Average Error: 10.0 → 0.1
Time: 2.4s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.40301390883669074 \cdot 10^{-70} \lor \neg \left(z \le 3.9888923668282164 \cdot 10^{-23}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, 1 \cdot \frac{x}{z} - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -4.40301390883669074 \cdot 10^{-70} \lor \neg \left(z \le 3.9888923668282164 \cdot 10^{-23}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, 1 \cdot \frac{x}{z} - x\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\

\end{array}
double f(double x, double y, double z) {
        double r635531 = x;
        double r635532 = y;
        double r635533 = z;
        double r635534 = r635532 - r635533;
        double r635535 = 1.0;
        double r635536 = r635534 + r635535;
        double r635537 = r635531 * r635536;
        double r635538 = r635537 / r635533;
        return r635538;
}
↓
double f(double x, double y, double z) {
        double r635539 = z;
        double r635540 = -4.403013908836691e-70;
        bool r635541 = r635539 <= r635540;
        double r635542 = 3.9888923668282164e-23;
        bool r635543 = r635539 <= r635542;
        double r635544 = !r635543;
        bool r635545 = r635541 || r635544;
        double r635546 = y;
        double r635547 = r635546 / r635539;
        double r635548 = x;
        double r635549 = 1.0;
        double r635550 = r635548 / r635539;
        double r635551 = r635549 * r635550;
        double r635552 = r635551 - r635548;
        double r635553 = fma(r635547, r635548, r635552);
        double r635554 = r635546 - r635539;
        double r635555 = r635554 + r635549;
        double r635556 = r635550 * r635555;
        double r635557 = r635545 ? r635553 : r635556;
        return r635557;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.0
Target0.4
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt -2.7148310671343599 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.87410881643954616 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -4.403013908836691e-70 or 3.9888923668282164e-23 < z

    1. Initial program 14.5

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
    2. Using strategy rm

    3. Applied associate-/l*0.1

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}}\]
    4. Using strategy rm

    5. Applied div-inv0.3

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

    6. Taylor expanded around 0 5.1

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

    7. Simplified0.2

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

    if -4.403013908836691e-70 < z < 3.9888923668282164e-23

    1. Initial program 0.1

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
    2. Using strategy rm

    3. Applied associate-/l*9.5

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}}\]
    4. Using strategy rm

    5. Applied div-inv9.6

      \[\leadsto \frac{x}{\color{blue}{z \cdot \frac{1}{\left(y - z\right) + 1}}}\]
    6. Using strategy rm

    7. Applied un-div-inv9.5

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

    8. Applied associate-/r/0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.40301390883669074 \cdot 10^{-70} \lor \neg \left(z \le 3.9888923668282164 \cdot 10^{-23}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, 1 \cdot \frac{x}{z} - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< x -2.71483106713436e-162) (- (* (+ 1 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1)) (/ 1 z)) (- (* (+ 1 y) (/ x z)) x)))

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