Average Error: 0.4 → 0.1
Time: 4.9s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
↓
\[\frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\left(\pi - \pi \cdot \left(v \cdot v\right)\right) \cdot \sqrt{2 - \left(v \cdot v\right) \cdot 6}}}{t} \]
(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
 (/
  (/
   (fma v (* v -5.0) 1.0)
   (* (- PI (* PI (* v v))) (sqrt (- 2.0 (* (* v v) 6.0)))))
  t))
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) {
	return (fma(v, (v * -5.0), 1.0) / ((((double) M_PI) - (((double) M_PI) * (v * v))) * sqrt((2.0 - ((v * v) * 6.0))))) / t;
}

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)
	return Float64(Float64(fma(v, Float64(v * -5.0), 1.0) / Float64(Float64(pi - Float64(pi * Float64(v * v))) * sqrt(Float64(2.0 - Float64(Float64(v * v) * 6.0))))) / t)
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_] := N[(N[(N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(Pi - N[(Pi * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 - N[(N[(v * v), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $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)}
↓
\frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\left(\pi - \pi \cdot \left(v \cdot v\right)\right) \cdot \sqrt{2 - \left(v \cdot v\right) \cdot 6}}}{t}

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. Taylor expanded in t around 0 0.4

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

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

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

  5. Applied times-frac_binary640.3

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

  6. Simplified0.3

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

  7. Applied associate-*l/_binary640.1

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2022137 
(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)))))