Falkner and Boettcher, Appendix B, 1

Average Error: 0.5 → 0.5

Time: 13.7s

Precision: binary64

Cost: 97344

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{\sqrt[3]{\log \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\\ e^{{\left(t_0 \cdot {t_0}^{2}\right)}^{3}} \end{array} \]

FPCore

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

↓

(FPCore (v)
 :precision binary64
 (let* ((t_0
         (cbrt (cbrt (log (acos (/ (fma v (* v -5.0) 1.0) (fma v v -1.0))))))))
   (exp (pow (* t_0 (pow t_0 2.0)) 3.0))))

C

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

↓

double code(double v) {
	double t_0 = cbrt(cbrt(log(acos((fma(v, (v * -5.0), 1.0) / fma(v, v, -1.0))))));
	return exp(pow((t_0 * pow(t_0, 2.0)), 3.0));
}

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 = cbrt(cbrt(log(acos(Float64(fma(v, Float64(v * -5.0), 1.0) / fma(v, v, -1.0))))))
	return exp((Float64(t_0 * (t_0 ^ 2.0)) ^ 3.0))
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[Power[N[Power[N[Log[N[ArcCos[N[(N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision]}, N[Exp[N[Power[N[(t$95$0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $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

   \[\leadsto \color{blue}{e^{\log \cos^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}} \]

3. Applied egg-rr 3.0

   \[\leadsto e^{\color{blue}{{\left(\sqrt[3]{\log \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{3}}} \]

4. Applied egg-rr 0.5

   \[\leadsto e^{{\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{\log \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}^{2} \cdot \sqrt[3]{\sqrt[3]{\log \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}}^{3}} \]

5. Final simplification 0.5

   \[\leadsto e^{{\left(\sqrt[3]{\sqrt[3]{\log \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}} \cdot {\left(\sqrt[3]{\sqrt[3]{\log \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}^{2}\right)}^{3}} \]

Alternatives

Alternative 1
Error	0.5
Cost	90944

\[\begin{array}{l} t_0 := \sqrt[3]{\log \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\\ e^{t_0 \cdot {\left({\left(\sqrt[3]{t_0}\right)}^{2}\right)}^{3}} \end{array} \]

Alternative 2
Error	0.5
Cost	13824

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

Alternative 3
Error	0.5
Cost	7232

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

Alternative 4
Error	1.3
Cost	6464

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

Reproduce

herbie shell --seed 2023066 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))