Average Error: 7.8 → 6.7
Time: 18.7s
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 r461819 = x;
        double r461820 = y;
        double r461821 = r461819 + r461820;
        double r461822 = 1.0;
        double r461823 = z;
        double r461824 = r461820 / r461823;
        double r461825 = r461822 - r461824;
        double r461826 = r461821 / r461825;
        return r461826;
}
↓
double f(double x, double y, double z) {
        double r461827 = x;
        double r461828 = y;
        double r461829 = r461827 + r461828;
        double r461830 = 1.0;
        double r461831 = z;
        double r461832 = r461828 / r461831;
        double r461833 = r461830 - r461832;
        double r461834 = r461829 / r461833;
        double r461835 = -7.693280280124389e-271;
        bool r461836 = r461834 <= r461835;
        double r461837 = -0.0;
        bool r461838 = r461834 <= r461837;
        double r461839 = !r461838;
        bool r461840 = r461836 || r461839;
        double r461841 = sqrt(r461830);
        double r461842 = sqrt(r461828);
        double r461843 = sqrt(r461831);
        double r461844 = r461842 / r461843;
        double r461845 = r461841 + r461844;
        double r461846 = r461829 / r461845;
        double r461847 = r461841 - r461844;
        double r461848 = r461846 / r461847;
        double r461849 = r461840 ? r461834 : r461848;
        return r461849;
}

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 0.1

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

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

    1. Initial program 57.2

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

    3. Applied add-sqr-sqrt59.5

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

    4. Applied add-sqr-sqrt61.7

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

    5. Applied times-frac61.7

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

    6. Applied add-sqr-sqrt61.7

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

    7. Applied difference-of-squares61.7

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

    8. Applied associate-/r*49.0

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