Average Error: 12.7 → 0.4
Time: 11.7s
Precision: binary64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{1 - v}\\ \left(\frac{\frac{2}{r}}{r} + -1.5\right) - \left(\left(0.375 - v \cdot 0.25\right) \cdot \frac{r \cdot w}{t_0 \cdot t_0}\right) \cdot \left(\left(r \cdot w\right) \cdot \sqrt[3]{\frac{1}{1 - v}}\right) \end{array} \]
\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
↓
\begin{array}{l}
t_0 := \sqrt[3]{1 - v}\\
\left(\frac{\frac{2}{r}}{r} + -1.5\right) - \left(\left(0.375 - v \cdot 0.25\right) \cdot \frac{r \cdot w}{t_0 \cdot t_0}\right) \cdot \left(\left(r \cdot w\right) \cdot \sqrt[3]{\frac{1}{1 - v}}\right)
\end{array}

(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
 (let* ((t_0 (cbrt (- 1.0 v))))
   (-
    (+ (/ (/ 2.0 r) r) -1.5)
    (*
     (* (- 0.375 (* v 0.25)) (/ (* r w) (* t_0 t_0)))
     (* (* r w) (cbrt (/ 1.0 (- 1.0 v))))))))
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) {
	double t_0 = cbrt(1.0 - v);
	return (((2.0 / r) / r) + -1.5) - (((0.375 - (v * 0.25)) * ((r * w) / (t_0 * t_0))) * ((r * w) * cbrt(1.0 / (1.0 - v))));
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 12.7

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

  2. Simplified9.0

    \[\leadsto \color{blue}{\left(\frac{2}{r \cdot r} + -1.5\right) - \left(0.375 + v \cdot -0.25\right) \cdot \frac{r}{\frac{1 - v}{r \cdot \left(w \cdot w\right)}}} \]
  3. Using strategy rm

  4. Applied associate-*r*_binary644.0

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

  5. Simplified4.0

    \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \left(0.375 + v \cdot -0.25\right) \cdot \frac{r}{\frac{1 - v}{\color{blue}{\left(w \cdot r\right)} \cdot w}} \]
  6. Using strategy rm

  7. Applied add-cube-cbrt_binary644.2

    \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \left(0.375 + v \cdot -0.25\right) \cdot \frac{r}{\frac{\color{blue}{\left(\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}\right) \cdot \sqrt[3]{1 - v}}}{\left(w \cdot r\right) \cdot w}} \]

  8. Applied times-frac_binary643.0

    \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \left(0.375 + v \cdot -0.25\right) \cdot \frac{r}{\color{blue}{\frac{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}}{w \cdot r} \cdot \frac{\sqrt[3]{1 - v}}{w}}} \]

  9. Applied *-un-lft-identity_binary643.0

    \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \left(0.375 + v \cdot -0.25\right) \cdot \frac{\color{blue}{1 \cdot r}}{\frac{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}}{w \cdot r} \cdot \frac{\sqrt[3]{1 - v}}{w}} \]

  10. Applied times-frac_binary640.8

    \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \left(0.375 + v \cdot -0.25\right) \cdot \color{blue}{\left(\frac{1}{\frac{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}}{w \cdot r}} \cdot \frac{r}{\frac{\sqrt[3]{1 - v}}{w}}\right)} \]

  11. Applied associate-*r*_binary640.7

    \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \color{blue}{\left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{1}{\frac{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}}{w \cdot r}}\right) \cdot \frac{r}{\frac{\sqrt[3]{1 - v}}{w}}} \]

  12. Simplified0.7

    \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \color{blue}{\left(\left(0.375 - v \cdot 0.25\right) \cdot \frac{w \cdot r}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}}\right)} \cdot \frac{r}{\frac{\sqrt[3]{1 - v}}{w}} \]

  13. Taylor expanded around 0 16.5

    \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \left(\left(0.375 - v \cdot 0.25\right) \cdot \frac{w \cdot r}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}}\right) \cdot \color{blue}{\left({\left(\frac{1}{1 - v}\right)}^{0.3333333333333333} \cdot \left(w \cdot r\right)\right)} \]

  14. Simplified0.4

    \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \left(\left(0.375 - v \cdot 0.25\right) \cdot \frac{w \cdot r}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}}\right) \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot \sqrt[3]{\frac{1}{1 - v}}\right)} \]
  15. Using strategy rm

  16. Applied associate-/r*_binary640.4

    \[\leadsto \left(\color{blue}{\frac{\frac{2}{r}}{r}} + -1.5\right) - \left(\left(0.375 - v \cdot 0.25\right) \cdot \frac{w \cdot r}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}}\right) \cdot \left(\left(w \cdot r\right) \cdot \sqrt[3]{\frac{1}{1 - v}}\right) \]

  17. Final simplification0.4

    \[\leadsto \left(\frac{\frac{2}{r}}{r} + -1.5\right) - \left(\left(0.375 - v \cdot 0.25\right) \cdot \frac{r \cdot w}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}}\right) \cdot \left(\left(r \cdot w\right) \cdot \sqrt[3]{\frac{1}{1 - v}}\right) \]

Reproduce

herbie shell --seed 2021205 
(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))