Average Error: 14.6 → 2.9
Time: 2.7s
Precision: 64
\[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -1.68209557109898762 \cdot 10^{256} \lor \neg \left(\frac{y}{z} \le -8.7578361305297263 \cdot 10^{-224}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]
x \cdot \frac{\frac{y}{z} \cdot t}{t}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y}{z} \le -1.68209557109898762 \cdot 10^{256} \lor \neg \left(\frac{y}{z} \le -8.7578361305297263 \cdot 10^{-224}\right):\\
\;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r113099 = x;
        double r113100 = y;
        double r113101 = z;
        double r113102 = r113100 / r113101;
        double r113103 = t;
        double r113104 = r113102 * r113103;
        double r113105 = r113104 / r113103;
        double r113106 = r113099 * r113105;
        return r113106;
}
↓
double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r113107 = y;
        double r113108 = z;
        double r113109 = r113107 / r113108;
        double r113110 = -1.6820955710989876e+256;
        bool r113111 = r113109 <= r113110;
        double r113112 = -8.757836130529726e-224;
        bool r113113 = r113109 <= r113112;
        double r113114 = !r113113;
        bool r113115 = r113111 || r113114;
        double r113116 = x;
        double r113117 = r113116 * r113107;
        double r113118 = 1.0;
        double r113119 = r113118 / r113108;
        double r113120 = r113117 * r113119;
        double r113121 = r113108 / r113107;
        double r113122 = r113116 / r113121;
        double r113123 = r113115 ? r113120 : r113122;
        return r113123;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (/ y z) < -1.6820955710989876e+256 or -8.757836130529726e-224 < (/ y z)

    1. Initial program 17.3

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

    2. Simplified9.1

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

    4. Applied div-inv9.2

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

    5. Applied associate-*r*4.3

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

    if -1.6820955710989876e+256 < (/ y z) < -8.757836130529726e-224

    1. Initial program 9.3

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

    2. Simplified0.2

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

    4. Applied associate-*r/8.9

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

    6. Applied associate-/l*0.2

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

  4. Final simplification2.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -1.68209557109898762 \cdot 10^{256} \lor \neg \left(\frac{y}{z} \le -8.7578361305297263 \cdot 10^{-224}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1"
  :precision binary64
  (* x (/ (* (/ y z) t) t)))