Average Error: 14.8 → 1.7
Time: 11.4s
Precision: 64
\[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -2.0989985598366828 \cdot 10^{-236}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \le 4.4303806781760739 \cdot 10^{-235}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 6.4284572525223976 \cdot 10^{303}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot y}}\\ \end{array}\]
x \cdot \frac{\frac{y}{z} \cdot t}{t}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y}{z} \le -2.0989985598366828 \cdot 10^{-236}:\\
\;\;\;\;\frac{x}{\frac{z}{y}}\\

\mathbf{elif}\;\frac{y}{z} \le 4.4303806781760739 \cdot 10^{-235}:\\
\;\;\;\;\frac{x \cdot y}{z}\\

\mathbf{elif}\;\frac{y}{z} \le 6.4284572525223976 \cdot 10^{303}:\\
\;\;\;\;\frac{x}{\frac{z}{y}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{z}{x \cdot y}}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r116639 = x;
        double r116640 = y;
        double r116641 = z;
        double r116642 = r116640 / r116641;
        double r116643 = t;
        double r116644 = r116642 * r116643;
        double r116645 = r116644 / r116643;
        double r116646 = r116639 * r116645;
        return r116646;
}
↓
double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r116647 = y;
        double r116648 = z;
        double r116649 = r116647 / r116648;
        double r116650 = -2.098998559836683e-236;
        bool r116651 = r116649 <= r116650;
        double r116652 = x;
        double r116653 = r116648 / r116647;
        double r116654 = r116652 / r116653;
        double r116655 = 4.430380678176074e-235;
        bool r116656 = r116649 <= r116655;
        double r116657 = r116652 * r116647;
        double r116658 = r116657 / r116648;
        double r116659 = 6.428457252522398e+303;
        bool r116660 = r116649 <= r116659;
        double r116661 = 1.0;
        double r116662 = r116648 / r116657;
        double r116663 = r116661 / r116662;
        double r116664 = r116660 ? r116654 : r116663;
        double r116665 = r116656 ? r116658 : r116664;
        double r116666 = r116651 ? r116654 : r116665;
        return r116666;
}

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 3 regimes

  2. if (/ y z) < -2.098998559836683e-236 or 4.430380678176074e-235 < (/ y z) < 6.428457252522398e+303

    1. Initial program 11.8

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

    2. Simplified2.3

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

    4. Applied pow12.3

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

    5. Applied pow12.3

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

    6. Applied pow-prod-down2.3

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

    7. Simplified8.2

      \[\leadsto {\color{blue}{\left(\frac{x \cdot y}{z}\right)}}^{1}\]
    8. Using strategy rm

    9. Applied associate-/l*2.2

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

    if -2.098998559836683e-236 < (/ y z) < 4.430380678176074e-235

    1. Initial program 18.7

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

    2. Simplified12.1

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

    4. Applied pow112.1

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

    5. Applied pow112.1

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

    6. Applied pow-prod-down12.1

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

    7. Simplified0.4

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

    if 6.428457252522398e+303 < (/ y z)

    1. Initial program 64.0

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

    2. Simplified61.9

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

    4. Applied pow161.9

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

    5. Applied pow161.9

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

    6. Applied pow-prod-down61.9

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

    7. Simplified0.2

      \[\leadsto {\color{blue}{\left(\frac{x \cdot y}{z}\right)}}^{1}\]
    8. Using strategy rm

    9. Applied clear-num0.4

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

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -2.0989985598366828 \cdot 10^{-236}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \le 4.4303806781760739 \cdot 10^{-235}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 6.4284572525223976 \cdot 10^{303}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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)))