Average Error: 7.6 → 6.1
Time: 14.2s
Precision: 64
\[\frac{x + y}{1.0 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y + x}{1.0 - \frac{y}{z}} \le -5.015331337804172 \cdot 10^{-270}:\\ \;\;\;\;\frac{y + x}{1.0 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{y + x}{1.0 - \frac{y}{z}} \le -0.0:\\ \;\;\;\;\frac{1}{\frac{\sqrt{y}}{\sqrt{z}} + \sqrt{1.0}} \cdot \frac{y + x}{\sqrt{1.0} - \frac{\sqrt{y}}{\sqrt{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{1.0 - \frac{y}{z}}\\ \end{array}\]
\frac{x + y}{1.0 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y + x}{1.0 - \frac{y}{z}} \le -5.015331337804172 \cdot 10^{-270}:\\
\;\;\;\;\frac{y + x}{1.0 - \frac{y}{z}}\\

\mathbf{elif}\;\frac{y + x}{1.0 - \frac{y}{z}} \le -0.0:\\
\;\;\;\;\frac{1}{\frac{\sqrt{y}}{\sqrt{z}} + \sqrt{1.0}} \cdot \frac{y + x}{\sqrt{1.0} - \frac{\sqrt{y}}{\sqrt{z}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{y + x}{1.0 - \frac{y}{z}}\\

\end{array}
double f(double x, double y, double z) {
        double r42160973 = x;
        double r42160974 = y;
        double r42160975 = r42160973 + r42160974;
        double r42160976 = 1.0;
        double r42160977 = z;
        double r42160978 = r42160974 / r42160977;
        double r42160979 = r42160976 - r42160978;
        double r42160980 = r42160975 / r42160979;
        return r42160980;
}
↓
double f(double x, double y, double z) {
        double r42160981 = y;
        double r42160982 = x;
        double r42160983 = r42160981 + r42160982;
        double r42160984 = 1.0;
        double r42160985 = z;
        double r42160986 = r42160981 / r42160985;
        double r42160987 = r42160984 - r42160986;
        double r42160988 = r42160983 / r42160987;
        double r42160989 = -5.015331337804172e-270;
        bool r42160990 = r42160988 <= r42160989;
        double r42160991 = -0.0;
        bool r42160992 = r42160988 <= r42160991;
        double r42160993 = 1.0;
        double r42160994 = sqrt(r42160981);
        double r42160995 = sqrt(r42160985);
        double r42160996 = r42160994 / r42160995;
        double r42160997 = sqrt(r42160984);
        double r42160998 = r42160996 + r42160997;
        double r42160999 = r42160993 / r42160998;
        double r42161000 = r42160997 - r42160996;
        double r42161001 = r42160983 / r42161000;
        double r42161002 = r42160999 * r42161001;
        double r42161003 = r42160992 ? r42161002 : r42160988;
        double r42161004 = r42160990 ? r42160988 : r42161003;
        return r42161004;
}

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

Original7.6
Target4.2
Herbie6.1
\[\begin{array}{l} \mathbf{if}\;y \lt -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1.0 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ x y) (- 1.0 (/ y z))) < -5.015331337804172e-270 or -0.0 < (/ (+ x y) (- 1.0 (/ y z)))

    1. Initial program 0.1

      \[\frac{x + y}{1.0 - \frac{y}{z}}\]

    if -5.015331337804172e-270 < (/ (+ x y) (- 1.0 (/ y z))) < -0.0

    1. Initial program 56.8

      \[\frac{x + y}{1.0 - \frac{y}{z}}\]
    2. Using strategy rm

    3. Applied clear-num56.8

      \[\leadsto \color{blue}{\frac{1}{\frac{1.0 - \frac{y}{z}}{x + y}}}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity56.8

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

    6. Applied add-sqr-sqrt58.4

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

    7. Applied add-sqr-sqrt60.4

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

    8. Applied times-frac60.4

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

    9. Applied add-sqr-sqrt60.4

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

    10. Applied difference-of-squares60.4

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

    11. Applied times-frac45.7

      \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{1.0} + \frac{\sqrt{y}}{\sqrt{z}}}{1} \cdot \frac{\sqrt{1.0} - \frac{\sqrt{y}}{\sqrt{z}}}{x + y}}}\]

    12. Applied add-cube-cbrt45.7

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{\sqrt{1.0} + \frac{\sqrt{y}}{\sqrt{z}}}{1} \cdot \frac{\sqrt{1.0} - \frac{\sqrt{y}}{\sqrt{z}}}{x + y}}\]

    13. Applied times-frac45.8

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\sqrt{1.0} + \frac{\sqrt{y}}{\sqrt{z}}}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{\sqrt{1.0} - \frac{\sqrt{y}}{\sqrt{z}}}{x + y}}}\]

    14. Simplified45.8

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{y}}{\sqrt{z}} + \sqrt{1.0}}} \cdot \frac{\sqrt[3]{1}}{\frac{\sqrt{1.0} - \frac{\sqrt{y}}{\sqrt{z}}}{x + y}}\]

    15. Simplified45.7

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1.0 - \frac{y}{z}} \le -5.015331337804172 \cdot 10^{-270}:\\ \;\;\;\;\frac{y + x}{1.0 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{y + x}{1.0 - \frac{y}{z}} \le -0.0:\\ \;\;\;\;\frac{1}{\frac{\sqrt{y}}{\sqrt{z}} + \sqrt{1.0}} \cdot \frac{y + x}{\sqrt{1.0} - \frac{\sqrt{y}}{\sqrt{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{1.0 - \frac{y}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"

  :herbie-target
  (if (< y -3.7429310762689856e+171) (* (/ (+ y x) (- y)) z) (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1.0 (/ y z))) (* (/ (+ y x) (- y)) z)))

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