Average Error: 15.4 → 1.0
Time: 15.6s
Precision: 64
\[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -1.057433920064963 \cdot 10^{133}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot y}}\\ \mathbf{elif}\;\frac{y}{z} \le -1.0665637347194774 \cdot 10^{-221} \lor \neg \left(\frac{y}{z} \le 1.5828563463705101 \cdot 10^{-127}\right) \land \frac{y}{z} \le 2.5905951762285947 \cdot 10^{176}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]
x \cdot \frac{\frac{y}{z} \cdot t}{t}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y}{z} \le -1.057433920064963 \cdot 10^{133}:\\
\;\;\;\;\frac{1}{\frac{z}{x \cdot y}}\\

\mathbf{elif}\;\frac{y}{z} \le -1.0665637347194774 \cdot 10^{-221} \lor \neg \left(\frac{y}{z} \le 1.5828563463705101 \cdot 10^{-127}\right) \land \frac{y}{z} \le 2.5905951762285947 \cdot 10^{176}:\\
\;\;\;\;\frac{x}{\frac{z}{y}}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r139656 = x;
        double r139657 = y;
        double r139658 = z;
        double r139659 = r139657 / r139658;
        double r139660 = t;
        double r139661 = r139659 * r139660;
        double r139662 = r139661 / r139660;
        double r139663 = r139656 * r139662;
        return r139663;
}
↓
double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r139664 = y;
        double r139665 = z;
        double r139666 = r139664 / r139665;
        double r139667 = -1.057433920064963e+133;
        bool r139668 = r139666 <= r139667;
        double r139669 = 1.0;
        double r139670 = x;
        double r139671 = r139670 * r139664;
        double r139672 = r139665 / r139671;
        double r139673 = r139669 / r139672;
        double r139674 = -1.0665637347194774e-221;
        bool r139675 = r139666 <= r139674;
        double r139676 = 1.5828563463705101e-127;
        bool r139677 = r139666 <= r139676;
        double r139678 = !r139677;
        double r139679 = 2.5905951762285947e+176;
        bool r139680 = r139666 <= r139679;
        bool r139681 = r139678 && r139680;
        bool r139682 = r139675 || r139681;
        double r139683 = r139665 / r139664;
        double r139684 = r139670 / r139683;
        double r139685 = r139671 / r139665;
        double r139686 = r139682 ? r139684 : r139685;
        double r139687 = r139668 ? r139673 : r139686;
        return r139687;
}

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) < -1.057433920064963e+133

    1. Initial program 34.0

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

    2. Simplified16.5

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

    4. Applied pow116.5

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

    5. Applied pow116.5

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

    6. Applied pow-prod-down16.5

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

    7. Simplified4.0

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

    9. Applied clear-num4.1

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

    if -1.057433920064963e+133 < (/ y z) < -1.0665637347194774e-221 or 1.5828563463705101e-127 < (/ y z) < 2.5905951762285947e+176

    1. Initial program 6.8

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

    2. Simplified0.2

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

    4. Applied pow10.2

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

    5. Applied pow10.2

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

    6. Applied pow-prod-down0.2

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

    7. Simplified10.3

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

    9. Applied associate-/l*0.2

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

    if -1.0665637347194774e-221 < (/ y z) < 1.5828563463705101e-127 or 2.5905951762285947e+176 < (/ y z)

    1. Initial program 22.0

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

    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. Simplified1.2

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -1.057433920064963 \cdot 10^{133}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot y}}\\ \mathbf{elif}\;\frac{y}{z} \le -1.0665637347194774 \cdot 10^{-221} \lor \neg \left(\frac{y}{z} \le 1.5828563463705101 \cdot 10^{-127}\right) \land \frac{y}{z} \le 2.5905951762285947 \cdot 10^{176}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +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)))