Rosa's TurbineBenchmark

Average Error: 12.8 → 0.6

Time: 9.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \sqrt{\mathsf{fma}\left(w \cdot \left(r \cdot r\right), \frac{w}{1 - v} \cdot \mathsf{fma}\left(v, -0.25, 0.375\right), 1.5\right)}\\ t_2 := t_0 - t_1 \cdot t_1\\ \mathbf{if}\;w \leq -2.5867080588929657 \cdot 10^{+212}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;w \leq 7.389519611649986 \cdot 10^{+131}:\\ \;\;\;\;t_0 - \mathsf{fma}\left(r, w \cdot \left(\left(w \cdot r\right) \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\right), 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (sqrt
          (fma (* w (* r r)) (* (/ w (- 1.0 v)) (fma v -0.25 0.375)) 1.5)))
        (t_2 (- t_0 (* t_1 t_1))))
   (if (<= w -2.5867080588929657e+212)
     t_2
     (if (<= w 7.389519611649986e+131)
       (- t_0 (fma r (* w (* (* w r) (/ (fma v -0.25 0.375) (- 1.0 v)))) 1.5))
       t_2))))

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 t_0 = 2.0 / (r * r);
	double t_1 = sqrt(fma((w * (r * r)), ((w / (1.0 - v)) * fma(v, -0.25, 0.375)), 1.5));
	double t_2 = t_0 - (t_1 * t_1);
	double tmp;
	if (w <= -2.5867080588929657e+212) {
		tmp = t_2;
	} else if (w <= 7.389519611649986e+131) {
		tmp = t_0 - fma(r, (w * ((w * r) * (fma(v, -0.25, 0.375) / (1.0 - v)))), 1.5);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

1. Split input into 2 regimes

2. if w < -2.5867080588929657e212 or 7.38951961164998592e131 < w

   1. Initial program 54.5

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

   2. Simplified 52.0

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

   3. Applied associate-*r*_binary64 32.5

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

   4. Applied associate-*l*_binary64 32.5

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

   5. Applied add-sqr-sqrt_binary64 32.5

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

   6. Simplified 32.5

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

   7. Simplified 0.5

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

   if -2.5867080588929657e212 < w < 7.38951961164998592e131

   1. Initial program 10.4

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

   2. Simplified 5.8

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

   3. Applied associate-*r*_binary64 0.6

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

   4. Taylor expanded in r around 0 9.5

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

   5. Simplified 0.6

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -2.5867080588929657 \cdot 10^{+212}:\\ \;\;\;\;\frac{2}{r \cdot r} - \sqrt{\mathsf{fma}\left(w \cdot \left(r \cdot r\right), \frac{w}{1 - v} \cdot \mathsf{fma}\left(v, -0.25, 0.375\right), 1.5\right)} \cdot \sqrt{\mathsf{fma}\left(w \cdot \left(r \cdot r\right), \frac{w}{1 - v} \cdot \mathsf{fma}\left(v, -0.25, 0.375\right), 1.5\right)}\\ \mathbf{elif}\;w \leq 7.389519611649986 \cdot 10^{+131}:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(r, w \cdot \left(\left(w \cdot r\right) \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\right), 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - \sqrt{\mathsf{fma}\left(w \cdot \left(r \cdot r\right), \frac{w}{1 - v} \cdot \mathsf{fma}\left(v, -0.25, 0.375\right), 1.5\right)} \cdot \sqrt{\mathsf{fma}\left(w \cdot \left(r \cdot r\right), \frac{w}{1 - v} \cdot \mathsf{fma}\left(v, -0.25, 0.375\right), 1.5\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(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))