Rosa's TurbineBenchmark

Average Error: 12.7 → 0.3

Time: 8.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;w \cdot w \leq 4.29896508473143 \cdot 10^{+278}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} - \mathsf{fma}\left(r, \left(w \cdot \left(w \cdot r\right)\right) \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(w \cdot \left(r \cdot r\right), \mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \frac{w}{1 - v}, 1.5\right)}\\ \frac{2}{r \cdot r} - t_0 \cdot t_0 \end{array}\\ \end{array} \]

FPCore

(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 w) 4.29896508473143e+278)
   (-
    (/ (/ 2.0 r) r)
    (fma r (* (* w (* w r)) (/ (fma v -0.25 0.375) (- 1.0 v))) 1.5))
   (let* ((t_0
           (sqrt
            (fma (* w (* r r)) (* (fma v -0.25 0.375) (/ w (- 1.0 v))) 1.5))))
     (- (/ 2.0 (* r r)) (* t_0 t_0)))))

C

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 * w) <= 4.29896508473143e+278) {
		tmp = ((2.0 / r) / r) - fma(r, ((w * (w * r)) * (fma(v, -0.25, 0.375) / (1.0 - v))), 1.5);
	} else {
		double t_0 = sqrt(fma((w * (r * r)), (fma(v, -0.25, 0.375) * (w / (1.0 - v))), 1.5));
		tmp = (2.0 / (r * r)) - (t_0 * t_0);
	}
	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) < 4.29896508473143004e278

   1. Initial program 8.7

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

   2. Simplified 4.3

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

   3. Applied associate-*r*_binary64 0.2

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

   4. Applied associate-/r*_binary64 0.2

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

   if 4.29896508473143004e278 < (*.f64 w w)

   1. Initial program 55.7

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

   2. Simplified 52.9

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

   3. Applied associate-*r*_binary64 24.9

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

   4. Applied associate-*l*_binary64 24.8

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

   5. Applied add-sqr-sqrt_binary64 24.9

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

   6. Simplified 24.8

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

   7. Simplified 0.5

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 4.29896508473143 \cdot 10^{+278}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} - \mathsf{fma}\left(r, \left(w \cdot \left(w \cdot r\right)\right) \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - \sqrt{\mathsf{fma}\left(w \cdot \left(r \cdot r\right), \mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \frac{w}{1 - v}, 1.5\right)} \cdot \sqrt{\mathsf{fma}\left(w \cdot \left(r \cdot r\right), \mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \frac{w}{1 - v}, 1.5\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021275 
(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))