Average Error: 10.4 → 0.3
Time: 2.6s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.2318558186692977 \cdot 10^{-61} \lor \neg \left(z \le 3.6265165215917426 \cdot 10^{-86}\right):\\ \;\;\;\;\left(\frac{x}{z} + y\right) - x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array}\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.2318558186692977 \cdot 10^{-61} \lor \neg \left(z \le 3.6265165215917426 \cdot 10^{-86}\right):\\
\;\;\;\;\left(\frac{x}{z} + y\right) - x \cdot \frac{y}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r863118 = x;
        double r863119 = y;
        double r863120 = z;
        double r863121 = r863120 - r863118;
        double r863122 = r863119 * r863121;
        double r863123 = r863118 + r863122;
        double r863124 = r863123 / r863120;
        return r863124;
}
↓
double f(double x, double y, double z) {
        double r863125 = z;
        double r863126 = -1.2318558186692977e-61;
        bool r863127 = r863125 <= r863126;
        double r863128 = 3.6265165215917426e-86;
        bool r863129 = r863125 <= r863128;
        double r863130 = !r863129;
        bool r863131 = r863127 || r863130;
        double r863132 = x;
        double r863133 = r863132 / r863125;
        double r863134 = y;
        double r863135 = r863133 + r863134;
        double r863136 = r863134 / r863125;
        double r863137 = r863132 * r863136;
        double r863138 = r863135 - r863137;
        double r863139 = r863125 - r863132;
        double r863140 = r863134 * r863139;
        double r863141 = r863132 + r863140;
        double r863142 = r863141 / r863125;
        double r863143 = r863131 ? r863138 : r863142;
        return r863143;
}

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

Original10.4
Target0.0
Herbie0.3
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.2318558186692977e-61 or 3.6265165215917426e-86 < z

    1. Initial program 14.2

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

    2. Taylor expanded around 0 4.8

      \[\leadsto \color{blue}{\left(\frac{x}{z} + y\right) - \frac{x \cdot y}{z}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity4.8

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

    5. Applied times-frac0.3

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

    6. Simplified0.3

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

    if -1.2318558186692977e-61 < z < 3.6265165215917426e-86

    1. Initial program 0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.2318558186692977 \cdot 10^{-61} \lor \neg \left(z \le 3.6265165215917426 \cdot 10^{-86}\right):\\ \;\;\;\;\left(\frac{x}{z} + y\right) - x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

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

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