Average Error: 0.6 → 0.6
Time: 16.0s
Precision: binary64
Cost: 97344
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
↓
\[\begin{array}{l} t_0 := {\left(\sqrt{\sqrt[3]{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}}\right)}^{3}\\ \mathsf{expm1}\left(t_0 \cdot t_0\right) \end{array} \]
(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))
↓
(FPCore (v)
 :precision binary64
 (let* ((t_0
         (pow
          (sqrt
           (cbrt (log1p (acos (/ (fma v (* v -5.0) 1.0) (fma v v -1.0))))))
          3.0)))
   (expm1 (* t_0 t_0))))
double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}
↓
double code(double v) {
	double t_0 = pow(sqrt(cbrt(log1p(acos((fma(v, (v * -5.0), 1.0) / fma(v, v, -1.0)))))), 3.0);
	return expm1((t_0 * t_0));
}

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 = sqrt(cbrt(log1p(acos(Float64(fma(v, Float64(v * -5.0), 1.0) / fma(v, v, -1.0)))))) ^ 3.0
	return expm1(Float64(t_0 * t_0))
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[Power[N[Sqrt[N[Power[N[Log[1 + 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]], $MachinePrecision], 3.0], $MachinePrecision]}, N[(Exp[N[(t$95$0 * t$95$0), $MachinePrecision]] - 1), $MachinePrecision]]

\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\begin{array}{l}
t_0 := {\left(\sqrt{\sqrt[3]{\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)}}\right)}^{3}\\
\mathsf{expm1}\left(t_0 \cdot t_0\right)
\end{array}

Error

Derivation

  1. Initial program 0.6

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

  2. Applied egg-rr0.6

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

  3. Applied egg-rr0.6

    \[\leadsto \mathsf{expm1}\left(\color{blue}{{\left(\sqrt{\sqrt[3]{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}}\right)}^{3} \cdot {\left(\sqrt{\sqrt[3]{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}}\right)}^{3}}\right) \]

  4. Final simplification0.6

    \[\leadsto \mathsf{expm1}\left({\left(\sqrt{\sqrt[3]{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}}\right)}^{3} \cdot {\left(\sqrt{\sqrt[3]{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}}\right)}^{3}\right) \]

Alternatives

Alternative 1
Error0.6
Cost84608
\[\begin{array}{l} t_0 := \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\\ \mathsf{expm1}\left({\left(\sqrt{\sqrt[3]{\mathsf{log1p}\left(t_0\right)}}\right)}^{3} \cdot \sqrt{\log \left(1 + t_0\right)}\right) \end{array} \]
Alternative 2
Error0.6
Cost51968
\[\sqrt[3]{{\left(e^{3 \cdot \log \left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}\right)}^{3}} \]
Alternative 3
Error0.6
Cost32704
\[\mathsf{expm1}\left(\log \left(1 + \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \]
Alternative 4
Error0.6
Cost32576
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \]
Alternative 5
Error0.8
Cost7232
\[\cos^{-1} \left(-1 + 4 \cdot \left(\left(v \cdot v\right) \cdot \left(1 + v \cdot v\right)\right)\right) \]
Alternative 6
Error0.6
Cost7232
\[\cos^{-1} \left(\frac{1 + -5 \cdot \left(v \cdot v\right)}{-1 + v \cdot v}\right) \]
Alternative 7
Error0.9
Cost6848
\[\cos^{-1} \left(-1 + \left(v \cdot v\right) \cdot 4\right) \]
Alternative 8
Error1.4
Cost6464
\[\cos^{-1} -1 \]

Error

Reproduce

herbie shell --seed 2022329 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))