Average Error: 10.2 → 0.1
Time: 5.9s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.4374981009777875 \cdot 10^{27}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right) - x\\ \mathbf{elif}\;x \le 8.649219799147649 \cdot 10^{-36}:\\ \;\;\;\;\left(\left(x \cdot y\right) \cdot \frac{1}{z} + \frac{x}{z} \cdot 1\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]
\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(y + 1\right) - x\\

\mathbf{elif}\;x \le 8.649219799147649 \cdot 10^{-36}:\\
\;\;\;\;\left(\left(x \cdot y\right) \cdot \frac{1}{z} + \frac{x}{z} \cdot 1\right) - x\\

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

\end{array}
double f(double x, double y, double z) {
        double r788514 = x;
        double r788515 = y;
        double r788516 = z;
        double r788517 = r788515 - r788516;
        double r788518 = 1.0;
        double r788519 = r788517 + r788518;
        double r788520 = r788514 * r788519;
        double r788521 = r788520 / r788516;
        return r788521;
}
↓
double f(double x, double y, double z) {
        double r788522 = x;
        double r788523 = -3.4374981009777875e+27;
        bool r788524 = r788522 <= r788523;
        double r788525 = z;
        double r788526 = r788522 / r788525;
        double r788527 = y;
        double r788528 = 1.0;
        double r788529 = r788527 + r788528;
        double r788530 = r788526 * r788529;
        double r788531 = r788530 - r788522;
        double r788532 = 8.649219799147649e-36;
        bool r788533 = r788522 <= r788532;
        double r788534 = r788522 * r788527;
        double r788535 = 1.0;
        double r788536 = r788535 / r788525;
        double r788537 = r788534 * r788536;
        double r788538 = r788526 * r788528;
        double r788539 = r788537 + r788538;
        double r788540 = r788539 - r788522;
        double r788541 = r788527 - r788525;
        double r788542 = r788541 + r788528;
        double r788543 = r788525 / r788542;
        double r788544 = r788522 / r788543;
        double r788545 = r788533 ? r788540 : r788544;
        double r788546 = r788524 ? r788531 : r788545;
        return r788546;
}

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. Simplified0.1

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

    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. Simplified3.0

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

    5. Applied distribute-lft-in3.0

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

    6. Simplified0.1

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

    8. Applied div-inv0.1

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

    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

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

Reproduce

herbie shell --seed 2020046 +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))