\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]

Alternative 6
Error	29.3
Cost	14288

\[\begin{array}{l} t_0 := a \cdot a + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left({\pi}^{2} \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{if}\;angle \leq -2.3 \cdot 10^{+56}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;angle \leq -9 \cdot 10^{-161}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;angle \leq 6.8 \cdot 10^{-133}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;angle \leq 3.6 \cdot 10^{+42}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]

Alternative 7
Error	32.9
Cost	192

\[a \cdot a \]

Reproduce

herbie shell --seed 2022325 
(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)))