Average Error: 0.6 → 0.6
Time: 8.0s
Precision: binary64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
↓
\[{\left({\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}\right)}^{0.3333333333333333} \]
(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))
↓
(FPCore (v)
 :precision binary64
 (pow
  (pow (acos (/ (fma (* v v) -5.0 1.0) (fma v v -1.0))) 3.0)
  0.3333333333333333))
double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}
↓
double code(double v) {
	return pow(pow(acos((fma((v * v), -5.0, 1.0) / fma(v, v, -1.0))), 3.0), 0.3333333333333333);
}

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 (acos(Float64(fma(Float64(v * v), -5.0, 1.0) / fma(v, v, -1.0))) ^ 3.0) ^ 0.3333333333333333
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[Power[N[Power[N[ArcCos[N[(N[(N[(v * v), $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]

\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
{\left({\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}\right)}^{0.3333333333333333}

Error

Bits error versus v

Derivation

  1. Initial program 0.6

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

  2. Simplified0.6

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

  3. Applied egg-rr0.6

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

  4. Applied egg-rr0.6

    \[\leadsto \color{blue}{{\left({\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

  5. Final simplification0.6

    \[\leadsto {\left({\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}\right)}^{0.3333333333333333} \]

Reproduce

herbie shell --seed 2022131 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))