Falkner and Boettcher, Appendix B, 1

Average Error: 0.5 → 0.5

Time: 13.4s

Precision: binary64

Cost: 68416

Math

\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]

↓

\[\begin{array}{l} t_0 := \sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right)\\ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {t_0}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(t_0 \cdot t_0 + \left(\pi \cdot 0.5\right) \cdot t_0\right)} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))

↓

(FPCore (v)
 :precision binary64
 (let* ((t_0 (asin (/ (+ 1.0 (* (* v v) -5.0)) (+ (* v v) -1.0)))))
   (/
    (- (pow (* PI 0.5) 3.0) (pow t_0 3.0))
    (+ (* (* PI 0.5) (* PI 0.5)) (+ (* t_0 t_0) (* (* PI 0.5) t_0))))))

C

double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

↓

double code(double v) {
	double t_0 = asin(((1.0 + ((v * v) * -5.0)) / ((v * v) + -1.0)));
	return (pow((((double) M_PI) * 0.5), 3.0) - pow(t_0, 3.0)) / (((((double) M_PI) * 0.5) * (((double) M_PI) * 0.5)) + ((t_0 * t_0) + ((((double) M_PI) * 0.5) * t_0)));
}

Java

public static double code(double v) {
	return Math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

↓

public static double code(double v) {
	double t_0 = Math.asin(((1.0 + ((v * v) * -5.0)) / ((v * v) + -1.0)));
	return (Math.pow((Math.PI * 0.5), 3.0) - Math.pow(t_0, 3.0)) / (((Math.PI * 0.5) * (Math.PI * 0.5)) + ((t_0 * t_0) + ((Math.PI * 0.5) * t_0)));
}

Python

def code(v):
	return math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)))

↓

def code(v):
	t_0 = math.asin(((1.0 + ((v * v) * -5.0)) / ((v * v) + -1.0)))
	return (math.pow((math.pi * 0.5), 3.0) - math.pow(t_0, 3.0)) / (((math.pi * 0.5) * (math.pi * 0.5)) + ((t_0 * t_0) + ((math.pi * 0.5) * t_0)))

Julia

function code(v)
	return acos(Float64(Float64(1.0 - Float64(5.0 * Float64(v * v))) / Float64(Float64(v * v) - 1.0)))
end

↓

function code(v)
	t_0 = asin(Float64(Float64(1.0 + Float64(Float64(v * v) * -5.0)) / Float64(Float64(v * v) + -1.0)))
	return Float64(Float64((Float64(pi * 0.5) ^ 3.0) - (t_0 ^ 3.0)) / Float64(Float64(Float64(pi * 0.5) * Float64(pi * 0.5)) + Float64(Float64(t_0 * t_0) + Float64(Float64(pi * 0.5) * t_0))))
end

MATLAB

function tmp = code(v)
	tmp = acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
end

↓

function tmp = code(v)
	t_0 = asin(((1.0 + ((v * v) * -5.0)) / ((v * v) + -1.0)));
	tmp = (((pi * 0.5) ^ 3.0) - (t_0 ^ 3.0)) / (((pi * 0.5) * (pi * 0.5)) + ((t_0 * t_0) + ((pi * 0.5) * t_0)));
end

Wolfram

code[v_] := N[ArcCos[N[(N[(1.0 - N[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(v * v), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[v_] := Block[{t$95$0 = N[ArcSin[N[(N[(1.0 + N[(N[(v * v), $MachinePrecision] * -5.0), $MachinePrecision]), $MachinePrecision] / N[(N[(v * v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Power[N[(Pi * 0.5), $MachinePrecision], 3.0], $MachinePrecision] - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(Pi * 0.5), $MachinePrecision] * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[(Pi * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 0.5

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]

2. Applied egg-rr 0.5

   \[\leadsto \color{blue}{\frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}} \]

3. Final simplification 0.5

   \[\leadsto \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right) \cdot \sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right)\right)} \]

Alternatives

Alternative 1
Error	0.5
Cost	20288

\[e^{\left(3 \cdot \log \cos^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right)\right) \cdot 0.3333333333333333} \]

Alternative 2
Error	0.5
Cost	13824

\[\pi \cdot 0.5 - \sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right) \]

Alternative 3
Error	0.5
Cost	7232

\[\cos^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right) \]

Alternative 4
Error	1.4
Cost	6464

\[\cos^{-1} -1 \]

Reproduce

herbie shell --seed 2023011 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))