Average Error: 12.7 → 0.5
Time: 8.4s
Precision: binary64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
↓
\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 1.748319656204719 \cdot 10^{+25}:\\ \;\;\;\;t_0 - \mathsf{fma}\left(r, w \cdot \left(\left(w \cdot r\right) \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\right), 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - w \cdot \left(r \cdot \left(\frac{w}{1 - v} \cdot \left(r \cdot \mathsf{fma}\left(v, -0.25, 0.375\right)\right)\right)\right)\\ \end{array} \]
\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 := \frac{2}{r \cdot r}\\
\mathbf{if}\;w \cdot w \leq 1.748319656204719 \cdot 10^{+25}:\\
\;\;\;\;t_0 - \mathsf{fma}\left(r, w \cdot \left(\left(w \cdot r\right) \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\right), 1.5\right)\\

\mathbf{else}:\\
\;\;\;\;t_0 - w \cdot \left(r \cdot \left(\frac{w}{1 - v} \cdot \left(r \cdot \mathsf{fma}\left(v, -0.25, 0.375\right)\right)\right)\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 (/ 2.0 (* r r))))
   (if (<= (* w w) 1.748319656204719e+25)
     (- t_0 (fma r (* w (* (* w r) (/ (fma v -0.25 0.375) (- 1.0 v)))) 1.5))
     (- t_0 (* w (* r (* (/ w (- 1.0 v)) (* r (fma v -0.25 0.375)))))))))
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 ((w * w) <= 1.748319656204719e+25) {
		tmp = t_0 - fma(r, (w * ((w * r) * (fma(v, -0.25, 0.375) / (1.0 - v)))), 1.5);
	} else {
		tmp = t_0 - (w * (r * ((w / (1.0 - v)) * (r * fma(v, -0.25, 0.375)))));
	}
	return tmp;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 w w) < 1.748319656204719e25

    1. Initial program 8.4

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

    2. Simplified5.3

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

    3. Applied associate-*r*_binary640.2

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

    4. Taylor expanded in r around 0 5.9

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

    5. Simplified0.2

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

    if 1.748319656204719e25 < (*.f64 w w)

    1. Initial program 25.9

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

    2. Simplified18.0

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

    3. Applied associate-*r*_binary649.7

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

    4. Taylor expanded in r around 0 30.5

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

    5. Simplified9.6

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

    6. Taylor expanded in r around inf 29.7

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

    7. Simplified1.4

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 1.748319656204719 \cdot 10^{+25}:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(r, w \cdot \left(\left(w \cdot r\right) \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\right), 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - w \cdot \left(r \cdot \left(\frac{w}{1 - v} \cdot \left(r \cdot \mathsf{fma}\left(v, -0.25, 0.375\right)\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021313 
(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))