?

Average Error: 0.5 → 0.1
Time: 11.4s
Precision: binary64
Cost: 27328

?

\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
↓
\[\frac{\frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}}{\pi}}{t \cdot \left(1 - {v}^{4}\right)} \cdot \left(1 + v \cdot v\right) \]
(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) (sqrt (+ 2.0 (* (* v v) -6.0)))) PI)
   (* t (- 1.0 (pow v 4.0))))
  (+ 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) {
	return (((fma(v, (v * -5.0), 1.0) / sqrt((2.0 + ((v * v) * -6.0)))) / ((double) M_PI)) / (t * (1.0 - pow(v, 4.0)))) * (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)
	return Float64(Float64(Float64(Float64(fma(v, Float64(v * -5.0), 1.0) / sqrt(Float64(2.0 + Float64(Float64(v * v) * -6.0)))) / pi) / Float64(t * Float64(1.0 - (v ^ 4.0)))) * 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_] := N[(N[(N[(N[(N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[N[(2.0 + N[(N[(v * v), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] / N[(t * N[(1.0 - N[Power[v, 4.0], $MachinePrecision]), $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)}
↓
\frac{\frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}}{\pi}}{t \cdot \left(1 - {v}^{4}\right)} \cdot \left(1 + v \cdot v\right)

Error?

Derivation?

  1. Initial program 0.5

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

  2. Simplified0.5

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

    [Start]0.5

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

    cancel-sign-sub-inv [=>]0.5

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

    *-commutative [=>]0.5

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

    metadata-eval [=>]0.5

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

    *-commutative [=>]0.5

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

    associate-*l* [=>]0.5

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

    cancel-sign-sub-inv [=>]0.5

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

    distribute-lft-in [=>]0.5

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

    metadata-eval [=>]0.5

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

    *-commutative [=>]0.5

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

    metadata-eval [=>]0.5

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

  3. Applied egg-rr0.1

    \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}}{\pi}}{t \cdot \left(1 - {v}^{4}\right)} \cdot \left(v \cdot v + 1\right)} \]

  4. Final simplification0.1

    \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}}{\pi}}{t \cdot \left(1 - {v}^{4}\right)} \cdot \left(1 + v \cdot v\right) \]

Alternatives

Alternative 1
Error0.4
Cost20608
\[\frac{\frac{-1 + \left(v \cdot v\right) \cdot 5}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}{\left(\pi \cdot t\right) \cdot \left(v \cdot v + -1\right)} \]
Alternative 2
Error0.4
Cost20608
\[\frac{\frac{-1 + \left(v \cdot v\right) \cdot 5}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}{t \cdot \left(\pi \cdot \left(v \cdot v + -1\right)\right)} \]
Alternative 3
Error0.5
Cost14464
\[\frac{1 + -5 \cdot \left(v \cdot v\right)}{\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(\left(1 - v \cdot v\right) \cdot \left(\pi \cdot t\right)\right)} \]
Alternative 4
Error0.9
Cost13696
\[\left(-1 - v \cdot \left(v \cdot -5\right)\right) \cdot \frac{\frac{-1}{\pi \cdot \sqrt{2}}}{t} \]
Alternative 5
Error1.5
Cost13184
\[\frac{\frac{1}{t}}{\pi} \cdot \sqrt{0.5} \]
Alternative 6
Error1.3
Cost13184
\[\frac{1}{\pi \cdot \left(t \cdot \sqrt{2}\right)} \]
Alternative 7
Error1.3
Cost13184
\[\frac{1}{\left(\pi \cdot t\right) \cdot \sqrt{2}} \]
Alternative 8
Error0.9
Cost13184
\[\frac{\frac{1}{\pi \cdot \sqrt{2}}}{t} \]
Alternative 9
Error1.5
Cost13056
\[\frac{\sqrt{0.5}}{\pi \cdot t} \]
Alternative 10
Error1.5
Cost13056
\[\frac{\frac{\sqrt{0.5}}{t}}{\pi} \]

Error

Reproduce?

herbie shell --seed 2023039 
(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)))))