Average Error: 7.7 → 6.1
Time: 20.1s
Precision: 64
\[\frac{x + y}{1.0 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y + x}{1.0 - \frac{y}{z}} \le -6.619731314741027 \cdot 10^{-303}:\\ \;\;\;\;\frac{y + x}{1.0 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{y + x}{1.0 - \frac{y}{z}} \le -0.0:\\ \;\;\;\;\frac{y + x}{\sqrt{1.0} - \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{1}{\sqrt{1.0} + \frac{\sqrt{y}}{\sqrt{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{1.0 - \frac{y}{z}}\\ \end{array}\]
\frac{x + y}{1.0 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y + x}{1.0 - \frac{y}{z}} \le -6.619731314741027 \cdot 10^{-303}:\\
\;\;\;\;\frac{y + x}{1.0 - \frac{y}{z}}\\

\mathbf{elif}\;\frac{y + x}{1.0 - \frac{y}{z}} \le -0.0:\\
\;\;\;\;\frac{y + x}{\sqrt{1.0} - \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{1}{\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 r27338675 = x;
        double r27338676 = y;
        double r27338677 = r27338675 + r27338676;
        double r27338678 = 1.0;
        double r27338679 = z;
        double r27338680 = r27338676 / r27338679;
        double r27338681 = r27338678 - r27338680;
        double r27338682 = r27338677 / r27338681;
        return r27338682;
}
↓
double f(double x, double y, double z) {
        double r27338683 = y;
        double r27338684 = x;
        double r27338685 = r27338683 + r27338684;
        double r27338686 = 1.0;
        double r27338687 = z;
        double r27338688 = r27338683 / r27338687;
        double r27338689 = r27338686 - r27338688;
        double r27338690 = r27338685 / r27338689;
        double r27338691 = -6.619731314741027e-303;
        bool r27338692 = r27338690 <= r27338691;
        double r27338693 = -0.0;
        bool r27338694 = r27338690 <= r27338693;
        double r27338695 = sqrt(r27338686);
        double r27338696 = sqrt(r27338683);
        double r27338697 = sqrt(r27338687);
        double r27338698 = r27338696 / r27338697;
        double r27338699 = r27338695 - r27338698;
        double r27338700 = r27338685 / r27338699;
        double r27338701 = 1.0;
        double r27338702 = r27338695 + r27338698;
        double r27338703 = r27338701 / r27338702;
        double r27338704 = r27338700 * r27338703;
        double r27338705 = r27338694 ? r27338704 : r27338690;
        double r27338706 = r27338692 ? r27338690 : r27338705;
        return r27338706;
}

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
Target4.0
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))) < -6.619731314741027e-303 or -0.0 < (/ (+ x y) (- 1.0 (/ y z)))

    1. Initial program 0.1

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

    if -6.619731314741027e-303 < (/ (+ x y) (- 1.0 (/ y z))) < -0.0

    1. Initial program 59.7

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

    3. Applied add-sqr-sqrt61.0

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

    4. Applied add-sqr-sqrt61.9

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

    5. Applied times-frac61.9

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

    6. Applied add-sqr-sqrt61.9

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

    7. Applied difference-of-squares61.9

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

    8. Applied *-un-lft-identity61.9

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

    9. Applied times-frac47.0

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1.0 - \frac{y}{z}} \le -6.619731314741027 \cdot 10^{-303}:\\ \;\;\;\;\frac{y + x}{1.0 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{y + x}{1.0 - \frac{y}{z}} \le -0.0:\\ \;\;\;\;\frac{y + x}{\sqrt{1.0} - \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{1}{\sqrt{1.0} + \frac{\sqrt{y}}{\sqrt{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{1.0 - \frac{y}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(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))))