ab-angle->ABCF C

Average Error: 20.4 → 20.4

Time: 20.5s

Precision: binary64

Cost: 26368

Math

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

↓

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

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 1.0) 2.0)
  (pow (* b (sin (* angle (* 0.005555555555555556 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 * 1.0), 2.0) + pow((b * sin((angle * (0.005555555555555556 * ((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 * 1.0), 2.0) + Math.pow((b * Math.sin((angle * (0.005555555555555556 * 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 * 1.0), 2.0) + math.pow((b * math.sin((angle * (0.005555555555555556 * 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((Float64(a * 1.0) ^ 2.0) + (Float64(b * sin(Float64(angle * Float64(0.005555555555555556 * 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 * 1.0) ^ 2.0) + ((b * sin((angle * (0.005555555555555556 * 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[N[(a * 1.0), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.4

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

2. Taylor expanded in angle around 0 20.4

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

3. Taylor expanded in b around 0 20.5

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

4. Simplified 20.4

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

   [Start] 20.5	\[ {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} \]
   rational.json-simplify-2 [=>] 20.5	\[ {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \color{blue}{\left(\pi \cdot angle\right)}\right)\right)}^{2} \]
   rational.json-simplify-43 [<=] 20.4	\[ {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right)}^{2} \]

5. Final simplification 20.4

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

Alternatives

Alternative 1
Error	20.5
Cost	26368

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

Alternative 2
Error	22.2
Cost	20232

\[\begin{array}{l} \mathbf{if}\;angle \leq -6.2 \cdot 10^{+56}:\\ \;\;\;\;{a}^{2}\\ \mathbf{elif}\;angle \leq 2.4 \cdot 10^{+19}:\\ \;\;\;\;{\left(a \cdot 1\right)}^{2} + {\left(angle \cdot \left(\pi \cdot \left(b \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2}\\ \end{array} \]

Alternative 3
Error	22.2
Cost	20232

\[\begin{array}{l} \mathbf{if}\;angle \leq -4.8 \cdot 10^{+56}:\\ \;\;\;\;{a}^{2}\\ \mathbf{elif}\;angle \leq 1.75 \cdot 10^{+19}:\\ \;\;\;\;{\left(a \cdot 1\right)}^{2} + {\left(b \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2}\\ \end{array} \]

Alternative 4
Error	22.2
Cost	20104

\[\begin{array}{l} \mathbf{if}\;angle \leq -4.8 \cdot 10^{+56}:\\ \;\;\;\;{a}^{2}\\ \mathbf{elif}\;angle \leq 6.2 \cdot 10^{+19}:\\ \;\;\;\;{a}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(angle \cdot \left(b \cdot \pi\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2}\\ \end{array} \]

Alternative 5
Error	22.2
Cost	20104

\[\begin{array}{l} \mathbf{if}\;angle \leq -7.2 \cdot 10^{+56}:\\ \;\;\;\;{a}^{2}\\ \mathbf{elif}\;angle \leq 1.15 \cdot 10^{+20}:\\ \;\;\;\;{a}^{2} + {\left(angle \cdot \left(0.005555555555555556 \cdot \left(b \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2}\\ \end{array} \]

Alternative 6
Error	31.7
Cost	6528

\[{a}^{2} \]

Reproduce

herbie shell --seed 2023074 
(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)))