ab-angle->ABCF B

Average Error: 31.2 → 21.8

Time: 19.4s

Precision: binary64

Cost: 14340

Math

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

↓

\[\begin{array}{l} t_0 := \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -20:\\ \;\;\;\;\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(t_0 \cdot \left(b - a\right)\right)}{b - a}\\ \mathbf{elif}\;\frac{angle}{180} \leq 2.5 \cdot 10^{-23}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(b \cdot b - a \cdot a\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (*
  (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* PI (/ angle 180.0))))
  (cos (* PI (/ angle 180.0)))))

↓

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (sin (* 0.011111111111111112 (* angle PI)))))
   (if (<= (/ angle 180.0) -20.0)
     (/ (* (* (+ b a) (- b a)) (* t_0 (- b a))) (- b a))
     (if (<= (/ angle 180.0) 2.5e-23)
       (* (- b a) (* (+ b a) (* angle (* 0.011111111111111112 PI))))
       (* t_0 (- (* b b) (* a a)))))))

C

double code(double a, double b, double angle) {
	return ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin((((double) M_PI) * (angle / 180.0)))) * cos((((double) M_PI) * (angle / 180.0)));
}

↓

double code(double a, double b, double angle) {
	double t_0 = sin((0.011111111111111112 * (angle * ((double) M_PI))));
	double tmp;
	if ((angle / 180.0) <= -20.0) {
		tmp = (((b + a) * (b - a)) * (t_0 * (b - a))) / (b - a);
	} else if ((angle / 180.0) <= 2.5e-23) {
		tmp = (b - a) * ((b + a) * (angle * (0.011111111111111112 * ((double) M_PI))));
	} else {
		tmp = t_0 * ((b * b) - (a * a));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	return ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin((Math.PI * (angle / 180.0)))) * Math.cos((Math.PI * (angle / 180.0)));
}

↓

public static double code(double a, double b, double angle) {
	double t_0 = Math.sin((0.011111111111111112 * (angle * Math.PI)));
	double tmp;
	if ((angle / 180.0) <= -20.0) {
		tmp = (((b + a) * (b - a)) * (t_0 * (b - a))) / (b - a);
	} else if ((angle / 180.0) <= 2.5e-23) {
		tmp = (b - a) * ((b + a) * (angle * (0.011111111111111112 * Math.PI)));
	} else {
		tmp = t_0 * ((b * b) - (a * a));
	}
	return tmp;
}

Python

def code(a, b, angle):
	return ((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin((math.pi * (angle / 180.0)))) * math.cos((math.pi * (angle / 180.0)))

↓

def code(a, b, angle):
	t_0 = math.sin((0.011111111111111112 * (angle * math.pi)))
	tmp = 0
	if (angle / 180.0) <= -20.0:
		tmp = (((b + a) * (b - a)) * (t_0 * (b - a))) / (b - a)
	elif (angle / 180.0) <= 2.5e-23:
		tmp = (b - a) * ((b + a) * (angle * (0.011111111111111112 * math.pi)))
	else:
		tmp = t_0 * ((b * b) - (a * a))
	return tmp

Julia

function code(a, b, angle)
	return Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(Float64(pi * Float64(angle / 180.0)))) * cos(Float64(pi * Float64(angle / 180.0))))
end

↓

function code(a, b, angle)
	t_0 = sin(Float64(0.011111111111111112 * Float64(angle * pi)))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -20.0)
		tmp = Float64(Float64(Float64(Float64(b + a) * Float64(b - a)) * Float64(t_0 * Float64(b - a))) / Float64(b - a));
	elseif (Float64(angle / 180.0) <= 2.5e-23)
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * Float64(angle * Float64(0.011111111111111112 * pi))));
	else
		tmp = Float64(t_0 * Float64(Float64(b * b) - Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin((pi * (angle / 180.0)))) * cos((pi * (angle / 180.0)));
end

↓

function tmp_2 = code(a, b, angle)
	t_0 = sin((0.011111111111111112 * (angle * pi)));
	tmp = 0.0;
	if ((angle / 180.0) <= -20.0)
		tmp = (((b + a) * (b - a)) * (t_0 * (b - a))) / (b - a);
	elseif ((angle / 180.0) <= 2.5e-23)
		tmp = (b - a) * ((b + a) * (angle * (0.011111111111111112 * pi)));
	else
		tmp = t_0 * ((b * b) - (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, angle_] := Block[{t$95$0 = N[Sin[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -20.0], N[(N[(N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2.5e-23], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(angle * N[(0.011111111111111112 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -20

   1. Initial program 50.9

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

   2. Simplified 50.9

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

   3. Applied egg-rr 50.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(b + a, \left(b - a\right) \cdot \sin \left(\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot 2\right), \sin \left(\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot 2\right) \cdot \mathsf{fma}\left(a, -a, a \cdot a\right)\right)} \]

   4. Taylor expanded in a around 0 50.9

      \[\leadsto \mathsf{fma}\left(b + a, \left(b - a\right) \cdot \sin \left(\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot 2\right), \sin \left(\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot 2\right) \cdot \color{blue}{0}\right) \]

   5. Applied egg-rr 50.9

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right) \cdot \left(b - a\right)} \]

   6. Applied egg-rr 52.6

      \[\leadsto \color{blue}{\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b - a\right)\right)}{b - a}} \]

   if -20 < (/.f64 angle 180) < 2.5000000000000001e-23

   1. Initial program 17.9

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

   2. Simplified 17.8

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

   3. Applied egg-rr 9.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(b + a, \left(b - a\right) \cdot \sin \left(\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot 2\right), \sin \left(\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot 2\right) \cdot \mathsf{fma}\left(a, -a, a \cdot a\right)\right)} \]

   4. Taylor expanded in a around 0 0.3

      \[\leadsto \mathsf{fma}\left(b + a, \left(b - a\right) \cdot \sin \left(\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot 2\right), \sin \left(\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot 2\right) \cdot \color{blue}{0}\right) \]

   5. Applied egg-rr 0.3

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right) \cdot \left(b - a\right)} \]

   6. Taylor expanded in angle around 0 0.6

      \[\leadsto \left(\left(b + a\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \left(b - a\right) \]

   7. Simplified 0.6

      \[\leadsto \left(\left(b + a\right) \cdot \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right)}\right) \cdot \left(b - a\right) \]

   if 2.5000000000000001e-23 < (/.f64 angle 180)

   1. Initial program 46.0

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

   2. Simplified 46.0

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

   3. Applied egg-rr 51.7

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

   4. Applied egg-rr 45.9

      \[\leadsto \color{blue}{{\left(\sqrt[3]{b \cdot b - a \cdot a}\right)}^{2} \cdot \left(\sqrt[3]{b \cdot b - a \cdot a} \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)} \]

   5. Taylor expanded in b around 0 46.0

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

   6. Simplified 46.0

      \[\leadsto \color{blue}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 21.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -20:\\ \;\;\;\;\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b - a\right)\right)}{b - a}\\ \mathbf{elif}\;\frac{angle}{180} \leq 2.5 \cdot 10^{-23}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	21.3
Cost	26368

\[\left(\left(b + a\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right) \cdot \left(b - a\right) \]

Alternative 2
Error	21.4
Cost	14088

\[\begin{array}{l} t_0 := \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{-28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{angle}{180} \leq 2.5 \cdot 10^{-23}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	22.8
Cost	13640

\[\begin{array}{l} \mathbf{if}\;angle \leq -3.15 \cdot 10^{-15}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \sin \left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right)\\ \mathbf{elif}\;angle \leq 3.8:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a \cdot \left(-a\right)\right)\\ \end{array} \]

Alternative 4
Error	22.7
Cost	13640

\[\begin{array}{l} t_0 := \sin \left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right)\\ \mathbf{if}\;angle \leq -3.15 \cdot 10^{-15}:\\ \;\;\;\;\left(b \cdot b\right) \cdot t_0\\ \mathbf{elif}\;angle \leq 3.8:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(a \cdot \left(-a\right)\right)\\ \end{array} \]

Alternative 5
Error	22.7
Cost	13576

\[\begin{array}{l} t_0 := \left(b \cdot b\right) \cdot \sin \left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right)\\ \mathbf{if}\;angle \leq -3.15 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;angle \leq 0.04:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	22.7
Cost	13576

\[\begin{array}{l} t_0 := angle \cdot \left(0.011111111111111112 \cdot \pi\right)\\ \mathbf{if}\;angle \leq -3.15 \cdot 10^{-15}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \sin \left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right)\\ \mathbf{elif}\;angle \leq 0.04:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \sin t_0\\ \end{array} \]

Alternative 7
Error	29.4
Cost	7432

\[\begin{array}{l} t_0 := 0.011111111111111112 \cdot \left(\left(b \cdot angle\right) \cdot \left(b \cdot \pi\right)\right)\\ \mathbf{if}\;b \leq -3.055656168447592 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.9272518405087043 \cdot 10^{+135}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	24.9
Cost	7300

\[\begin{array}{l} \mathbf{if}\;angle \leq -7.0883178645955424 \cdot 10^{+28}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(1 + \pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \]

Alternative 9
Error	24.8
Cost	7300

\[\begin{array}{l} \mathbf{if}\;angle \leq -7.0883178645955424 \cdot 10^{+28}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(1 + \pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 10
Error	38.7
Cost	7176

\[\begin{array}{l} t_0 := 0.011111111111111112 \cdot \left(\left(b \cdot angle\right) \cdot \left(b \cdot \pi\right)\right)\\ \mathbf{if}\;b \leq -1 \cdot 10^{+160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.9272518405087043 \cdot 10^{+135}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle \cdot \pi\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	38.6
Cost	7176

\[\begin{array}{l} t_0 := 0.011111111111111112 \cdot \left(\left(b \cdot angle\right) \cdot \left(b \cdot \pi\right)\right)\\ \mathbf{if}\;b \leq -1 \cdot 10^{+160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 8.385576574461971 \cdot 10^{+110}:\\ \;\;\;\;angle \cdot \left(b \cdot \left(b \cdot \left(0.011111111111111112 \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	34.1
Cost	7176

\[\begin{array}{l} t_0 := 0.011111111111111112 \cdot \left(\left(b \cdot angle\right) \cdot \left(b \cdot \pi\right)\right)\\ \mathbf{if}\;b \leq -1.0401569228476551 \cdot 10^{-140}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 8.93891492717829 \cdot 10^{+45}:\\ \;\;\;\;angle \cdot \left(-0.011111111111111112 \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	34.1
Cost	7176

\[\begin{array}{l} t_0 := 0.011111111111111112 \cdot \left(\left(b \cdot angle\right) \cdot \left(b \cdot \pi\right)\right)\\ \mathbf{if}\;b \leq -1.0401569228476551 \cdot 10^{-140}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 8.93891492717829 \cdot 10^{+45}:\\ \;\;\;\;\pi \cdot \left(angle \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 14
Error	34.1
Cost	7176

\[\begin{array}{l} t_0 := 0.011111111111111112 \cdot \left(\left(b \cdot angle\right) \cdot \left(b \cdot \pi\right)\right)\\ \mathbf{if}\;b \leq -1.0401569228476551 \cdot 10^{-140}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 8.93891492717829 \cdot 10^{+45}:\\ \;\;\;\;angle \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 15
Error	43.3
Cost	6912

\[0.011111111111111112 \cdot \left(\left(angle \cdot \pi\right) \cdot \left(b \cdot b\right)\right) \]

Reproduce

herbie shell --seed 2022228 
(FPCore (a b angle)
  :name "ab-angle->ABCF B"
  :precision binary64
  (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* PI (/ angle 180.0)))) (cos (* PI (/ angle 180.0)))))