Average Error: 7.8 → 6.7
Time: 18.9s
Precision: 64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -7.693280280124389469819150824952027040683 \cdot 10^{-271} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + y}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -7.693280280124389469819150824952027040683 \cdot 10^{-271} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\
\;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\

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

\end{array}
double f(double x, double y, double z) {
        double r357929 = x;
        double r357930 = y;
        double r357931 = r357929 + r357930;
        double r357932 = 1.0;
        double r357933 = z;
        double r357934 = r357930 / r357933;
        double r357935 = r357932 - r357934;
        double r357936 = r357931 / r357935;
        return r357936;
}
↓
double f(double x, double y, double z) {
        double r357937 = x;
        double r357938 = y;
        double r357939 = r357937 + r357938;
        double r357940 = 1.0;
        double r357941 = z;
        double r357942 = r357938 / r357941;
        double r357943 = r357940 - r357942;
        double r357944 = r357939 / r357943;
        double r357945 = -7.693280280124389e-271;
        bool r357946 = r357944 <= r357945;
        double r357947 = -0.0;
        bool r357948 = r357944 <= r357947;
        double r357949 = !r357948;
        bool r357950 = r357946 || r357949;
        double r357951 = sqrt(r357940);
        double r357952 = sqrt(r357938);
        double r357953 = sqrt(r357941);
        double r357954 = r357952 / r357953;
        double r357955 = r357951 + r357954;
        double r357956 = r357939 / r357955;
        double r357957 = r357951 - r357954;
        double r357958 = r357956 / r357957;
        double r357959 = r357950 ? r357944 : r357958;
        return r357959;
}

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

Original7.8
Target4.0
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;y \lt -3.742931076268985646434612946949172132145 \cdot 10^{171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.553466245608673435460441960303815115662 \cdot 10^{168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ x y) (- 1.0 (/ y z))) < -7.693280280124389e-271 or -0.0 < (/ (+ x y) (- 1.0 (/ y z)))

    1. Initial program 4.3

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

    if -7.693280280124389e-271 < (/ (+ x y) (- 1.0 (/ y z))) < -0.0

    1. Initial program 54.4

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

    3. Applied add-sqr-sqrt55.5

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

    4. Applied add-sqr-sqrt59.7

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

    5. Applied times-frac59.7

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

    6. Applied add-sqr-sqrt59.7

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

    7. Applied difference-of-squares59.7

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

    8. Applied associate-/r*37.1

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

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -7.693280280124389469819150824952027040683 \cdot 10^{-271} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + y}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +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))))