ab-angle->ABCF C

Average Accuracy: 68.3% → 68.2%

Time: 17.9s

Precision: binary64

Cost: 32832

Math

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

↓

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

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 (* (pow (/ 1.0 angle) -1.0) (* PI 0.005555555555555556))))
   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((pow((1.0 / angle), -1.0) * (((double) M_PI) * 0.005555555555555556)))), 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((Math.pow((1.0 / angle), -1.0) * (Math.PI * 0.005555555555555556)))), 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((math.pow((1.0 / angle), -1.0) * (math.pi * 0.005555555555555556)))), 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((Float64(1.0 / angle) ^ -1.0) * Float64(pi * 0.005555555555555556)))) ^ 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((((1.0 / angle) ^ -1.0) * (pi * 0.005555555555555556)))) ^ 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[(N[Power[N[(1.0 / angle), $MachinePrecision], -1.0], $MachinePrecision] * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.3%

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

2. Applied egg-rr 68.3%

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

   [Start] 68.3	\[ {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   associate-*r/ [=>] 68.3	\[ {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\frac{\pi \cdot angle}{180}\right)}\right)}^{2} \]
   clear-num [=>] 68.2	\[ {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\frac{1}{\frac{180}{\pi \cdot angle}}\right)}\right)}^{2} \]
   *-commutative [=>] 68.2	\[ {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{1}{\frac{180}{\color{blue}{angle \cdot \pi}}}\right)\right)}^{2} \]
   associate-/r* [=>] 68.3	\[ {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{1}{\color{blue}{\frac{\frac{180}{angle}}{\pi}}}\right)\right)}^{2} \]

3. Taylor expanded in angle around 0 68.1%

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

4. Taylor expanded in angle around 0 68.1%

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

5. Simplified 68.2%

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

   [Start] 68.1	\[ {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)\right)}^{2} \]
   associate-/l/ [<=] 68.2	\[ {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\frac{1}{\color{blue}{\frac{\frac{180}{\pi}}{angle}}}\right)\right)}^{2} \]

6. Applied egg-rr 68.2%

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

   [Start] 68.2	\[ {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\frac{1}{\frac{\frac{180}{\pi}}{angle}}\right)\right)}^{2} \]
   inv-pow [=>] 68.2	\[ {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left({\left(\frac{\frac{180}{\pi}}{angle}\right)}^{-1}\right)}\right)}^{2} \]
   div-inv [=>] 68.1	\[ {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left({\color{blue}{\left(\frac{180}{\pi} \cdot \frac{1}{angle}\right)}}^{-1}\right)\right)}^{2} \]
   metadata-eval [<=] 68.1	\[ {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left({\left(\frac{180}{\pi} \cdot \frac{1}{angle}\right)}^{\color{blue}{\left(-1\right)}}\right)\right)}^{2} \]
   unpow-prod-down [=>] 68.2	\[ {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left({\left(\frac{180}{\pi}\right)}^{\left(-1\right)} \cdot {\left(\frac{1}{angle}\right)}^{\left(-1\right)}\right)}\right)}^{2} \]
   pow-flip [<=] 68.2	\[ {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\color{blue}{\frac{1}{{\left(\frac{180}{\pi}\right)}^{1}}} \cdot {\left(\frac{1}{angle}\right)}^{\left(-1\right)}\right)\right)}^{2} \]
   pow1 [<=] 68.2	\[ {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\frac{1}{\color{blue}{\frac{180}{\pi}}} \cdot {\left(\frac{1}{angle}\right)}^{\left(-1\right)}\right)\right)}^{2} \]
   clear-num [<=] 68.2	\[ {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\color{blue}{\frac{\pi}{180}} \cdot {\left(\frac{1}{angle}\right)}^{\left(-1\right)}\right)\right)}^{2} \]
   div-inv [=>] 68.2	\[ {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\color{blue}{\left(\pi \cdot \frac{1}{180}\right)} \cdot {\left(\frac{1}{angle}\right)}^{\left(-1\right)}\right)\right)}^{2} \]
   metadata-eval [=>] 68.2	\[ {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\left(\pi \cdot \color{blue}{0.005555555555555556}\right) \cdot {\left(\frac{1}{angle}\right)}^{\left(-1\right)}\right)\right)}^{2} \]
   metadata-eval [=>] 68.2	\[ {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\left(\pi \cdot 0.005555555555555556\right) \cdot {\left(\frac{1}{angle}\right)}^{\color{blue}{-1}}\right)\right)}^{2} \]

7. Final simplification 68.2%

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

Alternatives

Alternative 1
Accuracy	68.1%
Cost	26368

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

Alternative 2
Accuracy	68.2%
Cost	26240

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

Alternative 3
Accuracy	68.2%
Cost	26240

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

Alternative 4
Accuracy	67.8%
Cost	20425

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.005 \lor \neg \left(angle \leq 2.6 \cdot 10^{-16}\right):\\ \;\;\;\;{a}^{2} + \frac{b}{\frac{2}{b}} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(\left(b \cdot angle\right) \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{2}\\ \end{array} \]

Alternative 5
Accuracy	63.1%
Cost	20361

\[\begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{+153} \lor \neg \left(b \leq 10^{+119}\right):\\ \;\;\;\;{a}^{2} + {\left(\left(b \cdot angle\right) \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(angle \cdot \left(b \cdot \left(b \cdot {\pi}^{2}\right)\right)\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	59.1%
Cost	19840

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

Alternative 7
Accuracy	59.1%
Cost	19840

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

Alternative 8
Accuracy	59.1%
Cost	19840

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

Alternative 9
Accuracy	59.2%
Cost	19840

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

Alternative 10
Accuracy	59.2%
Cost	19840

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

Reproduce

herbie shell --seed 2023135 
(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)))