Average Error: 0.5 → 0.5
Time: 6.4s
Precision: binary64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
↓
\[\begin{array}{l} t_0 := 1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\\ t_1 := {t_0}^{0.16666666666666666}\\ -1 + t_1 \cdot \left(\sqrt[3]{{t_0}^{2}} \cdot t_1\right) \end{array} \]
(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))
↓
(FPCore (v)
 :precision binary64
 (let* ((t_0 (+ 1.0 (acos (/ (fma v (* v -5.0) 1.0) (fma v v -1.0)))))
        (t_1 (pow t_0 0.16666666666666666)))
   (+ -1.0 (* t_1 (* (cbrt (pow t_0 2.0)) t_1)))))
double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}
↓
double code(double v) {
	double t_0 = 1.0 + acos((fma(v, (v * -5.0), 1.0) / fma(v, v, -1.0)));
	double t_1 = pow(t_0, 0.16666666666666666);
	return -1.0 + (t_1 * (cbrt(pow(t_0, 2.0)) * t_1));
}

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 = Float64(1.0 + acos(Float64(fma(v, Float64(v * -5.0), 1.0) / fma(v, v, -1.0))))
	t_1 = t_0 ^ 0.16666666666666666
	return Float64(-1.0 + Float64(t_1 * Float64(cbrt((t_0 ^ 2.0)) * t_1)))
end

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[(1.0 + N[ArcCos[N[(N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 0.16666666666666666], $MachinePrecision]}, N[(-1.0 + N[(t$95$1 * N[(N[Power[N[Power[t$95$0, 2.0], $MachinePrecision], 1/3], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\begin{array}{l}
t_0 := 1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\\
t_1 := {t_0}^{0.16666666666666666}\\
-1 + t_1 \cdot \left(\sqrt[3]{{t_0}^{2}} \cdot t_1\right)
\end{array}

Error

Derivation

  1. Initial program 0.5

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

  2. Simplified0.5

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

  3. Applied egg-rr0.5

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} - 1} \]

  4. Applied egg-rr2.0

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

  5. Applied egg-rr0.5

    \[\leadsto \color{blue}{\left(\sqrt[3]{{\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{2}} \cdot {\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{0.16666666666666666}\right) \cdot {\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{0.16666666666666666}} - 1 \]

  6. Final simplification0.5

    \[\leadsto -1 + {\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{0.16666666666666666} \cdot \left(\sqrt[3]{{\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{2}} \cdot {\left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{0.16666666666666666}\right) \]

Reproduce

herbie shell --seed 2022213 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))