ab-angle->ABCF C

Average Error: 20.7 → 21.1

Time: 17.1s

Precision: binary64

Cost: 20424

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;angle \leq -5.5 \cdot 10^{+14}:\\ \;\;\;\;{a}^{2} + \frac{1 - \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)}{\frac{2}{b \cdot b}}\\ \mathbf{elif}\;angle \leq 5.5 \cdot 10^{-14}:\\ \;\;\;\;{a}^{2} + {\left(\pi \cdot \left(b \cdot angle\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + \frac{b}{\frac{2}{b}} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]

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
 (if (<= angle -5.5e+14)
   (+
    (pow a 2.0)
    (/ (- 1.0 (cos (* (* PI angle) 0.011111111111111112))) (/ 2.0 (* b b))))
   (if (<= angle 5.5e-14)
     (+ (pow a 2.0) (* (pow (* PI (* b angle)) 2.0) 3.08641975308642e-5))
     (+
      (pow a 2.0)
      (*
       (/ b (/ 2.0 b))
       (- 1.0 (cos (* angle (* PI 0.011111111111111112)))))))))

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) {
	double tmp;
	if (angle <= -5.5e+14) {
		tmp = pow(a, 2.0) + ((1.0 - cos(((((double) M_PI) * angle) * 0.011111111111111112))) / (2.0 / (b * b)));
	} else if (angle <= 5.5e-14) {
		tmp = pow(a, 2.0) + (pow((((double) M_PI) * (b * angle)), 2.0) * 3.08641975308642e-5);
	} else {
		tmp = pow(a, 2.0) + ((b / (2.0 / b)) * (1.0 - cos((angle * (((double) M_PI) * 0.011111111111111112)))));
	}
	return tmp;
}

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) {
	double tmp;
	if (angle <= -5.5e+14) {
		tmp = Math.pow(a, 2.0) + ((1.0 - Math.cos(((Math.PI * angle) * 0.011111111111111112))) / (2.0 / (b * b)));
	} else if (angle <= 5.5e-14) {
		tmp = Math.pow(a, 2.0) + (Math.pow((Math.PI * (b * angle)), 2.0) * 3.08641975308642e-5);
	} else {
		tmp = Math.pow(a, 2.0) + ((b / (2.0 / b)) * (1.0 - Math.cos((angle * (Math.PI * 0.011111111111111112)))));
	}
	return tmp;
}

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):
	tmp = 0
	if angle <= -5.5e+14:
		tmp = math.pow(a, 2.0) + ((1.0 - math.cos(((math.pi * angle) * 0.011111111111111112))) / (2.0 / (b * b)))
	elif angle <= 5.5e-14:
		tmp = math.pow(a, 2.0) + (math.pow((math.pi * (b * angle)), 2.0) * 3.08641975308642e-5)
	else:
		tmp = math.pow(a, 2.0) + ((b / (2.0 / b)) * (1.0 - math.cos((angle * (math.pi * 0.011111111111111112)))))
	return tmp

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)
	tmp = 0.0
	if (angle <= -5.5e+14)
		tmp = Float64((a ^ 2.0) + Float64(Float64(1.0 - cos(Float64(Float64(pi * angle) * 0.011111111111111112))) / Float64(2.0 / Float64(b * b))));
	elseif (angle <= 5.5e-14)
		tmp = Float64((a ^ 2.0) + Float64((Float64(pi * Float64(b * angle)) ^ 2.0) * 3.08641975308642e-5));
	else
		tmp = Float64((a ^ 2.0) + Float64(Float64(b / Float64(2.0 / b)) * Float64(1.0 - cos(Float64(angle * Float64(pi * 0.011111111111111112))))));
	end
	return tmp
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_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= -5.5e+14)
		tmp = (a ^ 2.0) + ((1.0 - cos(((pi * angle) * 0.011111111111111112))) / (2.0 / (b * b)));
	elseif (angle <= 5.5e-14)
		tmp = (a ^ 2.0) + (((pi * (b * angle)) ^ 2.0) * 3.08641975308642e-5);
	else
		tmp = (a ^ 2.0) + ((b / (2.0 / b)) * (1.0 - cos((angle * (pi * 0.011111111111111112)))));
	end
	tmp_2 = tmp;
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_] := If[LessEqual[angle, -5.5e+14], N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[(1.0 - N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 5.5e-14], N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[Power[N[(Pi * N[(b * angle), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[(b / N[(2.0 / b), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < -5.5e14

   1. Initial program 46.9

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

   2. Taylor expanded in angle around 0 46.2

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

   3. Applied egg-rr 46.2

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot b\right)}^{2}\right)\right)} \]

   4. Applied egg-rr 46.1

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\frac{\left(b \cdot b\right) \cdot \left(\cos 0 - \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)}{2}} \]

   5. Simplified 46.1

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\frac{1 - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}{\frac{2}{b \cdot b}}} \]
      Proof

      [Start] 46.1	\[ {\left(a \cdot 1\right)}^{2} + \frac{\left(b \cdot b\right) \cdot \left(\cos 0 - \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)}{2} \]
      *-commutative [=>] 46.1	\[ {\left(a \cdot 1\right)}^{2} + \frac{\color{blue}{\left(\cos 0 - \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot \left(b \cdot b\right)}}{2} \]
      associate-/l* [=>] 46.1	\[ {\left(a \cdot 1\right)}^{2} + \color{blue}{\frac{\cos 0 - \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)}{\frac{2}{b \cdot b}}} \]
      cos-0 [=>] 46.1	\[ {\left(a \cdot 1\right)}^{2} + \frac{\color{blue}{1} - \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)}{\frac{2}{b \cdot b}} \]
      *-commutative [=>] 46.1	\[ {\left(a \cdot 1\right)}^{2} + \frac{1 - \cos \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot 0.011111111111111112\right)}{\frac{2}{b \cdot b}} \]

   if -5.5e14 < angle < 5.49999999999999991e-14

   1. Initial program 0.5

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

   2. Taylor expanded in angle around 0 0.8

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

   3. Taylor expanded in angle around 0 1.2

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

   4. Applied egg-rr 1.3

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{{\left(b \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}} \]

   5. Applied egg-rr 1.3

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left({\left(\pi \cdot \left(b \cdot angle\right)\right)}^{2} \cdot 1\right)} \cdot 3.08641975308642 \cdot 10^{-5} \]

   if 5.49999999999999991e-14 < angle

   1. Initial program 42.9

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

   2. Taylor expanded in angle around 0 42.8

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

   3. Applied egg-rr 43.7

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\frac{\left(b \cdot b\right) \cdot \left(\cos \left(\left(\pi \cdot angle\right) \cdot 0\right) - \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)}{2}} \]

   4. Simplified 43.7

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\frac{b}{\frac{2}{b}} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)} \]
      Proof

      [Start] 43.7	\[ {\left(a \cdot 1\right)}^{2} + \frac{\left(b \cdot b\right) \cdot \left(\cos \left(\left(\pi \cdot angle\right) \cdot 0\right) - \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)}{2} \]
      unpow2 [<=] 43.7	\[ {\left(a \cdot 1\right)}^{2} + \frac{\color{blue}{{b}^{2}} \cdot \left(\cos \left(\left(\pi \cdot angle\right) \cdot 0\right) - \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)}{2} \]
      associate-*l/ [<=] 43.7	\[ {\left(a \cdot 1\right)}^{2} + \color{blue}{\frac{{b}^{2}}{2} \cdot \left(\cos \left(\left(\pi \cdot angle\right) \cdot 0\right) - \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)} \]
      unpow2 [=>] 43.7	\[ {\left(a \cdot 1\right)}^{2} + \frac{\color{blue}{b \cdot b}}{2} \cdot \left(\cos \left(\left(\pi \cdot angle\right) \cdot 0\right) - \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \]
      associate-/l* [=>] 43.7	\[ {\left(a \cdot 1\right)}^{2} + \color{blue}{\frac{b}{\frac{2}{b}}} \cdot \left(\cos \left(\left(\pi \cdot angle\right) \cdot 0\right) - \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \]
      mul0-rgt [=>] 43.7	\[ {\left(a \cdot 1\right)}^{2} + \frac{b}{\frac{2}{b}} \cdot \left(\cos \color{blue}{0} - \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \]
      cos-0 [=>] 43.7	\[ {\left(a \cdot 1\right)}^{2} + \frac{b}{\frac{2}{b}} \cdot \left(\color{blue}{1} - \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \]
      *-commutative [=>] 43.7	\[ {\left(a \cdot 1\right)}^{2} + \frac{b}{\frac{2}{b}} \cdot \left(1 - \cos \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot 0.011111111111111112\right)\right) \]
      associate-*l* [=>] 43.7	\[ {\left(a \cdot 1\right)}^{2} + \frac{b}{\frac{2}{b}} \cdot \left(1 - \cos \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 21.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq -5.5 \cdot 10^{+14}:\\ \;\;\;\;{a}^{2} + \frac{1 - \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)}{\frac{2}{b \cdot b}}\\ \mathbf{elif}\;angle \leq 5.5 \cdot 10^{-14}:\\ \;\;\;\;{a}^{2} + {\left(\pi \cdot \left(b \cdot angle\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + \frac{b}{\frac{2}{b}} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	20.7
Cost	26240

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

Alternative 2
Error	20.6
Cost	26240

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

Alternative 3
Error	21.1
Cost	20425

\[\begin{array}{l} \mathbf{if}\;angle \leq -5.5 \cdot 10^{+14} \lor \neg \left(angle \leq 5.5 \cdot 10^{-14}\right):\\ \;\;\;\;{a}^{2} + \frac{b}{\frac{2}{b}} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(\pi \cdot \left(b \cdot angle\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]

Alternative 4
Error	24.0
Cost	20360

\[\begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+153}:\\ \;\;\;\;{a}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(angle \cdot \left(b \cdot \pi\right)\right)}^{2}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-38}:\\ \;\;\;\;{a}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(angle \cdot \left(b \cdot \left(b \cdot {\pi}^{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}\\ \end{array} \]

Alternative 5
Error	26.6
Cost	19840

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

Alternative 6
Error	26.6
Cost	19840

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

Alternative 7
Error	26.5
Cost	19840

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

Alternative 8
Error	26.6
Cost	19840

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

Reproduce

herbie shell --seed 2023066 
(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)))