Average Error: 10.2 → 0.1
Time: 10.0s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.250199817382656908466842526905793420156 \cdot 10^{-4}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{z - 1}{z}\right)\\ \mathbf{elif}\;z \le 2.343353023250889138543349043669121873388 \cdot 10^{-16}:\\ \;\;\;\;\left(y - \left(z - 1\right)\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{z - 1}{z}\right)\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.250199817382656908466842526905793420156 \cdot 10^{-4}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} - \frac{z - 1}{z}\right)\\

\mathbf{elif}\;z \le 2.343353023250889138543349043669121873388 \cdot 10^{-16}:\\
\;\;\;\;\left(y - \left(z - 1\right)\right) \cdot \frac{x}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r31125106 = x;
        double r31125107 = y;
        double r31125108 = z;
        double r31125109 = r31125107 - r31125108;
        double r31125110 = 1.0;
        double r31125111 = r31125109 + r31125110;
        double r31125112 = r31125106 * r31125111;
        double r31125113 = r31125112 / r31125108;
        return r31125113;
}
↓
double f(double x, double y, double z) {
        double r31125114 = z;
        double r31125115 = -0.0001250199817382657;
        bool r31125116 = r31125114 <= r31125115;
        double r31125117 = x;
        double r31125118 = y;
        double r31125119 = r31125118 / r31125114;
        double r31125120 = 1.0;
        double r31125121 = r31125114 - r31125120;
        double r31125122 = r31125121 / r31125114;
        double r31125123 = r31125119 - r31125122;
        double r31125124 = r31125117 * r31125123;
        double r31125125 = 2.343353023250889e-16;
        bool r31125126 = r31125114 <= r31125125;
        double r31125127 = r31125118 - r31125121;
        double r31125128 = r31125117 / r31125114;
        double r31125129 = r31125127 * r31125128;
        double r31125130 = r31125126 ? r31125129 : r31125124;
        double r31125131 = r31125116 ? r31125124 : r31125130;
        return r31125131;
}

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

Derivation

  1. Split input into 2 regimes

  2. if z < -0.0001250199817382657 or 2.343353023250889e-16 < z

    1. Initial program 16.3

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

    3. Applied *-un-lft-identity16.3

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

    4. Applied times-frac0.1

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

    5. Simplified0.1

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

    7. Applied associate-+l-0.1

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

    8. Applied div-sub0.1

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

    if -0.0001250199817382657 < z < 2.343353023250889e-16

    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-frac8.8

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

    5. Simplified8.8

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

    7. Applied associate-+l-8.8

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

    8. Applied div-sub8.8

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

    10. Applied div-inv8.8

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

    11. Applied div-inv8.8

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

    12. Applied distribute-rgt-out--8.8

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

    13. Applied associate-*r*0.3

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

    14. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.250199817382656908466842526905793420156 \cdot 10^{-4}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{z - 1}{z}\right)\\ \mathbf{elif}\;z \le 2.343353023250889138543349043669121873388 \cdot 10^{-16}:\\ \;\;\;\;\left(y - \left(z - 1\right)\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{z - 1}{z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(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))