Average Error: 9.9 → 0.2
Time: 10.4s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -13905844898214332094159488174298157285380 \lor \neg \left(x \le 9.371800790310021716388672147044997948808 \cdot 10^{-61}\right):\\ \;\;\;\;\left(\frac{1 \cdot x}{z} - x\right) + \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \left(x \cdot y\right) + \left(\frac{1 \cdot x}{z} - x\right)\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -13905844898214332094159488174298157285380 \lor \neg \left(x \le 9.371800790310021716388672147044997948808 \cdot 10^{-61}\right):\\
\;\;\;\;\left(\frac{1 \cdot x}{z} - x\right) + \frac{y}{\frac{z}{x}}\\

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

\end{array}
double f(double x, double y, double z) {
        double r611951 = x;
        double r611952 = y;
        double r611953 = z;
        double r611954 = r611952 - r611953;
        double r611955 = 1.0;
        double r611956 = r611954 + r611955;
        double r611957 = r611951 * r611956;
        double r611958 = r611957 / r611953;
        return r611958;
}
↓
double f(double x, double y, double z) {
        double r611959 = x;
        double r611960 = -1.3905844898214332e+40;
        bool r611961 = r611959 <= r611960;
        double r611962 = 9.371800790310022e-61;
        bool r611963 = r611959 <= r611962;
        double r611964 = !r611963;
        bool r611965 = r611961 || r611964;
        double r611966 = 1.0;
        double r611967 = r611966 * r611959;
        double r611968 = z;
        double r611969 = r611967 / r611968;
        double r611970 = r611969 - r611959;
        double r611971 = y;
        double r611972 = r611968 / r611959;
        double r611973 = r611971 / r611972;
        double r611974 = r611970 + r611973;
        double r611975 = 1.0;
        double r611976 = r611975 / r611968;
        double r611977 = r611959 * r611971;
        double r611978 = r611976 * r611977;
        double r611979 = r611978 + r611970;
        double r611980 = r611965 ? r611974 : r611979;
        return r611980;
}

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

Original9.9
Target0.5
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt -2.714831067134359919650240696134672137284 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.874108816439546156869494499878029491333 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.3905844898214332e+40 or 9.371800790310022e-61 < x

    1. Initial program 23.3

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

    2. Simplified0.4

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

    3. Taylor expanded around 0 7.9

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

    4. Simplified0.1

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

    if -1.3905844898214332e+40 < x < 9.371800790310022e-61

    1. Initial program 0.3

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

    2. Simplified14.8

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

    3. Taylor expanded around 0 0.2

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

    4. Simplified3.1

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

    6. Applied div-inv3.1

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

    7. Applied *-un-lft-identity3.1

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

    8. Applied times-frac0.3

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

    9. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -13905844898214332094159488174298157285380 \lor \neg \left(x \le 9.371800790310021716388672147044997948808 \cdot 10^{-61}\right):\\ \;\;\;\;\left(\frac{1 \cdot x}{z} - x\right) + \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \left(x \cdot y\right) + \left(\frac{1 \cdot x}{z} - x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"

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

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