Average Error: 12.9 → 0.4
Time: 34.5s
Precision: binary64
Cost: 14272
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
↓
\[-\left(1.5 + \left({\left(w \cdot r\right)}^{2} \cdot \frac{0.125 \cdot \mathsf{fma}\left(2, v, -3\right)}{v + -1} - \frac{\frac{2}{r}}{r}\right)\right) \]
(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
 (-
  (+
   1.5
   (-
    (* (pow (* w r) 2.0) (/ (* 0.125 (fma 2.0 v -3.0)) (+ v -1.0)))
    (/ (/ 2.0 r) r)))))
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 -(1.5 + ((pow((w * r), 2.0) * ((0.125 * fma(2.0, v, -3.0)) / (v + -1.0))) - ((2.0 / r) / r)));
}

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(1.5 + Float64(Float64((Float64(w * r) ^ 2.0) * Float64(Float64(0.125 * fma(2.0, v, -3.0)) / Float64(v + -1.0))) - Float64(Float64(2.0 / r) / r))))
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[(1.5 + N[(N[(N[Power[N[(w * r), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(0.125 * N[(2.0 * v + -3.0), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(2.0 / r), $MachinePrecision] / r), $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
↓
-\left(1.5 + \left({\left(w \cdot r\right)}^{2} \cdot \frac{0.125 \cdot \mathsf{fma}\left(2, v, -3\right)}{v + -1} - \frac{\frac{2}{r}}{r}\right)\right)

Error

Derivation

  1. Initial program 12.9

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

  2. Simplified0.4

    \[\leadsto \color{blue}{-\left(1.5 + \mathsf{fma}\left({\left(w \cdot r\right)}^{2}, \frac{-0.125 \cdot \mathsf{fma}\left(2, v, -3\right)}{1 - v}, \frac{-2}{r \cdot r}\right)\right)} \]
    Proof

  3. Applied egg-rr0.4

    \[\leadsto -\left(1.5 + \color{blue}{\left({\left(w \cdot r\right)}^{2} \cdot \frac{0.125 \cdot \mathsf{fma}\left(2, v, -3\right)}{v + -1} - \frac{\frac{2}{r}}{r}\right)}\right) \]

Alternatives

Alternative 1
Error0.4
Cost7428
\[\begin{array}{l} t_0 := \left(w \cdot r\right) \cdot \left(w \cdot r\right)\\ t_1 := \frac{2}{r \cdot r}\\ t_2 := 3 + t_1\\ \mathbf{if}\;v \leq -3.8 \cdot 10^{+14}:\\ \;\;\;\;t_1 - \left({\left(\left(w \cdot r\right) \cdot 0.5\right)}^{2} - -1.5\right)\\ \mathbf{elif}\;v \leq 1950000:\\ \;\;\;\;\left(t_2 - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot t_0}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t_2 - 0.25 \cdot t_0\right) - 4.5\\ \end{array} \]
Alternative 2
Error0.4
Cost2120
\[\begin{array}{l} t_0 := \left(w \cdot r\right) \cdot \left(w \cdot r\right)\\ t_1 := 3 + \frac{2}{r \cdot r}\\ t_2 := \left(t_1 - 0.25 \cdot t_0\right) - 4.5\\ \mathbf{if}\;v \leq -8.4 \cdot 10^{+14}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;v \leq 1950000:\\ \;\;\;\;\left(t_1 - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot t_0}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 3
Error17.4
Cost1672
\[\begin{array}{l} t_0 := \frac{\frac{2}{r}}{r}\\ t_1 := -1.5 + t_0\\ \mathbf{if}\;w \cdot w \leq 3.7 \cdot 10^{-272}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;w \cdot w \leq \infty:\\ \;\;\;\;-\left(1.5 + \left(0.375 \cdot \left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right) - t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 4
Error0.7
Cost1672
\[\begin{array}{l} t_0 := \left(w \cdot r\right) \cdot \left(w \cdot r\right)\\ t_1 := \left(\left(3 + \frac{2}{r \cdot r}\right) - 0.25 \cdot t_0\right) - 4.5\\ \mathbf{if}\;v \leq -45000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq 1:\\ \;\;\;\;-\left(1.5 + \left(t_0 \cdot \left(0.125 \cdot v + 0.375\right) - \frac{\frac{2}{r}}{r}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 5
Error17.4
Cost1608
\[\begin{array}{l} t_0 := -1.5 + \frac{\frac{2}{r}}{r}\\ \mathbf{if}\;w \cdot w \leq 5 \cdot 10^{-283}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;w \cdot w \leq \infty:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -0.25 \cdot \left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)\right) - 1.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 6
Error5.0
Cost1480
\[\begin{array}{l} t_0 := \left(\left(3 + \frac{2}{r \cdot r}\right) - 0.25 \cdot \left(\left(r \cdot \left(w \cdot w\right)\right) \cdot r\right)\right) - 4.5\\ \mathbf{if}\;v \leq -78000000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;v \leq 6.5 \cdot 10^{+46}:\\ \;\;\;\;-\left(1.5 + \left(\left(0.375 \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right) - \frac{\frac{2}{r}}{r}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 7
Error0.8
Cost1480
\[\begin{array}{l} t_0 := \left(\left(3 + \frac{2}{r \cdot r}\right) - 0.25 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right) - 4.5\\ \mathbf{if}\;v \leq -45000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;v \leq 1.55:\\ \;\;\;\;-\left(1.5 + \left(\left(0.375 \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right) - \frac{\frac{2}{r}}{r}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 8
Error11.7
Cost1152
\[-\left(1.5 + \left(w \cdot \left(0.375 \cdot \left(\left(w \cdot r\right) \cdot r\right)\right) - \frac{\frac{2}{r}}{r}\right)\right) \]
Alternative 9
Error9.2
Cost1152
\[-\left(1.5 + \left(\left(0.375 \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right) - \frac{\frac{2}{r}}{r}\right)\right) \]
Alternative 10
Error21.1
Cost448
\[-1.5 + \frac{2}{r \cdot r} \]
Alternative 11
Error21.1
Cost448
\[-1.5 + \frac{\frac{2}{r}}{r} \]

Error

Reproduce

herbie shell --seed 2023010 
(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))