Average Error: 13.1 → 0.3
Time: 10.0s
Precision: binary64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
↓
\[\frac{2}{r \cdot r} - \left(1.5 + {\left(r \cdot w\right)}^{2} \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\right) \]
\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
↓
\frac{2}{r \cdot r} - \left(1.5 + {\left(r \cdot w\right)}^{2} \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\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
 (-
  (/ 2.0 (* r r))
  (+ 1.5 (* (pow (* r w) 2.0) (/ (fma v -0.25 0.375) (- 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) {
	return (2.0 / (r * r)) - (1.5 + (pow((r * w), 2.0) * (fma(v, -0.25, 0.375) / (1.0 - v))));
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 13.1

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

  2. Simplified8.6

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

  3. Applied egg-rr8.8

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

  4. Applied egg-rr2.4

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

  5. Applied egg-rr0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2022130 
(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))