Average Error: 12.8 → 3.6
Time: 2.1s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.35557924277947119 \cdot 10^{-194}:\\ \;\;\;\;\frac{x \cdot y}{z} + x\\ \mathbf{elif}\;z \le 3.6732880554028338 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{x}}{y}} + x\\ \mathbf{elif}\;z \le 9085.26537501298844:\\ \;\;\;\;\frac{x \cdot y}{z} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \end{array}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.35557924277947119 \cdot 10^{-194}:\\
\;\;\;\;\frac{x \cdot y}{z} + x\\

\mathbf{elif}\;z \le 3.6732880554028338 \cdot 10^{-308}:\\
\;\;\;\;\frac{1}{\frac{\frac{z}{x}}{y}} + x\\

\mathbf{elif}\;z \le 9085.26537501298844:\\
\;\;\;\;\frac{x \cdot y}{z} + x\\

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

\end{array}
double f(double x, double y, double z) {
        double r420479 = x;
        double r420480 = y;
        double r420481 = z;
        double r420482 = r420480 + r420481;
        double r420483 = r420479 * r420482;
        double r420484 = r420483 / r420481;
        return r420484;
}
↓
double f(double x, double y, double z) {
        double r420485 = z;
        double r420486 = -1.3555792427794712e-194;
        bool r420487 = r420485 <= r420486;
        double r420488 = x;
        double r420489 = y;
        double r420490 = r420488 * r420489;
        double r420491 = r420490 / r420485;
        double r420492 = r420491 + r420488;
        double r420493 = 3.673288055402834e-308;
        bool r420494 = r420485 <= r420493;
        double r420495 = 1.0;
        double r420496 = r420485 / r420488;
        double r420497 = r420496 / r420489;
        double r420498 = r420495 / r420497;
        double r420499 = r420498 + r420488;
        double r420500 = 9085.265375012988;
        bool r420501 = r420485 <= r420500;
        double r420502 = r420489 + r420485;
        double r420503 = r420485 / r420502;
        double r420504 = r420488 / r420503;
        double r420505 = r420501 ? r420492 : r420504;
        double r420506 = r420494 ? r420499 : r420505;
        double r420507 = r420487 ? r420492 : r420506;
        return r420507;
}

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

Original12.8
Target2.8
Herbie3.6
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.3555792427794712e-194 or 3.673288055402834e-308 < z < 9085.265375012988

    1. Initial program 11.0

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

    2. Taylor expanded around 0 4.3

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

    if -1.3555792427794712e-194 < z < 3.673288055402834e-308

    1. Initial program 13.1

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

    2. Taylor expanded around 0 9.2

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

    4. Applied clear-num9.3

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

    6. Applied associate-/r*11.8

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

    if 9085.265375012988 < z

    1. Initial program 16.9

      \[\frac{x \cdot \left(y + z\right)}{z}\]
    2. Using strategy rm

    3. Applied associate-/l*0.1

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

  4. Final simplification3.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.35557924277947119 \cdot 10^{-194}:\\ \;\;\;\;\frac{x \cdot y}{z} + x\\ \mathbf{elif}\;z \le 3.6732880554028338 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{x}}{y}} + x\\ \mathbf{elif}\;z \le 9085.26537501298844:\\ \;\;\;\;\frac{x \cdot y}{z} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"
  :precision binary64

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

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