Average Error: 7.4 → 2.8
Time: 7.2s
Precision: 64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.58006598525956814 \cdot 10^{-96} \lor \neg \left(z \le 4.069990541399454 \cdot 10^{-76}\right):\\ \;\;\;\;\frac{x + y}{1 \cdot \left(1 - \frac{y}{z}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{y}{\left(x + y\right) \cdot z}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.58006598525956814 \cdot 10^{-96} \lor \neg \left(z \le 4.069990541399454 \cdot 10^{-76}\right):\\
\;\;\;\;\frac{x + y}{1 \cdot \left(1 - \frac{y}{z}\right)}\\

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

\end{array}
double f(double x, double y, double z) {
        double r584385 = x;
        double r584386 = y;
        double r584387 = r584385 + r584386;
        double r584388 = 1.0;
        double r584389 = z;
        double r584390 = r584386 / r584389;
        double r584391 = r584388 - r584390;
        double r584392 = r584387 / r584391;
        return r584392;
}
↓
double f(double x, double y, double z) {
        double r584393 = z;
        double r584394 = -2.580065985259568e-96;
        bool r584395 = r584393 <= r584394;
        double r584396 = 4.069990541399454e-76;
        bool r584397 = r584393 <= r584396;
        double r584398 = !r584397;
        bool r584399 = r584395 || r584398;
        double r584400 = x;
        double r584401 = y;
        double r584402 = r584400 + r584401;
        double r584403 = 1.0;
        double r584404 = 1.0;
        double r584405 = r584401 / r584393;
        double r584406 = r584404 - r584405;
        double r584407 = r584403 * r584406;
        double r584408 = r584402 / r584407;
        double r584409 = r584404 / r584402;
        double r584410 = r584402 * r584393;
        double r584411 = r584401 / r584410;
        double r584412 = r584409 - r584411;
        double r584413 = r584403 / r584412;
        double r584414 = r584399 ? r584408 : r584413;
        return r584414;
}

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

Original7.4
Target3.9
Herbie2.8
\[\begin{array}{l} \mathbf{if}\;y \lt -3.74293107626898565 \cdot 10^{171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.55346624560867344 \cdot 10^{168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.580065985259568e-96 or 4.069990541399454e-76 < z

    1. Initial program 0.9

      \[\frac{x + y}{1 - \frac{y}{z}}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity0.9

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

    if -2.580065985259568e-96 < z < 4.069990541399454e-76

    1. Initial program 18.7

      \[\frac{x + y}{1 - \frac{y}{z}}\]
    2. Using strategy rm

    3. Applied clear-num18.8

      \[\leadsto \color{blue}{\frac{1}{\frac{1 - \frac{y}{z}}{x + y}}}\]
    4. Using strategy rm

    5. Applied div-sub18.8

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

    7. Applied div-inv18.9

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

    8. Applied associate-/l*6.1

      \[\leadsto \frac{1}{\frac{1}{x + y} - \color{blue}{\frac{y}{\frac{x + y}{\frac{1}{z}}}}}\]

    9. Simplified6.1

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

  4. Final simplification2.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.58006598525956814 \cdot 10^{-96} \lor \neg \left(z \le 4.069990541399454 \cdot 10^{-76}\right):\\ \;\;\;\;\frac{x + y}{1 \cdot \left(1 - \frac{y}{z}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{y}{\left(x + y\right) \cdot z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :herbie-target
  (if (< y -3.7429310762689856e+171) (* (/ (+ y x) (- y)) z) (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1 (/ y z))) (* (/ (+ y x) (- y)) z)))

  (/ (+ x y) (- 1 (/ y z))))