Average Error: 7.8 → 0.1
Time: 25.3s
Precision: 64
\[\frac{x \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -3.752148075371786588524042595562668578523 \cdot 10^{46}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{elif}\;y \le 257574.272225504857487976551055908203125:\\ \;\;\;\;\frac{x \cdot y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \end{array}\]
\frac{x \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -3.752148075371786588524042595562668578523 \cdot 10^{46}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\

\mathbf{elif}\;y \le 257574.272225504857487976551055908203125:\\
\;\;\;\;\frac{x \cdot y}{1 + y}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\

\end{array}
double f(double x, double y) {
        double r30464567 = x;
        double r30464568 = y;
        double r30464569 = r30464567 * r30464568;
        double r30464570 = 1.0;
        double r30464571 = r30464568 + r30464570;
        double r30464572 = r30464569 / r30464571;
        return r30464572;
}
↓
double f(double x, double y) {
        double r30464573 = y;
        double r30464574 = -3.7521480753717866e+46;
        bool r30464575 = r30464573 <= r30464574;
        double r30464576 = x;
        double r30464577 = r30464576 / r30464573;
        double r30464578 = 1.0;
        double r30464579 = r30464578 / r30464573;
        double r30464580 = r30464579 - r30464578;
        double r30464581 = fma(r30464577, r30464580, r30464576);
        double r30464582 = 257574.27222550486;
        bool r30464583 = r30464573 <= r30464582;
        double r30464584 = r30464576 * r30464573;
        double r30464585 = r30464578 + r30464573;
        double r30464586 = r30464584 / r30464585;
        double r30464587 = r30464583 ? r30464586 : r30464581;
        double r30464588 = r30464575 ? r30464581 : r30464587;
        return r30464588;
}

Error

Bits error versus x

Bits error versus y

Target

Original7.8
Target0.0
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{x}{y \cdot y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;\frac{x \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -3.7521480753717866e+46 or 257574.27222550486 < y

    1. Initial program 16.8

      \[\frac{x \cdot y}{y + 1}\]

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(1 \cdot \frac{x}{{y}^{2}} + x\right) - 1 \cdot \frac{x}{y}}\]

    3. Simplified0

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

    if -3.7521480753717866e+46 < y < 257574.27222550486

    1. Initial program 0.2

      \[\frac{x \cdot y}{y + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.752148075371786588524042595562668578523 \cdot 10^{46}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{elif}\;y \le 257574.272225504857487976551055908203125:\\ \;\;\;\;\frac{x \cdot y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, B"

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ x (* y y)) (- (/ x y) x)) (if (< y 6799310503.41891) (/ (* x y) (+ y 1.0)) (- (/ x (* y y)) (- (/ x y) x))))

  (/ (* x y) (+ y 1.0)))