Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, D

Average Error: 22.7 → 0.3

Time: 2.7s

Precision: 64

Math

\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{y + 1} \le 0.822819537817507496 \lor \neg \left(\frac{\left(1 - x\right) \cdot y}{y + 1} \le 1.03361419767810836\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y + 1}, x - 1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r674457 = 1.0;
        double r674458 = x;
        double r674459 = r674457 - r674458;
        double r674460 = y;
        double r674461 = r674459 * r674460;
        double r674462 = r674460 + r674457;
        double r674463 = r674461 / r674462;
        double r674464 = r674457 - r674463;
        return r674464;
}

↓

double f(double x, double y) {
        double r674465 = 1.0;
        double r674466 = x;
        double r674467 = r674465 - r674466;
        double r674468 = y;
        double r674469 = r674467 * r674468;
        double r674470 = r674468 + r674465;
        double r674471 = r674469 / r674470;
        double r674472 = 0.8228195378175075;
        bool r674473 = r674471 <= r674472;
        double r674474 = 1.0336141976781084;
        bool r674475 = r674471 <= r674474;
        double r674476 = !r674475;
        bool r674477 = r674473 || r674476;
        double r674478 = r674468 / r674470;
        double r674479 = r674466 - r674465;
        double r674480 = fma(r674478, r674479, r674465);
        double r674481 = 1.0;
        double r674482 = r674481 / r674468;
        double r674483 = r674466 / r674468;
        double r674484 = r674482 - r674483;
        double r674485 = fma(r674465, r674484, r674466);
        double r674486 = r674477 ? r674480 : r674485;
        return r674486;
}

Error

Bits error versus x

Bits error versus y

Target

Original	22.7
Target	0.2
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (/ (* (- 1.0 x) y) (+ y 1.0)) < 0.8228195378175075 or 1.0336141976781084 < (/ (* (- 1.0 x) y) (+ y 1.0))

   1. Initial program 11.2

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]

   2. Simplified 0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{y + 1}, x - 1, 1\right)}\]

   if 0.8228195378175075 < (/ (* (- 1.0 x) y) (+ y 1.0)) < 1.0336141976781084

   1. Initial program 57.8

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]

   2. Taylor expanded around inf 1.2

      \[\leadsto \color{blue}{\left(x + 1 \cdot \frac{1}{y}\right) - 1 \cdot \frac{x}{y}}\]

   3. Simplified 1.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{y + 1} \le 0.822819537817507496 \lor \neg \left(\frac{\left(1 - x\right) \cdot y}{y + 1} \le 1.03361419767810836\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y + 1}, x - 1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x))))

  (- 1 (/ (* (- 1 x) y) (+ y 1))))