ab-angle->ABCF C

Average Error: 20.5 → 20.6

Time: 1.1min

Precision: binary64

Cost: 105984

Math

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

↓

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot -0.005555555555555556\right)\\ t_1 := \frac{t_0}{-2}\\ {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) + \frac{3 \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) - \left(\sin \left(\frac{t_0}{2}\right) \cdot \cos t_1 - \sin \left(\frac{\pi \cdot \left(1 - angle \cdot -0.005555555555555556\right)}{2}\right) \cdot \sin t_1\right)}{4}}{2}\right)}^{2} \end{array} \]

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
 (let* ((t_0 (* angle (* PI -0.005555555555555556))) (t_1 (/ t_0 -2.0)))
   (+
    (pow (* a (cos (* PI (/ angle 180.0)))) 2.0)
    (pow
     (*
      b
      (/
       (+
        (sin (* 0.005555555555555556 (* angle PI)))
        (/
         (-
          (* 3.0 (sin (* PI (* 0.005555555555555556 angle))))
          (-
           (* (sin (/ t_0 2.0)) (cos t_1))
           (*
            (sin (/ (* PI (- 1.0 (* angle -0.005555555555555556))) 2.0))
            (sin t_1))))
         4.0))
       2.0))
     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) {
	double t_0 = angle * (((double) M_PI) * -0.005555555555555556);
	double t_1 = t_0 / -2.0;
	return pow((a * cos((((double) M_PI) * (angle / 180.0)))), 2.0) + pow((b * ((sin((0.005555555555555556 * (angle * ((double) M_PI)))) + (((3.0 * sin((((double) M_PI) * (0.005555555555555556 * angle)))) - ((sin((t_0 / 2.0)) * cos(t_1)) - (sin(((((double) M_PI) * (1.0 - (angle * -0.005555555555555556))) / 2.0)) * sin(t_1)))) / 4.0)) / 2.0)), 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) {
	double t_0 = angle * (Math.PI * -0.005555555555555556);
	double t_1 = t_0 / -2.0;
	return Math.pow((a * Math.cos((Math.PI * (angle / 180.0)))), 2.0) + Math.pow((b * ((Math.sin((0.005555555555555556 * (angle * Math.PI))) + (((3.0 * Math.sin((Math.PI * (0.005555555555555556 * angle)))) - ((Math.sin((t_0 / 2.0)) * Math.cos(t_1)) - (Math.sin(((Math.PI * (1.0 - (angle * -0.005555555555555556))) / 2.0)) * Math.sin(t_1)))) / 4.0)) / 2.0)), 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):
	t_0 = angle * (math.pi * -0.005555555555555556)
	t_1 = t_0 / -2.0
	return math.pow((a * math.cos((math.pi * (angle / 180.0)))), 2.0) + math.pow((b * ((math.sin((0.005555555555555556 * (angle * math.pi))) + (((3.0 * math.sin((math.pi * (0.005555555555555556 * angle)))) - ((math.sin((t_0 / 2.0)) * math.cos(t_1)) - (math.sin(((math.pi * (1.0 - (angle * -0.005555555555555556))) / 2.0)) * math.sin(t_1)))) / 4.0)) / 2.0)), 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)
	t_0 = Float64(angle * Float64(pi * -0.005555555555555556))
	t_1 = Float64(t_0 / -2.0)
	return Float64((Float64(a * cos(Float64(pi * Float64(angle / 180.0)))) ^ 2.0) + (Float64(b * Float64(Float64(sin(Float64(0.005555555555555556 * Float64(angle * pi))) + Float64(Float64(Float64(3.0 * sin(Float64(pi * Float64(0.005555555555555556 * angle)))) - Float64(Float64(sin(Float64(t_0 / 2.0)) * cos(t_1)) - Float64(sin(Float64(Float64(pi * Float64(1.0 - Float64(angle * -0.005555555555555556))) / 2.0)) * sin(t_1)))) / 4.0)) / 2.0)) ^ 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)
	t_0 = angle * (pi * -0.005555555555555556);
	t_1 = t_0 / -2.0;
	tmp = ((a * cos((pi * (angle / 180.0)))) ^ 2.0) + ((b * ((sin((0.005555555555555556 * (angle * pi))) + (((3.0 * sin((pi * (0.005555555555555556 * angle)))) - ((sin((t_0 / 2.0)) * cos(t_1)) - (sin(((pi * (1.0 - (angle * -0.005555555555555556))) / 2.0)) * sin(t_1)))) / 4.0)) / 2.0)) ^ 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_] := Block[{t$95$0 = N[(angle * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / -2.0), $MachinePrecision]}, 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[(N[(N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(3.0 * N[Sin[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[N[(t$95$0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(N[(Pi * N[(1.0 - N[(angle * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 20.5

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

2. Applied egg-rr 38.2

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) + \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556 - 0.5\right)\right)}{2}}\right)}^{2} \]

3. Simplified 38.2

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) + \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle - 0.5\right)\right)}{2}}\right)}^{2} \]
   Proof

   [Start] 38.2	\[ {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \frac{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) + \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556 - 0.5\right)\right)}{2}\right)}^{2} \]
   rational_best-simplify-50 [<=] 38.2	\[ {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \frac{\sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} + \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556 - 0.5\right)\right)}{2}\right)}^{2} \]
   rational_best-simplify-1 [=>] 38.2	\[ {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) + \cos \left(\pi \cdot \left(\color{blue}{0.005555555555555556 \cdot angle} - 0.5\right)\right)}{2}\right)}^{2} \]

4. Applied egg-rr 20.6

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) + \color{blue}{\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) + 0\right)}}{2}\right)}^{2} \]

5. Simplified 20.6

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) + \color{blue}{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}}{2}\right)}^{2} \]
   Proof

   [Start] 20.6	\[ {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) + \left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) + 0\right)}{2}\right)}^{2} \]
   rational_best-simplify-3 [=>] 20.6	\[ {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) + \color{blue}{\left(0 + \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}}{2}\right)}^{2} \]
   rational_best-simplify-6 [=>] 20.6	\[ {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) + \color{blue}{\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}}{2}\right)}^{2} \]
   rational_best-simplify-1 [=>] 20.6	\[ {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) + \sin \left(\pi \cdot \color{blue}{\left(angle \cdot 0.005555555555555556\right)}\right)}{2}\right)}^{2} \]

6. Applied egg-rr 20.6

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) + \color{blue}{\left(\frac{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 3}{4} - \frac{\sin \left(-\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}{4}\right)}}{2}\right)}^{2} \]

7. Simplified 20.5

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) + \color{blue}{\frac{3 \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) - \sin \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)}{4}}}{2}\right)}^{2} \]
   Proof

   [Start] 20.6	\[ {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) + \left(\frac{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 3}{4} - \frac{\sin \left(-\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}{4}\right)}{2}\right)}^{2} \]
   metadata-eval [<=] 20.6	\[ {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) + \left(\frac{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 3}{\color{blue}{2 \cdot 2}} - \frac{\sin \left(-\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}{4}\right)}{2}\right)}^{2} \]
   rational_best-simplify-53 [<=] 20.6	\[ {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) + \left(\color{blue}{\frac{\frac{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 3}{2}}{2}} - \frac{\sin \left(-\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}{4}\right)}{2}\right)}^{2} \]
   rational_best-simplify-108 [=>] 20.6	\[ {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) + \left(\color{blue}{\frac{\frac{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 3}{2} + \frac{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 3}{2}}{2 + 2}} - \frac{\sin \left(-\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}{4}\right)}{2}\right)}^{2} \]
   metadata-eval [=>] 20.6	\[ {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) + \left(\frac{\frac{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 3}{2} + \frac{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 3}{2}}{\color{blue}{4}} - \frac{\sin \left(-\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}{4}\right)}{2}\right)}^{2} \]
   rational_best-simplify-66 [=>] 20.6	\[ {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) + \color{blue}{\frac{\left(\frac{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 3}{2} + \frac{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 3}{2}\right) - \sin \left(-\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}{4}}}{2}\right)}^{2} \]

8. Applied egg-rr 20.6

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) + \frac{3 \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) - \color{blue}{\left(\sin \left(\frac{angle \cdot \left(\pi \cdot -0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{angle \cdot \left(\pi \cdot -0.005555555555555556\right)}{-2}\right) - \sin \left(\frac{\pi \cdot \left(1 - angle \cdot -0.005555555555555556\right)}{2}\right) \cdot \sin \left(\frac{angle \cdot \left(\pi \cdot -0.005555555555555556\right)}{-2}\right)\right)}}{4}}{2}\right)}^{2} \]

9. Final simplification 20.6

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) + \frac{3 \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) - \left(\sin \left(\frac{angle \cdot \left(\pi \cdot -0.005555555555555556\right)}{2}\right) \cdot \cos \left(\frac{angle \cdot \left(\pi \cdot -0.005555555555555556\right)}{-2}\right) - \sin \left(\frac{\pi \cdot \left(1 - angle \cdot -0.005555555555555556\right)}{2}\right) \cdot \sin \left(\frac{angle \cdot \left(\pi \cdot -0.005555555555555556\right)}{-2}\right)\right)}{4}}{2}\right)}^{2} \]

Alternatives

Alternative 1
Error	20.5
Cost	65984

\[{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) + \frac{3 \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) - \sin \left(angle \cdot \left(-0.005555555555555556 \cdot \pi\right)\right)}{4}}{2}\right)}^{2} \]

Alternative 2
Error	20.6
Cost	52608

\[{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) + \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}{2}\right)}^{2} \]

Alternative 3
Error	20.5
Cost	39360

\[{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

Alternative 4
Error	20.5
Cost	39360

\[{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2} \]

Alternative 5
Error	22.4
Cost	26632

\[\begin{array}{l} t_0 := {\left(a \cdot 1\right)}^{2}\\ t_1 := t_0 + 0.25 \cdot {\left(\cos \left(-0.5 \cdot \pi\right) \cdot b\right)}^{2}\\ \mathbf{if}\;angle \leq -7 \cdot 10^{+40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;angle \leq 24000000000000:\\ \;\;\;\;t_0 + 3.08641975308642 \cdot 10^{-5} \cdot {\left(\left(angle \cdot \pi\right) \cdot b\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	20.7
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 7
Error	20.7
Cost	26368

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

Alternative 8
Error	26.3
Cost	19968

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

Alternative 9
Error	26.3
Cost	19968

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

Reproduce

herbie shell --seed 2023099 
(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)))