Average Error: 7.4 → 6.1
Time: 3.9s
Precision: 64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -2.7404041025427589 \cdot 10^{-299} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{x + y}}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}}{\frac{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}{\sqrt{x + y}}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -2.7404041025427589 \cdot 10^{-299} \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{\sqrt{x + y}}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}}{\frac{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}{\sqrt{x + y}}}\\

\end{array}
double f(double x, double y, double z) {
        double r527031 = x;
        double r527032 = y;
        double r527033 = r527031 + r527032;
        double r527034 = 1.0;
        double r527035 = z;
        double r527036 = r527032 / r527035;
        double r527037 = r527034 - r527036;
        double r527038 = r527033 / r527037;
        return r527038;
}
↓
double f(double x, double y, double z) {
        double r527039 = x;
        double r527040 = y;
        double r527041 = r527039 + r527040;
        double r527042 = 1.0;
        double r527043 = z;
        double r527044 = r527040 / r527043;
        double r527045 = r527042 - r527044;
        double r527046 = r527041 / r527045;
        double r527047 = -2.740404102542759e-299;
        bool r527048 = r527046 <= r527047;
        double r527049 = -0.0;
        bool r527050 = r527046 <= r527049;
        double r527051 = !r527050;
        bool r527052 = r527048 || r527051;
        double r527053 = sqrt(r527041);
        double r527054 = sqrt(r527042);
        double r527055 = sqrt(r527040);
        double r527056 = sqrt(r527043);
        double r527057 = r527055 / r527056;
        double r527058 = r527054 + r527057;
        double r527059 = r527053 / r527058;
        double r527060 = r527054 - r527057;
        double r527061 = r527060 / r527053;
        double r527062 = r527059 / r527061;
        double r527063 = r527052 ? r527046 : r527062;
        return r527063;
}

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

Original7.4
Target3.9
Herbie6.1
\[\begin{array}{l} \mathbf{if}\;y \lt -3.74293107626898565 \cdot 10^{171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.55346624560867344 \cdot 10^{168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ x y) (- 1.0 (/ y z))) < -2.740404102542759e-299 or -0.0 < (/ (+ x y) (- 1.0 (/ y z)))

    1. Initial program 4.1

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

    if -2.740404102542759e-299 < (/ (+ x y) (- 1.0 (/ y z))) < -0.0

    1. Initial program 59.5

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

    3. Applied clear-num59.5

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

    5. Applied add-sqr-sqrt61.8

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

    6. Applied add-sqr-sqrt61.9

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

    7. Applied add-sqr-sqrt61.9

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

    8. Applied times-frac61.9

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

    9. Applied add-sqr-sqrt61.9

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

    10. Applied difference-of-squares61.9

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

    11. Applied times-frac33.8

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

    12. Applied associate-/r*33.8

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

    13. Simplified33.7

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -2.7404041025427589 \cdot 10^{-299} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{x + y}}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}}{\frac{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}{\sqrt{x + y}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +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))))