Average Error: 7.7 → 6.4
Time: 16.1s
Precision: 64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \le -4.61291755031202367093527746970482296077 \cdot 10^{-270}:\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{y + x}{1 - \frac{y}{z}} \le 0.0:\\ \;\;\;\;\frac{1}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \left(\left(y + x\right) \cdot \frac{1}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \le -4.61291755031202367093527746970482296077 \cdot 10^{-270}:\\
\;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\

\mathbf{elif}\;\frac{y + x}{1 - \frac{y}{z}} \le 0.0:\\
\;\;\;\;\frac{1}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \left(\left(y + x\right) \cdot \frac{1}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\

\end{array}
double f(double x, double y, double z) {
        double r29418859 = x;
        double r29418860 = y;
        double r29418861 = r29418859 + r29418860;
        double r29418862 = 1.0;
        double r29418863 = z;
        double r29418864 = r29418860 / r29418863;
        double r29418865 = r29418862 - r29418864;
        double r29418866 = r29418861 / r29418865;
        return r29418866;
}
↓
double f(double x, double y, double z) {
        double r29418867 = y;
        double r29418868 = x;
        double r29418869 = r29418867 + r29418868;
        double r29418870 = 1.0;
        double r29418871 = z;
        double r29418872 = r29418867 / r29418871;
        double r29418873 = r29418870 - r29418872;
        double r29418874 = r29418869 / r29418873;
        double r29418875 = -4.612917550312024e-270;
        bool r29418876 = r29418874 <= r29418875;
        double r29418877 = 0.0;
        bool r29418878 = r29418874 <= r29418877;
        double r29418879 = 1.0;
        double r29418880 = sqrt(r29418870);
        double r29418881 = sqrt(r29418867);
        double r29418882 = sqrt(r29418871);
        double r29418883 = r29418881 / r29418882;
        double r29418884 = r29418880 + r29418883;
        double r29418885 = r29418879 / r29418884;
        double r29418886 = r29418880 - r29418883;
        double r29418887 = r29418879 / r29418886;
        double r29418888 = r29418869 * r29418887;
        double r29418889 = r29418885 * r29418888;
        double r29418890 = r29418878 ? r29418889 : r29418874;
        double r29418891 = r29418876 ? r29418874 : r29418890;
        return r29418891;
}

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

Original7.7
Target4.2
Herbie6.4
\[\begin{array}{l} \mathbf{if}\;y \lt -3.742931076268985646434612946949172132145 \cdot 10^{171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.553466245608673435460441960303815115662 \cdot 10^{168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ x y) (- 1.0 (/ y z))) < -4.612917550312024e-270 or 0.0 < (/ (+ x y) (- 1.0 (/ y z)))

    1. Initial program 0.1

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

    if -4.612917550312024e-270 < (/ (+ x y) (- 1.0 (/ y z))) < 0.0

    1. Initial program 57.1

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

    3. Applied clear-num57.1

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

    5. Applied div-inv57.1

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

    6. Applied associate-/r*57.1

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

    8. Applied *-un-lft-identity57.1

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

    9. Applied add-cube-cbrt57.1

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

    10. Applied times-frac57.1

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

    11. Applied add-sqr-sqrt59.5

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

    12. Applied add-sqr-sqrt61.9

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

    13. Applied times-frac61.9

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

    14. Applied add-sqr-sqrt61.9

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

    15. Applied difference-of-squares61.9

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

    16. Applied add-cube-cbrt61.9

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

    17. Applied times-frac61.2

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

    18. Applied times-frac47.7

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

    19. Simplified47.7

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

    20. Simplified47.7

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \le -4.61291755031202367093527746970482296077 \cdot 10^{-270}:\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{y + x}{1 - \frac{y}{z}} \le 0.0:\\ \;\;\;\;\frac{1}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \left(\left(y + x\right) \cdot \frac{1}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(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))))