ab-angle->ABCF A

Average Accuracy: 67.8% → 67.2%

Time: 21.1s

Precision: binary64

Cost: 20489

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;angle \leq -1.1 \cdot 10^{+14} \lor \neg \left(angle \leq 0.07\right):\\ \;\;\;\;{b}^{2} + \frac{a}{\frac{2}{a}} \cdot \left(1 - \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + 0.005555555555555556 \cdot \left(\left(angle \cdot \left(a \cdot \pi\right)\right) \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \left(a \cdot \pi\right)\right)\right)\right)\\ \end{array} \]

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
 (if (or (<= angle -1.1e+14) (not (<= angle 0.07)))
   (+
    (pow b 2.0)
    (* (/ a (/ 2.0 a)) (- 1.0 (cos (* PI (* angle 0.011111111111111112))))))
   (+
    (pow b 2.0)
    (*
     0.005555555555555556
     (* (* angle (* a PI)) (* angle (* 0.005555555555555556 (* a PI))))))))

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) {
	double tmp;
	if ((angle <= -1.1e+14) || !(angle <= 0.07)) {
		tmp = pow(b, 2.0) + ((a / (2.0 / a)) * (1.0 - cos((((double) M_PI) * (angle * 0.011111111111111112)))));
	} else {
		tmp = pow(b, 2.0) + (0.005555555555555556 * ((angle * (a * ((double) M_PI))) * (angle * (0.005555555555555556 * (a * ((double) M_PI))))));
	}
	return tmp;
}

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) {
	double tmp;
	if ((angle <= -1.1e+14) || !(angle <= 0.07)) {
		tmp = Math.pow(b, 2.0) + ((a / (2.0 / a)) * (1.0 - Math.cos((Math.PI * (angle * 0.011111111111111112)))));
	} else {
		tmp = Math.pow(b, 2.0) + (0.005555555555555556 * ((angle * (a * Math.PI)) * (angle * (0.005555555555555556 * (a * Math.PI)))));
	}
	return tmp;
}

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):
	tmp = 0
	if (angle <= -1.1e+14) or not (angle <= 0.07):
		tmp = math.pow(b, 2.0) + ((a / (2.0 / a)) * (1.0 - math.cos((math.pi * (angle * 0.011111111111111112)))))
	else:
		tmp = math.pow(b, 2.0) + (0.005555555555555556 * ((angle * (a * math.pi)) * (angle * (0.005555555555555556 * (a * math.pi)))))
	return tmp

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)
	tmp = 0.0
	if ((angle <= -1.1e+14) || !(angle <= 0.07))
		tmp = Float64((b ^ 2.0) + Float64(Float64(a / Float64(2.0 / a)) * Float64(1.0 - cos(Float64(pi * Float64(angle * 0.011111111111111112))))));
	else
		tmp = Float64((b ^ 2.0) + Float64(0.005555555555555556 * Float64(Float64(angle * Float64(a * pi)) * Float64(angle * Float64(0.005555555555555556 * Float64(a * pi))))));
	end
	return tmp
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_2 = code(a, b, angle)
	tmp = 0.0;
	if ((angle <= -1.1e+14) || ~((angle <= 0.07)))
		tmp = (b ^ 2.0) + ((a / (2.0 / a)) * (1.0 - cos((pi * (angle * 0.011111111111111112)))));
	else
		tmp = (b ^ 2.0) + (0.005555555555555556 * ((angle * (a * pi)) * (angle * (0.005555555555555556 * (a * pi)))));
	end
	tmp_2 = tmp;
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_] := If[Or[LessEqual[angle, -1.1e+14], N[Not[LessEqual[angle, 0.07]], $MachinePrecision]], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(a / N[(2.0 / a), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(0.005555555555555556 * N[(N[(angle * N[(a * Pi), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(0.005555555555555556 * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < -1.1e14 or 0.070000000000000007 < angle

   1. Initial program 28.4%

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

   2. Simplified 28.5%

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

      [Start] 28.4	\[ {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
      associate-*l/ [=>] 28.4	\[ {\left(a \cdot \sin \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
      associate-*r/ [<=] 28.4	\[ {\left(a \cdot \sin \color{blue}{\left(angle \cdot \frac{\pi}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
      associate-*l/ [=>] 28.3	\[ {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)}\right)}^{2} \]
      associate-*r/ [<=] 28.5	\[ {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(angle \cdot \frac{\pi}{180}\right)}\right)}^{2} \]

   3. Taylor expanded in angle around 0 28.8%

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

   4. Applied egg-rr 28.7%

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

      [Start] 28.8	\[ {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
      unpow2 [=>] 28.8	\[ \color{blue}{\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right) \cdot \left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
      swap-sqr [=>] 28.8	\[ \color{blue}{\left(a \cdot a\right) \cdot \left(\sin \left(angle \cdot \frac{\pi}{180}\right) \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
      sin-mult [=>] 28.8	\[ \left(a \cdot a\right) \cdot \color{blue}{\frac{\cos \left(angle \cdot \frac{\pi}{180} - angle \cdot \frac{\pi}{180}\right) - \cos \left(angle \cdot \frac{\pi}{180} + angle \cdot \frac{\pi}{180}\right)}{2}} + {\left(b \cdot 1\right)}^{2} \]
      associate-*r/ [=>] 28.7	\[ \color{blue}{\frac{\left(a \cdot a\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{180} - angle \cdot \frac{\pi}{180}\right) - \cos \left(angle \cdot \frac{\pi}{180} + angle \cdot \frac{\pi}{180}\right)\right)}{2}} + {\left(b \cdot 1\right)}^{2} \]

   5. Simplified 28.8%

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

      [Start] 28.7	\[ \frac{\left(a \cdot a\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot 0\right) - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)}{2} + {\left(b \cdot 1\right)}^{2} \]
      unpow2 [<=] 28.7	\[ \frac{\color{blue}{{a}^{2}} \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot 0\right) - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)}{2} + {\left(b \cdot 1\right)}^{2} \]
      associate-*l/ [<=] 28.7	\[ \color{blue}{\frac{{a}^{2}}{2} \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot 0\right) - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
      unpow2 [=>] 28.7	\[ \frac{\color{blue}{a \cdot a}}{2} \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot 0\right) - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      associate-/l* [=>] 28.7	\[ \color{blue}{\frac{a}{\frac{2}{a}}} \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot 0\right) - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      mul0-rgt [=>] 28.7	\[ \frac{a}{\frac{2}{a}} \cdot \left(\cos \color{blue}{0} - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      cos-0 [=>] 28.7	\[ \frac{a}{\frac{2}{a}} \cdot \left(\color{blue}{1} - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      *-commutative [=>] 28.7	\[ \frac{a}{\frac{2}{a}} \cdot \left(1 - \cos \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot 0.011111111111111112\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      associate-*l* [=>] 28.8	\[ \frac{a}{\frac{2}{a}} \cdot \left(1 - \cos \color{blue}{\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

   if -1.1e14 < angle < 0.070000000000000007

   1. Initial program 99.2%

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

   2. Simplified 99.3%

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

      [Start] 99.2	\[ {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
      associate-*l/ [=>] 99.2	\[ {\left(a \cdot \sin \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
      associate-*r/ [<=] 99.3	\[ {\left(a \cdot \sin \color{blue}{\left(angle \cdot \frac{\pi}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
      associate-*l/ [=>] 99.3	\[ {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)}\right)}^{2} \]
      associate-*r/ [<=] 99.3	\[ {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(angle \cdot \frac{\pi}{180}\right)}\right)}^{2} \]

   3. Taylor expanded in angle around 0 98.7%

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

   4. Taylor expanded in angle around 0 97.8%

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

   5. Simplified 97.8%

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

      [Start] 97.8	\[ {\left(0.005555555555555556 \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
      *-commutative [=>] 97.8	\[ {\left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot a\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   6. Applied egg-rr 97.9%

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

      [Start] 97.8	\[ {\left(0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
      unpow2 [=>] 97.8	\[ \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right) \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
      associate-*l* [=>] 97.8	\[ \color{blue}{0.005555555555555556 \cdot \left(\left(angle \cdot \left(\pi \cdot a\right)\right) \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
      *-commutative [=>] 97.8	\[ \color{blue}{\left(\left(angle \cdot \left(\pi \cdot a\right)\right) \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right)\right) \cdot 0.005555555555555556} + {\left(b \cdot 1\right)}^{2} \]
      *-commutative [=>] 97.8	\[ \left(\left(angle \cdot \left(\pi \cdot a\right)\right) \cdot \color{blue}{\left(\left(angle \cdot \left(\pi \cdot a\right)\right) \cdot 0.005555555555555556\right)}\right) \cdot 0.005555555555555556 + {\left(b \cdot 1\right)}^{2} \]
      associate-*l* [=>] 97.9	\[ \left(\left(angle \cdot \left(\pi \cdot a\right)\right) \cdot \color{blue}{\left(angle \cdot \left(\left(\pi \cdot a\right) \cdot 0.005555555555555556\right)\right)}\right) \cdot 0.005555555555555556 + {\left(b \cdot 1\right)}^{2} \]
      *-commutative [=>] 97.9	\[ \left(\left(angle \cdot \left(\pi \cdot a\right)\right) \cdot \left(angle \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(\pi \cdot a\right)\right)}\right)\right) \cdot 0.005555555555555556 + {\left(b \cdot 1\right)}^{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq -1.1 \cdot 10^{+14} \lor \neg \left(angle \leq 0.07\right):\\ \;\;\;\;{b}^{2} + \frac{a}{\frac{2}{a}} \cdot \left(1 - \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + 0.005555555555555556 \cdot \left(\left(angle \cdot \left(a \cdot \pi\right)\right) \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \left(a \cdot \pi\right)\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	67.9%
Cost	58624

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

Alternative 2
Accuracy	67.9%
Cost	45760

\[{\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(angle \cdot \left(\pi \cdot \sqrt[3]{-1.7146776406035666 \cdot 10^{-7}}\right)\right)\right)}^{2} \]

Alternative 3
Accuracy	67.6%
Cost	26240

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

Alternative 4
Accuracy	67.7%
Cost	26240

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

Alternative 5
Accuracy	67.7%
Cost	26240

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

Alternative 6
Accuracy	67.3%
Cost	20425

\[\begin{array}{l} \mathbf{if}\;angle \leq -1.1 \cdot 10^{+14} \lor \neg \left(angle \leq 0.07\right):\\ \;\;\;\;{b}^{2} + \frac{a}{\frac{2}{a}} \cdot \left(1 - \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(angle \cdot \left(a \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 7
Accuracy	58.5%
Cost	19840

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

Alternative 8
Accuracy	58.5%
Cost	19840

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

Alternative 9
Accuracy	58.6%
Cost	19840

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

Alternative 10
Accuracy	58.6%
Cost	19840

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

Reproduce

herbie shell --seed 2023135 
(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)))