Average Error: 9.8 → 0.3
Time: 9.5s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1.0\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7.814855256918435 \cdot 10^{-142}:\\ \;\;\;\;\left(1.0 \cdot \frac{x}{z} + \frac{x}{z} \cdot y\right) - x\\ \mathbf{elif}\;x \le 4.0170515075320604 \cdot 10^{-115}:\\ \;\;\;\;\frac{1}{z} \cdot \left(x \cdot \left(\left(y - z\right) + 1.0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1.0 \cdot \frac{x}{z} + \frac{x}{z} \cdot y\right) - x\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1.0\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -7.814855256918435 \cdot 10^{-142}:\\
\;\;\;\;\left(1.0 \cdot \frac{x}{z} + \frac{x}{z} \cdot y\right) - x\\

\mathbf{elif}\;x \le 4.0170515075320604 \cdot 10^{-115}:\\
\;\;\;\;\frac{1}{z} \cdot \left(x \cdot \left(\left(y - z\right) + 1.0\right)\right)\\

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

\end{array}
double f(double x, double y, double z) {
        double r34545958 = x;
        double r34545959 = y;
        double r34545960 = z;
        double r34545961 = r34545959 - r34545960;
        double r34545962 = 1.0;
        double r34545963 = r34545961 + r34545962;
        double r34545964 = r34545958 * r34545963;
        double r34545965 = r34545964 / r34545960;
        return r34545965;
}
↓
double f(double x, double y, double z) {
        double r34545966 = x;
        double r34545967 = -7.814855256918435e-142;
        bool r34545968 = r34545966 <= r34545967;
        double r34545969 = 1.0;
        double r34545970 = z;
        double r34545971 = r34545966 / r34545970;
        double r34545972 = r34545969 * r34545971;
        double r34545973 = y;
        double r34545974 = r34545971 * r34545973;
        double r34545975 = r34545972 + r34545974;
        double r34545976 = r34545975 - r34545966;
        double r34545977 = 4.0170515075320604e-115;
        bool r34545978 = r34545966 <= r34545977;
        double r34545979 = 1.0;
        double r34545980 = r34545979 / r34545970;
        double r34545981 = r34545973 - r34545970;
        double r34545982 = r34545981 + r34545969;
        double r34545983 = r34545966 * r34545982;
        double r34545984 = r34545980 * r34545983;
        double r34545985 = r34545978 ? r34545984 : r34545976;
        double r34545986 = r34545968 ? r34545976 : r34545985;
        return r34545986;
}

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
Target0.4
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1.0\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -7.814855256918435e-142 or 4.0170515075320604e-115 < x

    1. Initial program 15.4

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

    3. Applied associate-/l*1.0

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

    4. Taylor expanded around 0 5.3

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

    5. Simplified0.2

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

    if -7.814855256918435e-142 < x < 4.0170515075320604e-115

    1. Initial program 0.2

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

    3. Applied associate-/l*6.6

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1.0}}}\]
    4. Using strategy rm

    5. Applied div-inv6.7

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

    6. Applied *-un-lft-identity6.7

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

    7. Applied times-frac0.3

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

    8. Simplified0.3

      \[\leadsto \frac{1}{z} \cdot \color{blue}{\left(\left(\left(y - z\right) + 1.0\right) \cdot x\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.814855256918435 \cdot 10^{-142}:\\ \;\;\;\;\left(1.0 \cdot \frac{x}{z} + \frac{x}{z} \cdot y\right) - x\\ \mathbf{elif}\;x \le 4.0170515075320604 \cdot 10^{-115}:\\ \;\;\;\;\frac{1}{z} \cdot \left(x \cdot \left(\left(y - z\right) + 1.0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1.0 \cdot \frac{x}{z} + \frac{x}{z} \cdot y\right) - x\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(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))