Average Error: 10.1 → 0.2
Time: 15.6s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.8170706341410064 \cdot 10^{29}:\\ \;\;\;\;x \cdot \frac{1 + y}{z} - x\\ \mathbf{elif}\;x \le 9.5183273331390796 \cdot 10^{52}:\\ \;\;\;\;\left(\frac{x}{z} \cdot 1 + \frac{x \cdot y}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.8170706341410064 \cdot 10^{29}:\\
\;\;\;\;x \cdot \frac{1 + y}{z} - x\\

\mathbf{elif}\;x \le 9.5183273331390796 \cdot 10^{52}:\\
\;\;\;\;\left(\frac{x}{z} \cdot 1 + \frac{x \cdot y}{z}\right) - x\\

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

\end{array}
double f(double x, double y, double z) {
        double r716589 = x;
        double r716590 = y;
        double r716591 = z;
        double r716592 = r716590 - r716591;
        double r716593 = 1.0;
        double r716594 = r716592 + r716593;
        double r716595 = r716589 * r716594;
        double r716596 = r716595 / r716591;
        return r716596;
}
↓
double f(double x, double y, double z) {
        double r716597 = x;
        double r716598 = -2.8170706341410064e+29;
        bool r716599 = r716597 <= r716598;
        double r716600 = 1.0;
        double r716601 = y;
        double r716602 = r716600 + r716601;
        double r716603 = z;
        double r716604 = r716602 / r716603;
        double r716605 = r716597 * r716604;
        double r716606 = r716605 - r716597;
        double r716607 = 9.51832733313908e+52;
        bool r716608 = r716597 <= r716607;
        double r716609 = r716597 / r716603;
        double r716610 = r716609 * r716600;
        double r716611 = r716597 * r716601;
        double r716612 = r716611 / r716603;
        double r716613 = r716610 + r716612;
        double r716614 = r716613 - r716597;
        double r716615 = r716601 - r716603;
        double r716616 = r716615 + r716600;
        double r716617 = r716603 / r716616;
        double r716618 = r716597 / r716617;
        double r716619 = r716608 ? r716614 : r716618;
        double r716620 = r716599 ? r716606 : r716619;
        return r716620;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.1
Target0.4
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt -2.7148310671343599 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.87410881643954616 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -2.8170706341410064e+29

    1. Initial program 29.1

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

    2. Taylor expanded around 0 10.1

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

    3. Simplified0.1

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

    5. Applied div-inv0.1

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

    6. Applied associate-*l*0.1

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

    7. Simplified0.1

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

    if -2.8170706341410064e+29 < x < 9.51832733313908e+52

    1. Initial program 0.5

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified2.4

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

    5. Applied distribute-lft-in2.4

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

    6. Simplified0.2

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

    if 9.51832733313908e+52 < x

    1. Initial program 30.6

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

    3. Applied associate-/l*0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.8170706341410064 \cdot 10^{29}:\\ \;\;\;\;x \cdot \frac{1 + y}{z} - x\\ \mathbf{elif}\;x \le 9.5183273331390796 \cdot 10^{52}:\\ \;\;\;\;\left(\frac{x}{z} \cdot 1 + \frac{x \cdot y}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +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))