?

Average Error: 0.6 → 0.6
Time: 15.0s
Precision: binary64
Cost: 45504

?

\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
↓
\[e^{\log \left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right) \cdot 9} \]
(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))
↓
(FPCore (v)
 :precision binary64
 (exp
  (*
   (log (cbrt (cbrt (acos (/ (fma v (* v -5.0) 1.0) (fma v v -1.0))))))
   9.0)))
double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}
↓
double code(double v) {
	return exp((log(cbrt(cbrt(acos((fma(v, (v * -5.0), 1.0) / fma(v, v, -1.0)))))) * 9.0));
}

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 exp(Float64(log(cbrt(cbrt(acos(Float64(fma(v, Float64(v * -5.0), 1.0) / fma(v, v, -1.0)))))) * 9.0))
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[(N[Log[N[Power[N[Power[N[ArcCos[N[(N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision] * 9.0), $MachinePrecision]], $MachinePrecision]

\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
e^{\log \left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right) \cdot 9}

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-rr1.5

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

  3. Applied egg-rr2.1

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

  4. Applied egg-rr0.6

    \[\leadsto \color{blue}{e^{\log \left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right) \cdot 9}} \]

  5. Final simplification0.6

    \[\leadsto e^{\log \left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right) \cdot 9} \]

Alternatives

Alternative 1
Error0.6
Cost7232
\[\cos^{-1} \left(\frac{1 + -5 \cdot \left(v \cdot v\right)}{-1 + v \cdot v}\right) \]
Alternative 2
Error1.3
Cost6464
\[\cos^{-1} -1 \]

Error

Reproduce?

herbie shell --seed 2023083 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))