?

Average Error: 19.49% → 0.42%
Time: 16.6s
Precision: binary64
Cost: 7872

?

\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
↓
\[\frac{2}{r \cdot r} + \left(-1.5 - \frac{w}{\frac{\frac{1 - v}{\mathsf{fma}\left(v, -0.25, 0.375\right)}}{r}} \cdot \left(r \cdot w\right)\right) \]
(FPCore (v w r)
 :precision binary64
 (-
  (-
   (+ 3.0 (/ 2.0 (* r r)))
   (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
  4.5))
↓
(FPCore (v w r)
 :precision binary64
 (+
  (/ 2.0 (* r r))
  (- -1.5 (* (/ w (/ (/ (- 1.0 v) (fma v -0.25 0.375)) r)) (* r w)))))
double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}
↓
double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (-1.5 - ((w / (((1.0 - v) / fma(v, -0.25, 0.375)) / r)) * (r * w)));
}

function code(v, w, r)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
end
↓
function code(v, w, r)
	return Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 - Float64(Float64(w / Float64(Float64(Float64(1.0 - v) / fma(v, -0.25, 0.375)) / r)) * Float64(r * w))))
end

code[v_, w_, r_] := N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]
↓
code[v_, w_, r_] := N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(N[(w / N[(N[(N[(1.0 - v), $MachinePrecision] / N[(v * -0.25 + 0.375), $MachinePrecision]), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5
↓
\frac{2}{r \cdot r} + \left(-1.5 - \frac{w}{\frac{\frac{1 - v}{\mathsf{fma}\left(v, -0.25, 0.375\right)}}{r}} \cdot \left(r \cdot w\right)\right)

Error?

Derivation?

  1. Initial program 19.49

    \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

  2. Simplified0.42

    \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \frac{w}{\frac{\frac{1 - v}{\mathsf{fma}\left(v, -0.25, 0.375\right)}}{r}} \cdot \left(r \cdot w\right)\right)} \]
    Proof

    [Start]19.49

    \[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

    sub-neg [=>]19.49

    \[ \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)} - 4.5 \]

    +-commutative [=>]19.49

    \[ \color{blue}{\left(\left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) + \left(3 + \frac{2}{r \cdot r}\right)\right)} - 4.5 \]

    associate--l+ [=>]19.48

    \[ \color{blue}{\left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) + \left(\left(3 + \frac{2}{r \cdot r}\right) - 4.5\right)} \]

    +-commutative [=>]19.48

    \[ \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - 4.5\right) + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)} \]

    sub-neg [=>]19.48

    \[ \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) + \left(-4.5\right)\right)} + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]

    +-commutative [=>]19.48

    \[ \color{blue}{\left(\left(-4.5\right) + \left(3 + \frac{2}{r \cdot r}\right)\right)} + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]

    associate-+r+ [=>]19.47

    \[ \color{blue}{\left(\left(\left(-4.5\right) + 3\right) + \frac{2}{r \cdot r}\right)} + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]

    +-commutative [<=]19.47

    \[ \color{blue}{\left(\frac{2}{r \cdot r} + \left(\left(-4.5\right) + 3\right)\right)} + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]

    associate-+r+ [<=]19.47

    \[ \color{blue}{\frac{2}{r \cdot r} + \left(\left(\left(-4.5\right) + 3\right) + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)} \]

  3. Final simplification0.42

    \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{w}{\frac{\frac{1 - v}{\mathsf{fma}\left(v, -0.25, 0.375\right)}}{r}} \cdot \left(r \cdot w\right)\right) \]

Alternatives

Alternative 1
Error2.59%
Cost3400
\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := w \cdot \left(r \cdot w\right)\\ \mathbf{if}\;v \leq -25000000:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{4}{r} + \frac{2}{r \cdot v}}\right)\\ \mathbf{elif}\;v \leq 1:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{2.6666666666666665}{r} + \frac{v \cdot -0.8888888888888888}{r}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 + r \cdot \left(\left(\frac{t_1}{\frac{v}{0.125}} - 0.25 \cdot \left(\frac{r}{v} \cdot \frac{w \cdot w}{v} + t_1\right)\right) + \frac{\left(w \cdot w\right) \cdot \left(r \cdot 0.375\right)}{v \cdot v}\right)\right)\\ \end{array} \]
Alternative 2
Error0.98%
Cost1737
\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -25000000 \lor \neg \left(v \leq 1\right):\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{4}{r}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{2.6666666666666665}{r} + \frac{v \cdot -0.8888888888888888}{r}}\right)\\ \end{array} \]
Alternative 3
Error0.94%
Cost1736
\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -25000000:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{4}{r} + \frac{2}{r \cdot v}}\right)\\ \mathbf{elif}\;v \leq 1:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{2.6666666666666665}{r} + \frac{v \cdot -0.8888888888888888}{r}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{4}{r}}\right)\\ \end{array} \]
Alternative 4
Error7.44%
Cost1353
\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -3.1 \cdot 10^{+35} \lor \neg \left(v \leq 1.15\right):\\ \;\;\;\;t_0 + \left(-1.5 + r \cdot \left(\left(w \cdot w\right) \cdot \left(r \cdot -0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.375\right)\right)\right)\\ \end{array} \]
Alternative 5
Error1.22%
Cost1353
\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -25000000 \lor \neg \left(v \leq 1\right):\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.375\right)\right)\right)\\ \end{array} \]
Alternative 6
Error1.2%
Cost1353
\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -25000000 \lor \neg \left(v \leq 1\right):\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{4}{r}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.375\right)\right)\right)\\ \end{array} \]
Alternative 7
Error34.4%
Cost1096
\[\begin{array}{l} \mathbf{if}\;w \cdot w \leq 10^{+140}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;w \cdot w \leq 10^{+266}:\\ \;\;\;\;\left(w \cdot w\right) \cdot \left(\left(r \cdot r\right) \cdot -0.375\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \end{array} \]
Alternative 8
Error16.06%
Cost1088
\[\frac{2}{r \cdot r} + \left(-1.5 + -0.375 \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right) \]
Alternative 9
Error13.8%
Cost1088
\[\frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.375\right)\right)\right) \]
Alternative 10
Error32.55%
Cost448
\[\frac{2}{r \cdot r} + -1.5 \]
Alternative 11
Error59.36%
Cost320
\[\frac{2}{r \cdot r} \]

Error

Reproduce?

herbie shell --seed 2023102 
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  :precision binary64
  (- (- (+ 3.0 (/ 2.0 (* r r))) (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v))) 4.5))