Average Error: 10.4 → 0.4
Time: 18.2s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.894138962339348575688367605867883657369 \cdot 10^{-104}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{elif}\;x \le 5.333921896352304070857289463437993050447 \cdot 10^{-71}:\\ \;\;\;\;\frac{1}{z} \cdot \left(x \cdot \left(\left(y - z\right) + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.894138962339348575688367605867883657369 \cdot 10^{-104}:\\
\;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\

\mathbf{elif}\;x \le 5.333921896352304070857289463437993050447 \cdot 10^{-71}:\\
\;\;\;\;\frac{1}{z} \cdot \left(x \cdot \left(\left(y - z\right) + 1\right)\right)\\

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

\end{array}
double f(double x, double y, double z) {
        double r413642 = x;
        double r413643 = y;
        double r413644 = z;
        double r413645 = r413643 - r413644;
        double r413646 = 1.0;
        double r413647 = r413645 + r413646;
        double r413648 = r413642 * r413647;
        double r413649 = r413648 / r413644;
        return r413649;
}
↓
double f(double x, double y, double z) {
        double r413650 = x;
        double r413651 = -1.8941389623393486e-104;
        bool r413652 = r413650 <= r413651;
        double r413653 = z;
        double r413654 = y;
        double r413655 = r413654 - r413653;
        double r413656 = 1.0;
        double r413657 = r413655 + r413656;
        double r413658 = r413653 / r413657;
        double r413659 = r413650 / r413658;
        double r413660 = 5.333921896352304e-71;
        bool r413661 = r413650 <= r413660;
        double r413662 = 1.0;
        double r413663 = r413662 / r413653;
        double r413664 = r413650 * r413657;
        double r413665 = r413663 * r413664;
        double r413666 = r413650 / r413653;
        double r413667 = r413656 + r413654;
        double r413668 = r413666 * r413667;
        double r413669 = r413668 - r413650;
        double r413670 = r413661 ? r413665 : r413669;
        double r413671 = r413652 ? r413659 : r413670;
        return r413671;
}

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

Original10.4
Target0.4
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \lt -2.714831067134359919650240696134672137284 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.874108816439546156869494499878029491333 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.8941389623393486e-104

    1. Initial program 17.3

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

    3. Applied associate-/l*0.9

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

    if -1.8941389623393486e-104 < x < 5.333921896352304e-71

    1. Initial program 0.2

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

    3. Applied associate-/l*6.9

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

    5. Applied div-inv7.0

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

    6. Applied *-un-lft-identity7.0

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

    7. Applied times-frac0.3

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

    8. Simplified0.3

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

    if 5.333921896352304e-71 < x

    1. Initial program 19.4

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

    2. Taylor expanded around 0 6.2

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

    3. Simplified0.1

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.894138962339348575688367605867883657369 \cdot 10^{-104}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{elif}\;x \le 5.333921896352304070857289463437993050447 \cdot 10^{-71}:\\ \;\;\;\;\frac{1}{z} \cdot \left(x \cdot \left(\left(y - z\right) + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(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))