Average Error: 0.4 → 0.1
Time: 3.9s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
↓
\[\begin{array}{l} t_1 := {\left(\mathsf{fma}\left(v \cdot v, -6, 2\right)\right)}^{0.25}\\ \frac{\frac{\frac{\sqrt[3]{{\pi}^{-2}}}{t_1} \cdot \frac{\sqrt[3]{\frac{1}{\pi}}}{t_1}}{\frac{t}{\mathsf{fma}\left(v, v \cdot -5, 1\right)}}}{1 - v \cdot v} \end{array} \]
(FPCore (v t)
 :precision binary64
 (/
  (- 1.0 (* 5.0 (* v v)))
  (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))
↓
(FPCore (v t)
 :precision binary64
 (let* ((t_1 (pow (fma (* v v) -6.0 2.0) 0.25)))
   (/
    (/
     (* (/ (cbrt (pow PI -2.0)) t_1) (/ (cbrt (/ 1.0 PI)) t_1))
     (/ t (fma v (* v -5.0) 1.0)))
    (- 1.0 (* v v)))))
double code(double v, double t) {
	return (1.0 - (5.0 * (v * v))) / (((((double) M_PI) * t) * sqrt((2.0 * (1.0 - (3.0 * (v * v)))))) * (1.0 - (v * v)));
}
↓
double code(double v, double t) {
	double t_1 = pow(fma((v * v), -6.0, 2.0), 0.25);
	return (((cbrt(pow(((double) M_PI), -2.0)) / t_1) * (cbrt((1.0 / ((double) M_PI))) / t_1)) / (t / fma(v, (v * -5.0), 1.0))) / (1.0 - (v * v));
}

function code(v, t)
	return Float64(Float64(1.0 - Float64(5.0 * Float64(v * v))) / Float64(Float64(Float64(pi * t) * sqrt(Float64(2.0 * Float64(1.0 - Float64(3.0 * Float64(v * v)))))) * Float64(1.0 - Float64(v * v))))
end
↓
function code(v, t)
	t_1 = fma(Float64(v * v), -6.0, 2.0) ^ 0.25
	return Float64(Float64(Float64(Float64(cbrt((pi ^ -2.0)) / t_1) * Float64(cbrt(Float64(1.0 / pi)) / t_1)) / Float64(t / fma(v, Float64(v * -5.0), 1.0))) / Float64(1.0 - Float64(v * v)))
end

code[v_, t_] := N[(N[(1.0 - N[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(Pi * t), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(1.0 - N[(3.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[v_, t_] := Block[{t$95$1 = N[Power[N[(N[(v * v), $MachinePrecision] * -6.0 + 2.0), $MachinePrecision], 0.25], $MachinePrecision]}, N[(N[(N[(N[(N[Power[N[Power[Pi, -2.0], $MachinePrecision], 1/3], $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[Power[N[(1.0 / Pi), $MachinePrecision], 1/3], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t / N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}
↓
\begin{array}{l}
t_1 := {\left(\mathsf{fma}\left(v \cdot v, -6, 2\right)\right)}^{0.25}\\
\frac{\frac{\frac{\sqrt[3]{{\pi}^{-2}}}{t_1} \cdot \frac{\sqrt[3]{\frac{1}{\pi}}}{t_1}}{\frac{t}{\mathsf{fma}\left(v, v \cdot -5, 1\right)}}}{1 - v \cdot v}
\end{array}

Error

Derivation

  1. Initial program 0.4

    \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]

  2. Simplified0.4

    \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{t \cdot \left(\pi \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}\right)}}{1 - v \cdot v}} \]

  3. Applied egg-rr0.3

    \[\leadsto \frac{\color{blue}{\frac{1}{\pi \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \cdot \frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{t}}}{1 - v \cdot v} \]

  4. Applied egg-rr0.1

    \[\leadsto \frac{\color{blue}{\frac{\frac{{\pi}^{-1}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{\frac{t}{\mathsf{fma}\left(v, v \cdot -5, 1\right)}}}}{1 - v \cdot v} \]

  5. Applied egg-rr0.1

    \[\leadsto \frac{\frac{\color{blue}{\frac{\sqrt[3]{{\pi}^{-2}}}{{\left(\mathsf{fma}\left(v \cdot v, -6, 2\right)\right)}^{0.25}} \cdot \frac{\sqrt[3]{\frac{1}{\pi}}}{{\left(\mathsf{fma}\left(v \cdot v, -6, 2\right)\right)}^{0.25}}}}{\frac{t}{\mathsf{fma}\left(v, v \cdot -5, 1\right)}}}{1 - v \cdot v} \]

  6. Final simplification0.1

    \[\leadsto \frac{\frac{\frac{\sqrt[3]{{\pi}^{-2}}}{{\left(\mathsf{fma}\left(v \cdot v, -6, 2\right)\right)}^{0.25}} \cdot \frac{\sqrt[3]{\frac{1}{\pi}}}{{\left(\mathsf{fma}\left(v \cdot v, -6, 2\right)\right)}^{0.25}}}{\frac{t}{\mathsf{fma}\left(v, v \cdot -5, 1\right)}}}{1 - v \cdot v} \]

Reproduce

herbie shell --seed 2022207 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  :precision binary64
  (/ (- 1.0 (* 5.0 (* v v))) (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))