Average Error: 10.2 → 1.6
Time: 2.5s
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 r660112 = x;
        double r660113 = y;
        double r660114 = z;
        double r660115 = r660113 - r660114;
        double r660116 = 1.0;
        double r660117 = r660115 + r660116;
        double r660118 = r660112 * r660117;
        double r660119 = r660118 / r660114;
        return r660119;
}
↓
double f(double x, double y, double z) {
        double r660120 = x;
        double r660121 = y;
        double r660122 = z;
        double r660123 = r660121 - r660122;
        double r660124 = 1.0;
        double r660125 = r660123 + r660124;
        double r660126 = r660120 * r660125;
        double r660127 = r660126 / r660122;
        double r660128 = -1.0708239262793344e-97;
        bool r660129 = r660127 <= r660128;
        double r660130 = r660120 / r660122;
        double r660131 = r660124 + r660121;
        double r660132 = r660130 * r660131;
        double r660133 = r660132 - r660120;
        double r660134 = 1.782934444176898e+140;
        bool r660135 = r660127 <= r660134;
        double r660136 = r660122 / r660125;
        double r660137 = r660120 / r660136;
        double r660138 = r660135 ? r660127 : r660137;
        double r660139 = r660129 ? r660133 : r660138;
        return r660139;
}

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. Simplified4.5

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

    4. Taylor expanded around 0 4.5

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

    5. 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 +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))