Average Error: 15.1 → 6.0
Time: 14.1s
Precision: 64
\[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.456893094179093804320322909204701959778 \cdot 10^{-222} \lor \neg \left(z \le -1.998153801628733335171500404420370674566 \cdot 10^{-247} \lor \neg \left(z \le 8.447680461540099603836085647567339379828 \cdot 10^{-193}\right) \land z \le 1.21944214684399880068881121045408236628 \cdot 10^{120}\right):\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array}\]
x \cdot \frac{\frac{y}{z} \cdot t}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.456893094179093804320322909204701959778 \cdot 10^{-222} \lor \neg \left(z \le -1.998153801628733335171500404420370674566 \cdot 10^{-247} \lor \neg \left(z \le 8.447680461540099603836085647567339379828 \cdot 10^{-193}\right) \land z \le 1.21944214684399880068881121045408236628 \cdot 10^{120}\right):\\
\;\;\;\;\frac{x \cdot y}{z}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r70370 = x;
        double r70371 = y;
        double r70372 = z;
        double r70373 = r70371 / r70372;
        double r70374 = t;
        double r70375 = r70373 * r70374;
        double r70376 = r70375 / r70374;
        double r70377 = r70370 * r70376;
        return r70377;
}
↓
double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r70378 = z;
        double r70379 = -5.456893094179094e-222;
        bool r70380 = r70378 <= r70379;
        double r70381 = -1.9981538016287333e-247;
        bool r70382 = r70378 <= r70381;
        double r70383 = 8.4476804615401e-193;
        bool r70384 = r70378 <= r70383;
        double r70385 = !r70384;
        double r70386 = 1.2194421468439988e+120;
        bool r70387 = r70378 <= r70386;
        bool r70388 = r70385 && r70387;
        bool r70389 = r70382 || r70388;
        double r70390 = !r70389;
        bool r70391 = r70380 || r70390;
        double r70392 = x;
        double r70393 = y;
        double r70394 = r70392 * r70393;
        double r70395 = r70394 / r70378;
        double r70396 = r70378 / r70392;
        double r70397 = r70393 / r70396;
        double r70398 = r70391 ? r70395 : r70397;
        return r70398;
}

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 z < -5.456893094179094e-222 or -1.9981538016287333e-247 < z < 8.4476804615401e-193 or 1.2194421468439988e+120 < z

    1. Initial program 15.2

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

    2. Simplified6.7

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

    4. Applied *-un-lft-identity6.7

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

    5. Applied times-frac6.8

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

    6. Simplified6.8

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

    8. Applied *-un-lft-identity6.8

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

    9. Applied associate-*l*6.8

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

    10. Simplified6.7

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

    if -5.456893094179094e-222 < z < -1.9981538016287333e-247 or 8.4476804615401e-193 < z < 1.2194421468439988e+120

    1. Initial program 14.8

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

    2. Simplified5.0

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

    4. Applied *-un-lft-identity5.0

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

    5. Applied times-frac4.7

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

    6. Simplified4.7

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

    7. Taylor expanded around 0 5.0

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

    8. Simplified3.9

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

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.456893094179093804320322909204701959778 \cdot 10^{-222} \lor \neg \left(z \le -1.998153801628733335171500404420370674566 \cdot 10^{-247} \lor \neg \left(z \le 8.447680461540099603836085647567339379828 \cdot 10^{-193}\right) \land z \le 1.21944214684399880068881121045408236628 \cdot 10^{120}\right):\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1"
  (* x (/ (* (/ y z) t) t)))