Average Error: 12.8 → 0.9
Time: 20.8s
Precision: binary64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\begin{array}{l} \mathbf{if}\;w \leq -5.003095036691503 \cdot 10^{+160}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \left(0.375 + v \cdot -0.25\right) \cdot \left(\frac{\sqrt[3]{r} \cdot \sqrt[3]{r}}{\frac{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}}{w \cdot r}} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{r}}{\frac{\sqrt[3]{1 - v}}{w}}} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{r}}{\frac{\sqrt[3]{1 - v}}{w}}} \cdot \sqrt[3]{\frac{\sqrt[3]{r}}{\frac{\sqrt[3]{1 - v}}{w}}}\right)\right)\right)\\ \mathbf{elif}\;w \leq 1.8322291642935536 \cdot 10^{-68}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \left(0.375 + v \cdot -0.25\right) \cdot \left(\frac{1}{1 - v} \cdot \left(r \cdot \left(w \cdot \left(w \cdot r\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \left(0.375 + v \cdot -0.25\right) \cdot \frac{r}{\frac{\frac{1 - v}{w \cdot r}}{w}}\\ \end{array}\]
\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}
\mathbf{if}\;w \leq -5.003095036691503 \cdot 10^{+160}:\\
\;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \left(0.375 + v \cdot -0.25\right) \cdot \left(\frac{\sqrt[3]{r} \cdot \sqrt[3]{r}}{\frac{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}}{w \cdot r}} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{r}}{\frac{\sqrt[3]{1 - v}}{w}}} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{r}}{\frac{\sqrt[3]{1 - v}}{w}}} \cdot \sqrt[3]{\frac{\sqrt[3]{r}}{\frac{\sqrt[3]{1 - v}}{w}}}\right)\right)\right)\\

\mathbf{elif}\;w \leq 1.8322291642935536 \cdot 10^{-68}:\\
\;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \left(0.375 + v \cdot -0.25\right) \cdot \left(\frac{1}{1 - v} \cdot \left(r \cdot \left(w \cdot \left(w \cdot r\right)\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \left(0.375 + v \cdot -0.25\right) \cdot \frac{r}{\frac{\frac{1 - v}{w \cdot r}}{w}}\\

\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
 (if (<= w -5.003095036691503e+160)
   (-
    (+ (/ 2.0 (* r r)) -1.5)
    (*
     (+ 0.375 (* v -0.25))
     (*
      (/
       (* (cbrt r) (cbrt r))
       (/ (* (cbrt (- 1.0 v)) (cbrt (- 1.0 v))) (* w r)))
      (*
       (cbrt (/ (cbrt r) (/ (cbrt (- 1.0 v)) w)))
       (*
        (cbrt (/ (cbrt r) (/ (cbrt (- 1.0 v)) w)))
        (cbrt (/ (cbrt r) (/ (cbrt (- 1.0 v)) w))))))))
   (if (<= w 1.8322291642935536e-68)
     (-
      (+ (/ 2.0 (* r r)) -1.5)
      (* (+ 0.375 (* v -0.25)) (* (/ 1.0 (- 1.0 v)) (* r (* w (* w r))))))
     (-
      (+ (/ 2.0 (* r r)) -1.5)
      (* (+ 0.375 (* v -0.25)) (/ r (/ (/ (- 1.0 v) (* w 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) {
	double tmp;
	if (w <= -5.003095036691503e+160) {
		tmp = ((2.0 / (r * r)) + -1.5) - ((0.375 + (v * -0.25)) * (((cbrt(r) * cbrt(r)) / ((cbrt(1.0 - v) * cbrt(1.0 - v)) / (w * r))) * (cbrt(cbrt(r) / (cbrt(1.0 - v) / w)) * (cbrt(cbrt(r) / (cbrt(1.0 - v) / w)) * cbrt(cbrt(r) / (cbrt(1.0 - v) / w))))));
	} else if (w <= 1.8322291642935536e-68) {
		tmp = ((2.0 / (r * r)) + -1.5) - ((0.375 + (v * -0.25)) * ((1.0 / (1.0 - v)) * (r * (w * (w * r)))));
	} else {
		tmp = ((2.0 / (r * r)) + -1.5) - ((0.375 + (v * -0.25)) * (r / (((1.0 - v) / (w * r)) / w)));
	}
	return tmp;
}

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. Split input into 3 regimes

  2. if w < -5.00309503669150277e160

    1. Initial program 64.0

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

    2. Simplified64.0

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

    4. Applied associate-*r*_binary6437.1

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

    6. Applied add-cube-cbrt_binary6437.2

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

    7. Applied times-frac_binary6419.7

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

    8. Applied add-cube-cbrt_binary6419.9

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

    9. Applied times-frac_binary641.0

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

    11. Applied add-cube-cbrt_binary641.0

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

    if -5.00309503669150277e160 < w < 1.8322291642935536e-68

    1. Initial program 9.2

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

    2. Simplified6.1

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

    4. Applied associate-*r*_binary642.3

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

    6. Applied div-inv_binary642.3

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

    7. Applied *-un-lft-identity_binary642.3

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

    8. Applied times-frac_binary640.3

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

    9. Simplified0.2

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

    if 1.8322291642935536e-68 < w

    1. Initial program 19.2

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

    2. Simplified11.1

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

    4. Applied associate-*r*_binary646.0

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

    6. Applied associate-/r*_binary643.1

      \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \left(0.375 + v \cdot -0.25\right) \cdot \frac{r}{\color{blue}{\frac{\frac{1 - v}{r \cdot w}}{w}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -5.003095036691503 \cdot 10^{+160}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \left(0.375 + v \cdot -0.25\right) \cdot \left(\frac{\sqrt[3]{r} \cdot \sqrt[3]{r}}{\frac{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}}{w \cdot r}} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{r}}{\frac{\sqrt[3]{1 - v}}{w}}} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{r}}{\frac{\sqrt[3]{1 - v}}{w}}} \cdot \sqrt[3]{\frac{\sqrt[3]{r}}{\frac{\sqrt[3]{1 - v}}{w}}}\right)\right)\right)\\ \mathbf{elif}\;w \leq 1.8322291642935536 \cdot 10^{-68}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \left(0.375 + v \cdot -0.25\right) \cdot \left(\frac{1}{1 - v} \cdot \left(r \cdot \left(w \cdot \left(w \cdot r\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \left(0.375 + v \cdot -0.25\right) \cdot \frac{r}{\frac{\frac{1 - v}{w \cdot r}}{w}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020231 
(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))