Average Error: 0.5 → 0.5
Time: 6.1s
Precision: binary64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
↓
\[\mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}^{2}\right) \]
(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))
↓
(FPCore (v)
 :precision binary64
 (expm1
  (pow (sqrt (log1p (acos (/ (fma v (* v -5.0) 1.0) (fma v v -1.0))))) 2.0)))
double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}
↓
double code(double v) {
	return expm1(pow(sqrt(log1p(acos((fma(v, (v * -5.0), 1.0) / fma(v, v, -1.0))))), 2.0));
}

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)
	return expm1((sqrt(log1p(acos(Float64(fma(v, Float64(v * -5.0), 1.0) / fma(v, v, -1.0))))) ^ 2.0))
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_] := N[(Exp[N[Power[N[Sqrt[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]], $MachinePrecision], 2.0], $MachinePrecision]] - 1), $MachinePrecision]

\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\mathsf{expm1}\left({\left(\sqrt{\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)}^{2}\right)

Error

Bits error versus v

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

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

  4. Applied egg-rr0.5

    \[\leadsto \mathsf{expm1}\left(\color{blue}{{\left(\sqrt{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}^{2}}\right) \]

  5. Final simplification0.5

    \[\leadsto \mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}^{2}\right) \]

Reproduce

herbie shell --seed 2022162 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))