Average Error: 12.0 → 2.1
Time: 15.0s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.089911330877771451248451588593061755111 \cdot 10^{-129} \lor \neg \left(z \le -1.26295748591940369208875411960667007374 \cdot 10^{-259}\right) \land \left(z \le 9.920339495877500547742673232912797278818 \cdot 10^{-231} \lor \neg \left(z \le 1.217132990787068613477600742069466164471 \cdot 10^{-83}\right)\right):\\ \;\;\;\;x + \frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{z} \cdot \left(x \cdot y\right)\\ \end{array}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.089911330877771451248451588593061755111 \cdot 10^{-129} \lor \neg \left(z \le -1.26295748591940369208875411960667007374 \cdot 10^{-259}\right) \land \left(z \le 9.920339495877500547742673232912797278818 \cdot 10^{-231} \lor \neg \left(z \le 1.217132990787068613477600742069466164471 \cdot 10^{-83}\right)\right):\\
\;\;\;\;x + \frac{x}{\frac{z}{y}}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{1}{z} \cdot \left(x \cdot y\right)\\

\end{array}
double f(double x, double y, double z) {
        double r413444 = x;
        double r413445 = y;
        double r413446 = z;
        double r413447 = r413445 + r413446;
        double r413448 = r413444 * r413447;
        double r413449 = r413448 / r413446;
        return r413449;
}
↓
double f(double x, double y, double z) {
        double r413450 = z;
        double r413451 = -2.0899113308777715e-129;
        bool r413452 = r413450 <= r413451;
        double r413453 = -1.2629574859194037e-259;
        bool r413454 = r413450 <= r413453;
        double r413455 = !r413454;
        double r413456 = 9.9203394958775e-231;
        bool r413457 = r413450 <= r413456;
        double r413458 = 1.2171329907870686e-83;
        bool r413459 = r413450 <= r413458;
        double r413460 = !r413459;
        bool r413461 = r413457 || r413460;
        bool r413462 = r413455 && r413461;
        bool r413463 = r413452 || r413462;
        double r413464 = x;
        double r413465 = y;
        double r413466 = r413450 / r413465;
        double r413467 = r413464 / r413466;
        double r413468 = r413464 + r413467;
        double r413469 = 1.0;
        double r413470 = r413469 / r413450;
        double r413471 = r413464 * r413465;
        double r413472 = r413470 * r413471;
        double r413473 = r413464 + r413472;
        double r413474 = r413463 ? r413468 : r413473;
        return r413474;
}

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

Original12.0
Target2.9
Herbie2.1
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.0899113308777715e-129 or -1.2629574859194037e-259 < z < 9.9203394958775e-231 or 1.2171329907870686e-83 < z

    1. Initial program 12.9

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

    2. Simplified3.6

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

    4. Applied fma-udef3.6

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

    5. Simplified1.6

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

    if -2.0899113308777715e-129 < z < -1.2629574859194037e-259 or 9.9203394958775e-231 < z < 1.2171329907870686e-83

    1. Initial program 8.1

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

    2. Simplified10.3

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

    4. Applied fma-udef10.3

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

    5. Simplified8.3

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

    7. Applied div-inv9.1

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

    8. Simplified9.1

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

    10. Applied div-inv9.1

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

    11. Applied associate-*r*3.9

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

    12. Simplified3.9

      \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{1}{z} + x\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.089911330877771451248451588593061755111 \cdot 10^{-129} \lor \neg \left(z \le -1.26295748591940369208875411960667007374 \cdot 10^{-259}\right) \land \left(z \le 9.920339495877500547742673232912797278818 \cdot 10^{-231} \lor \neg \left(z \le 1.217132990787068613477600742069466164471 \cdot 10^{-83}\right)\right):\\ \;\;\;\;x + \frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{z} \cdot \left(x \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +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))