Average Error: 9.9 → 1.1
Time: 11.3s
Precision: binary64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
↓
\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq -7.614131752839713 \cdot 10^{-100} \lor \neg \left(r \leq 2.4253524215214347 \cdot 10^{-119}\right):\\ \;\;\;\;t_0 - \mathsf{fma}\left(r, \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - 1.5\\ \end{array} \]
\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 := \frac{2}{r \cdot r}\\
\mathbf{if}\;r \leq -7.614131752839713 \cdot 10^{-100} \lor \neg \left(r \leq 2.4253524215214347 \cdot 10^{-119}\right):\\
\;\;\;\;t_0 - \mathsf{fma}\left(r, \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}, 1.5\right)\\

\mathbf{else}:\\
\;\;\;\;t_0 - 1.5\\


\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 (/ 2.0 (* r r))))
   (if (or (<= r -7.614131752839713e-100) (not (<= r 2.4253524215214347e-119)))
     (- t_0 (fma r (* (* w (* r w)) (/ (fma v -0.25 0.375) (- 1.0 v))) 1.5))
     (- t_0 1.5))))
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 = 2.0 / (r * r);
	double tmp;
	if ((r <= -7.614131752839713e-100) || !(r <= 2.4253524215214347e-119)) {
		tmp = t_0 - fma(r, ((w * (r * w)) * (fma(v, -0.25, 0.375) / (1.0 - v))), 1.5);
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Split input into 2 regimes

  2. if r < -7.6141317528397128e-100 or 2.42535242152143467e-119 < r

    1. Initial program 7.1

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

    2. Simplified4.4

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

    3. Applied associate-*r*_binary640.9

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

    if -7.6141317528397128e-100 < r < 2.42535242152143467e-119

    1. Initial program 15.6

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

    2. Simplified15.6

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

    3. Taylor expanded in r around 0 1.4

      \[\leadsto \frac{2}{r \cdot r} - \color{blue}{1.5} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq -7.614131752839713 \cdot 10^{-100} \lor \neg \left(r \leq 2.4253524215214347 \cdot 10^{-119}\right):\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(r, \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - 1.5\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(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))