Average Error: 10.0 → 0.1
Time: 2.3s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.04087632150993631 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \mathbf{elif}\;x \le 3.8730830355980695 \cdot 10^{-51}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + 1 \cdot \frac{x}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{\frac{z}{y}} + 1 \cdot \frac{x}{z}\right) - x\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -5.04087632150993631 \cdot 10^{-30}:\\
\;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\

\mathbf{elif}\;x \le 3.8730830355980695 \cdot 10^{-51}:\\
\;\;\;\;\left(\frac{x \cdot y}{z} + 1 \cdot \frac{x}{z}\right) - x\\

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

\end{array}
double f(double x, double y, double z) {
        double r708833 = x;
        double r708834 = y;
        double r708835 = z;
        double r708836 = r708834 - r708835;
        double r708837 = 1.0;
        double r708838 = r708836 + r708837;
        double r708839 = r708833 * r708838;
        double r708840 = r708839 / r708835;
        return r708840;
}
↓
double f(double x, double y, double z) {
        double r708841 = x;
        double r708842 = -5.040876321509936e-30;
        bool r708843 = r708841 <= r708842;
        double r708844 = z;
        double r708845 = r708841 / r708844;
        double r708846 = 1.0;
        double r708847 = y;
        double r708848 = r708846 + r708847;
        double r708849 = r708845 * r708848;
        double r708850 = r708849 - r708841;
        double r708851 = 3.8730830355980695e-51;
        bool r708852 = r708841 <= r708851;
        double r708853 = r708841 * r708847;
        double r708854 = r708853 / r708844;
        double r708855 = r708846 * r708845;
        double r708856 = r708854 + r708855;
        double r708857 = r708856 - r708841;
        double r708858 = r708844 / r708847;
        double r708859 = r708841 / r708858;
        double r708860 = r708859 + r708855;
        double r708861 = r708860 - r708841;
        double r708862 = r708852 ? r708857 : r708861;
        double r708863 = r708843 ? r708850 : r708862;
        return r708863;
}

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

Original10.0
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 < -5.040876321509936e-30

    1. Initial program 22.5

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

    2. Taylor expanded around 0 8.4

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

    3. Taylor expanded around 0 8.4

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

    4. Simplified0.1

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

    if -5.040876321509936e-30 < x < 3.8730830355980695e-51

    1. Initial program 0.1

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

    2. Taylor expanded around 0 0.1

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

    if 3.8730830355980695e-51 < x

    1. Initial program 20.6

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

    2. Taylor expanded around 0 6.9

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

    4. Applied associate-/l*0.3

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.04087632150993631 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \mathbf{elif}\;x \le 3.8730830355980695 \cdot 10^{-51}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + 1 \cdot \frac{x}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{\frac{z}{y}} + 1 \cdot \frac{x}{z}\right) - x\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(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))