Average Error: 10.2 → 1.6
Time: 2.4s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le -1.0708239262793344 \cdot 10^{-97}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \mathbf{elif}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le 1.78293444417689793 \cdot 10^{140}:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le -1.0708239262793344 \cdot 10^{-97}:\\
\;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\

\mathbf{elif}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le 1.78293444417689793 \cdot 10^{140}:\\
\;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r666849 = x;
        double r666850 = y;
        double r666851 = z;
        double r666852 = r666850 - r666851;
        double r666853 = 1.0;
        double r666854 = r666852 + r666853;
        double r666855 = r666849 * r666854;
        double r666856 = r666855 / r666851;
        return r666856;
}
↓
double f(double x, double y, double z) {
        double r666857 = x;
        double r666858 = y;
        double r666859 = z;
        double r666860 = r666858 - r666859;
        double r666861 = 1.0;
        double r666862 = r666860 + r666861;
        double r666863 = r666857 * r666862;
        double r666864 = r666863 / r666859;
        double r666865 = -1.0708239262793344e-97;
        bool r666866 = r666864 <= r666865;
        double r666867 = r666857 / r666859;
        double r666868 = r666861 + r666858;
        double r666869 = r666867 * r666868;
        double r666870 = r666869 - r666857;
        double r666871 = 1.782934444176898e+140;
        bool r666872 = r666864 <= r666871;
        double r666873 = r666859 / r666862;
        double r666874 = r666857 / r666873;
        double r666875 = r666872 ? r666864 : r666874;
        double r666876 = r666866 ? r666870 : r666875;
        return r666876;
}

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
Target0.4
Herbie1.6
\[\begin{array}{l} \mathbf{if}\;x \lt -2.7148310671343599 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.87410881643954616 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (* x (+ (- y z) 1.0)) z) < -1.0708239262793344e-97

    1. Initial program 13.3

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

    2. Taylor expanded around 0 4.5

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

    3. Taylor expanded around 0 4.5

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

    4. Simplified0.5

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

    if -1.0708239262793344e-97 < (/ (* x (+ (- y z) 1.0)) z) < 1.782934444176898e+140

    1. Initial program 0.1

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

    3. Applied *-un-lft-identity0.1

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

    4. Applied times-frac1.1

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

    5. Simplified1.1

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

    7. Applied associate-*r/0.1

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

    if 1.782934444176898e+140 < (/ (* x (+ (- y z) 1.0)) z)

    1. Initial program 27.6

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

    3. Applied associate-/l*7.1

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

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le -1.0708239262793344 \cdot 10^{-97}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \mathbf{elif}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le 1.78293444417689793 \cdot 10^{140}:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(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))