Average Error: 10.2 → 0.1
Time: 2.0s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.4374981009777875 \cdot 10^{27}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \mathbf{elif}\;x \le 8.649219799147649 \cdot 10^{-36}:\\ \;\;\;\;\left(\left(x \cdot y\right) \cdot \frac{1}{z} + 1 \cdot \frac{x}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.4374981009777875 \cdot 10^{27}:\\
\;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\

\mathbf{elif}\;x \le 8.649219799147649 \cdot 10^{-36}:\\
\;\;\;\;\left(\left(x \cdot y\right) \cdot \frac{1}{z} + 1 \cdot \frac{x}{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 r829537 = x;
        double r829538 = y;
        double r829539 = z;
        double r829540 = r829538 - r829539;
        double r829541 = 1.0;
        double r829542 = r829540 + r829541;
        double r829543 = r829537 * r829542;
        double r829544 = r829543 / r829539;
        return r829544;
}
↓
double f(double x, double y, double z) {
        double r829545 = x;
        double r829546 = -3.4374981009777875e+27;
        bool r829547 = r829545 <= r829546;
        double r829548 = z;
        double r829549 = r829545 / r829548;
        double r829550 = 1.0;
        double r829551 = y;
        double r829552 = r829550 + r829551;
        double r829553 = r829549 * r829552;
        double r829554 = r829553 - r829545;
        double r829555 = 8.649219799147649e-36;
        bool r829556 = r829545 <= r829555;
        double r829557 = r829545 * r829551;
        double r829558 = 1.0;
        double r829559 = r829558 / r829548;
        double r829560 = r829557 * r829559;
        double r829561 = r829550 * r829549;
        double r829562 = r829560 + r829561;
        double r829563 = r829562 - r829545;
        double r829564 = r829551 - r829548;
        double r829565 = r829564 + r829550;
        double r829566 = r829548 / r829565;
        double r829567 = r829545 / r829566;
        double r829568 = r829556 ? r829563 : r829567;
        double r829569 = r829547 ? r829554 : r829568;
        return r829569;
}

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

Original10.2
Target0.5
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 3 regimes

  2. if x < -3.4374981009777875e+27

    1. Initial program 28.2

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

    2. Taylor expanded around 0 9.1

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

    3. Taylor expanded around 0 9.1

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

    4. Simplified0.1

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

    if -3.4374981009777875e+27 < x < 8.649219799147649e-36

    1. Initial program 0.3

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

    2. Taylor expanded around 0 0.1

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

    4. Applied div-inv0.1

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

    if 8.649219799147649e-36 < x

    1. Initial program 21.3

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

    3. Applied associate-/l*0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.4374981009777875 \cdot 10^{27}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \mathbf{elif}\;x \le 8.649219799147649 \cdot 10^{-36}:\\ \;\;\;\;\left(\left(x \cdot y\right) \cdot \frac{1}{z} + 1 \cdot \frac{x}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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))