Average Error: 9.8 → 0.7
Time: 14.1s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1.0\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -29975784529430668.0:\\ \;\;\;\;\left(y + 1.0\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \le 1.4441879214959917 \cdot 10^{-288}:\\ \;\;\;\;\left(\frac{y \cdot x}{z} + 1.0 \cdot \frac{x}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\left(y + 1.0\right) \cdot \frac{x}{z} - x\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1.0\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -29975784529430668.0:\\
\;\;\;\;\left(y + 1.0\right) \cdot \frac{x}{z} - x\\

\mathbf{elif}\;x \le 1.4441879214959917 \cdot 10^{-288}:\\
\;\;\;\;\left(\frac{y \cdot x}{z} + 1.0 \cdot \frac{x}{z}\right) - x\\

\mathbf{else}:\\
\;\;\;\;\left(y + 1.0\right) \cdot \frac{x}{z} - x\\

\end{array}
double f(double x, double y, double z) {
        double r33153805 = x;
        double r33153806 = y;
        double r33153807 = z;
        double r33153808 = r33153806 - r33153807;
        double r33153809 = 1.0;
        double r33153810 = r33153808 + r33153809;
        double r33153811 = r33153805 * r33153810;
        double r33153812 = r33153811 / r33153807;
        return r33153812;
}
↓
double f(double x, double y, double z) {
        double r33153813 = x;
        double r33153814 = -29975784529430668.0;
        bool r33153815 = r33153813 <= r33153814;
        double r33153816 = y;
        double r33153817 = 1.0;
        double r33153818 = r33153816 + r33153817;
        double r33153819 = z;
        double r33153820 = r33153813 / r33153819;
        double r33153821 = r33153818 * r33153820;
        double r33153822 = r33153821 - r33153813;
        double r33153823 = 1.4441879214959917e-288;
        bool r33153824 = r33153813 <= r33153823;
        double r33153825 = r33153816 * r33153813;
        double r33153826 = r33153825 / r33153819;
        double r33153827 = r33153817 * r33153820;
        double r33153828 = r33153826 + r33153827;
        double r33153829 = r33153828 - r33153813;
        double r33153830 = r33153824 ? r33153829 : r33153822;
        double r33153831 = r33153815 ? r33153822 : r33153830;
        return r33153831;
}

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

Original9.8
Target0.5
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;x \lt -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1.0\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -29975784529430668.0 or 1.4441879214959917e-288 < x

    1. Initial program 14.6

      \[\frac{x \cdot \left(\left(y - z\right) + 1.0\right)}{z}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity14.6

      \[\leadsto \frac{x \cdot \left(\left(y - z\right) + 1.0\right)}{\color{blue}{1 \cdot z}}\]

    4. Applied times-frac2.4

      \[\leadsto \color{blue}{\frac{x}{1} \cdot \frac{\left(y - z\right) + 1.0}{z}}\]

    5. Simplified2.4

      \[\leadsto \color{blue}{x} \cdot \frac{\left(y - z\right) + 1.0}{z}\]

    6. Taylor expanded around 0 5.0

      \[\leadsto \color{blue}{\left(\frac{x \cdot y}{z} + 1.0 \cdot \frac{x}{z}\right) - x}\]

    7. Simplified1.0

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \left(y + 1.0\right) - x}\]

    if -29975784529430668.0 < x < 1.4441879214959917e-288

    1. Initial program 0.2

      \[\frac{x \cdot \left(\left(y - z\right) + 1.0\right)}{z}\]

    2. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{\left(\frac{x \cdot y}{z} + 1.0 \cdot \frac{x}{z}\right) - x}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -29975784529430668.0:\\ \;\;\;\;\left(y + 1.0\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \le 1.4441879214959917 \cdot 10^{-288}:\\ \;\;\;\;\left(\frac{y \cdot x}{z} + 1.0 \cdot \frac{x}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\left(y + 1.0\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"

  :herbie-target
  (if (< x -2.71483106713436e-162) (- (* (+ 1 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1.0)) (/ 1 z)) (- (* (+ 1 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1.0)) z))