Average Error: 12.7 → 2.0
Time: 42.7s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.047040197507521911880102455231371349848 \cdot 10^{123}:\\ \;\;\;\;x + \frac{\frac{\frac{x}{\sqrt[3]{z}}}{\sqrt[3]{z}}}{\frac{\sqrt[3]{z}}{y}}\\ \mathbf{elif}\;y \le 3.113004372037381436683458387632006392209 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -2.047040197507521911880102455231371349848 \cdot 10^{123}:\\
\;\;\;\;x + \frac{\frac{\frac{x}{\sqrt[3]{z}}}{\sqrt[3]{z}}}{\frac{\sqrt[3]{z}}{y}}\\

\mathbf{elif}\;y \le 3.113004372037381436683458387632006392209 \cdot 10^{-7}:\\
\;\;\;\;\frac{x}{\frac{z}{y}} + x\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\

\end{array}
double f(double x, double y, double z) {
        double r342022 = x;
        double r342023 = y;
        double r342024 = z;
        double r342025 = r342023 + r342024;
        double r342026 = r342022 * r342025;
        double r342027 = r342026 / r342024;
        return r342027;
}
↓
double f(double x, double y, double z) {
        double r342028 = y;
        double r342029 = -2.047040197507522e+123;
        bool r342030 = r342028 <= r342029;
        double r342031 = x;
        double r342032 = z;
        double r342033 = cbrt(r342032);
        double r342034 = r342031 / r342033;
        double r342035 = r342034 / r342033;
        double r342036 = r342033 / r342028;
        double r342037 = r342035 / r342036;
        double r342038 = r342031 + r342037;
        double r342039 = 3.1130043720373814e-07;
        bool r342040 = r342028 <= r342039;
        double r342041 = r342032 / r342028;
        double r342042 = r342031 / r342041;
        double r342043 = r342042 + r342031;
        double r342044 = r342031 / r342032;
        double r342045 = fma(r342044, r342028, r342031);
        double r342046 = r342040 ? r342043 : r342045;
        double r342047 = r342030 ? r342038 : r342046;
        return r342047;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original12.7
Target3.0
Herbie2.0
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -2.047040197507522e+123

    1. Initial program 12.7

      \[\frac{x \cdot \left(y + z\right)}{z}\]

    2. Simplified6.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{z}, y, x\right)}\]
    3. Using strategy rm

    4. Applied fma-udef6.7

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

    5. Simplified10.7

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

    7. Applied *-un-lft-identity10.7

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

    8. Applied add-cube-cbrt11.3

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

    9. Applied times-frac11.3

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

    10. Applied associate-/r*7.8

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

    11. Simplified7.8

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

    if -2.047040197507522e+123 < y < 3.1130043720373814e-07

    1. Initial program 12.9

      \[\frac{x \cdot \left(y + z\right)}{z}\]

    2. Simplified5.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{z}, y, x\right)}\]
    3. Using strategy rm

    4. Applied fma-udef5.1

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

    5. Simplified0.4

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

    if 3.1130043720373814e-07 < y

    1. Initial program 11.9

      \[\frac{x \cdot \left(y + z\right)}{z}\]

    2. Simplified3.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{z}, y, x\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification2.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.047040197507521911880102455231371349848 \cdot 10^{123}:\\ \;\;\;\;x + \frac{\frac{\frac{x}{\sqrt[3]{z}}}{\sqrt[3]{z}}}{\frac{\sqrt[3]{z}}{y}}\\ \mathbf{elif}\;y \le 3.113004372037381436683458387632006392209 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"

  :herbie-target
  (/ x (/ z (+ y z)))

  (/ (* x (+ y z)) z))