Average Error: 1.0 → 0.0
Time: 3.2s
Precision: binary64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
↓
\[\begin{array}{l} t_0 := \log \left(\sqrt{{\left(e^{\pi}\right)}^{\left(\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}\right)}}\right)\\ -1.3333333333333333 \cdot \frac{\frac{1}{\mathsf{fma}\left(v, v, -1\right)}}{t_0 + t_0} \end{array} \]
(FPCore (v)
 :precision binary64
 (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))
↓
(FPCore (v)
 :precision binary64
 (let* ((t_0 (log (sqrt (pow (exp PI) (sqrt (fma v (* v -6.0) 2.0)))))))
   (* -1.3333333333333333 (/ (/ 1.0 (fma v v -1.0)) (+ t_0 t_0)))))
double code(double v) {
	return 4.0 / (((3.0 * ((double) M_PI)) * (1.0 - (v * v))) * sqrt((2.0 - (6.0 * (v * v)))));
}
↓
double code(double v) {
	double t_0 = log(sqrt(pow(exp(((double) M_PI)), sqrt(fma(v, (v * -6.0), 2.0)))));
	return -1.3333333333333333 * ((1.0 / fma(v, v, -1.0)) / (t_0 + t_0));
}

function code(v)
	return Float64(4.0 / Float64(Float64(Float64(3.0 * pi) * Float64(1.0 - Float64(v * v))) * sqrt(Float64(2.0 - Float64(6.0 * Float64(v * v))))))
end
↓
function code(v)
	t_0 = log(sqrt((exp(pi) ^ sqrt(fma(v, Float64(v * -6.0), 2.0)))))
	return Float64(-1.3333333333333333 * Float64(Float64(1.0 / fma(v, v, -1.0)) / Float64(t_0 + t_0)))
end

code[v_] := N[(4.0 / N[(N[(N[(3.0 * Pi), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 - N[(6.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[v_] := Block[{t$95$0 = N[Log[N[Sqrt[N[Power[N[Exp[Pi], $MachinePrecision], N[Sqrt[N[(v * N[(v * -6.0), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[(-1.3333333333333333 * N[(N[(1.0 / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
↓
\begin{array}{l}
t_0 := \log \left(\sqrt{{\left(e^{\pi}\right)}^{\left(\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}\right)}}\right)\\
-1.3333333333333333 \cdot \frac{\frac{1}{\mathsf{fma}\left(v, v, -1\right)}}{t_0 + t_0}
\end{array}

Error

Derivation

  1. Initial program 1.0

    \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{\frac{-1.3333333333333333}{\mathsf{fma}\left(v, v, -1\right)}}{\pi \cdot \sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}} \]

  3. Applied egg-rr0.0

    \[\leadsto \color{blue}{-1.3333333333333333 \cdot \frac{\frac{1}{\mathsf{fma}\left(v, v, -1\right)}}{\pi \cdot \sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}} \]

  4. Applied egg-rr0.0

    \[\leadsto -1.3333333333333333 \cdot \frac{\frac{1}{\mathsf{fma}\left(v, v, -1\right)}}{\color{blue}{\log \left(\sqrt{{\left(e^{\pi}\right)}^{\left(\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}\right)}}\right) + \log \left(\sqrt{{\left(e^{\pi}\right)}^{\left(\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}\right)}}\right)}} \]

  5. Final simplification0.0

    \[\leadsto -1.3333333333333333 \cdot \frac{\frac{1}{\mathsf{fma}\left(v, v, -1\right)}}{\log \left(\sqrt{{\left(e^{\pi}\right)}^{\left(\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}\right)}}\right) + \log \left(\sqrt{{\left(e^{\pi}\right)}^{\left(\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}\right)}}\right)} \]

Reproduce

herbie shell --seed 2022209 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))