Average Error: 0.4 → 0.3
Time: 6.5s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
↓
\[\begin{array}{l} t_1 := \sqrt{\mathsf{fma}\left(v, v \cdot -5, 1\right)}\\ \frac{\frac{t_1}{\pi}}{t} \cdot \frac{\frac{t_1}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}{1 - v \cdot v} \end{array} \]
(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 (sqrt (fma v (* v -5.0) 1.0))))
   (*
    (/ (/ t_1 PI) t)
    (/ (/ t_1 (sqrt (fma v (* v -6.0) 2.0))) (- 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 = sqrt(fma(v, (v * -5.0), 1.0));
	return ((t_1 / ((double) M_PI)) / t) * ((t_1 / sqrt(fma(v, (v * -6.0), 2.0))) / (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 = sqrt(fma(v, Float64(v * -5.0), 1.0))
	return Float64(Float64(Float64(t_1 / pi) / t) * Float64(Float64(t_1 / sqrt(fma(v, Float64(v * -6.0), 2.0))) / 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[Sqrt[N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(t$95$1 / Pi), $MachinePrecision] / t), $MachinePrecision] * N[(N[(t$95$1 / N[Sqrt[N[(v * N[(v * -6.0), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(v * v), $MachinePrecision]), $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 := \sqrt{\mathsf{fma}\left(v, v \cdot -5, 1\right)}\\
\frac{\frac{t_1}{\pi}}{t} \cdot \frac{\frac{t_1}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}{1 - v \cdot v}
\end{array}

Error

Bits error versus v

Bits error versus t

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

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

  3. Applied associate-/r*_binary640.4

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

  4. Applied *-un-lft-identity_binary640.4

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

  5. Applied sqrt-prod_binary640.4

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

  6. Applied add-sqr-sqrt_binary640.4

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

  7. Applied times-frac_binary640.4

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

  8. Applied times-frac_binary640.5

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

  9. Simplified0.5

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

  10. Simplified0.5

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

  11. Applied associate-/r*_binary640.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2022131 
(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)))))