Rosa's TurbineBenchmark

Average Error: 12.7 → 0.3

Time: 13.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq -2.424091774928309 \cdot 10^{+34}:\\ \;\;\;\;t_0 - \mathsf{fma}\left(r, \left(r \cdot w\right) \cdot \left(\left(w \cdot \mathsf{fma}\left(v, -0.25, 0.375\right)\right) \cdot \frac{1}{1 - v}\right), 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_1 := \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\\ \mathbf{if}\;r \leq 5.256561651639415 \cdot 10^{-29}:\\ \;\;\;\;t_0 - \left(1.5 + \left(r \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot t_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} - \mathsf{fma}\left(r, t_1 \cdot \left(w \cdot \left(r \cdot w\right)\right), 1.5\right)\\ \end{array}\\ \end{array} \]

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
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r -2.424091774928309e+34)
     (-
      t_0
      (fma r (* (* r w) (* (* w (fma v -0.25 0.375)) (/ 1.0 (- 1.0 v)))) 1.5))
     (let* ((t_1 (/ (fma v -0.25 0.375) (- 1.0 v))))
       (if (<= r 5.256561651639415e-29)
         (- t_0 (+ 1.5 (* (* r (* r w)) (* w t_1))))
         (- (/ (/ 2.0 r) r) (fma r (* t_1 (* w (* r w))) 1.5)))))))

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 tmp;
	if (r <= -2.424091774928309e+34) {
		tmp = t_0 - fma(r, ((r * w) * ((w * fma(v, -0.25, 0.375)) * (1.0 / (1.0 - v)))), 1.5);
	} else {
		double t_1 = fma(v, -0.25, 0.375) / (1.0 - v);
		double tmp_1;
		if (r <= 5.256561651639415e-29) {
			tmp_1 = t_0 - (1.5 + ((r * (r * w)) * (w * t_1)));
		} else {
			tmp_1 = ((2.0 / r) / r) - fma(r, (t_1 * (w * (r * w))), 1.5);
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

1. Split input into 3 regimes

2. if r < -2.4240917749283089e34

   1. Initial program 16.0

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

   2. Simplified 8.6

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

   3. Applied 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-*l*_binary64 0.2

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

   5. Applied div-inv_binary64 0.2

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

   6. Applied associate-*r*_binary64 0.6

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

   if -2.4240917749283089e34 < r < 5.2565616516394145e-29

   1. Initial program 10.6

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

   2. Simplified 9.1

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

   3. Applied associate-*r*_binary64 4.7

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

   4. Applied associate-*l*_binary64 4.7

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

   5. Applied fma-udef_binary64 4.7

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

   6. Applied associate-*r*_binary64 0.3

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

   if 5.2565616516394145e-29 < r

   1. Initial program 13.3

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

   2. Simplified 7.1

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

   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 0.3

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq -2.424091774928309 \cdot 10^{+34}:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(r, \left(r \cdot w\right) \cdot \left(\left(w \cdot \mathsf{fma}\left(v, -0.25, 0.375\right)\right) \cdot \frac{1}{1 - v}\right), 1.5\right)\\ \mathbf{elif}\;r \leq 5.256561651639415 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{r \cdot r} - \left(1.5 + \left(r \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} - \mathsf{fma}\left(r, \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v} \cdot \left(w \cdot \left(r \cdot w\right)\right), 1.5\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022024 
(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))