Average Error: 14.8 → 2.4
Time: 21.2s
Precision: 64
\[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -3.636613989781371082537060282645406790623 \cdot 10^{235}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -4.268869399178432590236189660705002059275 \cdot 10^{-310}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{y} \cdot \frac{\frac{x}{\sqrt[3]{z}}}{\sqrt[3]{z}}\right) \cdot \sqrt[3]{y}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\\ \end{array}\]
x \cdot \frac{\frac{y}{z} \cdot t}{t}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y}{z} \le -3.636613989781371082537060282645406790623 \cdot 10^{235}:\\
\;\;\;\;\frac{x \cdot y}{z}\\

\mathbf{elif}\;\frac{y}{z} \le -4.268869399178432590236189660705002059275 \cdot 10^{-310}:\\
\;\;\;\;\frac{y}{z} \cdot x\\

\mathbf{else}:\\
\;\;\;\;\left(\left(\sqrt[3]{y} \cdot \frac{\frac{x}{\sqrt[3]{z}}}{\sqrt[3]{z}}\right) \cdot \sqrt[3]{y}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r25709288 = x;
        double r25709289 = y;
        double r25709290 = z;
        double r25709291 = r25709289 / r25709290;
        double r25709292 = t;
        double r25709293 = r25709291 * r25709292;
        double r25709294 = r25709293 / r25709292;
        double r25709295 = r25709288 * r25709294;
        return r25709295;
}
↓
double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r25709296 = y;
        double r25709297 = z;
        double r25709298 = r25709296 / r25709297;
        double r25709299 = -3.636613989781371e+235;
        bool r25709300 = r25709298 <= r25709299;
        double r25709301 = x;
        double r25709302 = r25709301 * r25709296;
        double r25709303 = r25709302 / r25709297;
        double r25709304 = -4.26886939917843e-310;
        bool r25709305 = r25709298 <= r25709304;
        double r25709306 = r25709298 * r25709301;
        double r25709307 = cbrt(r25709296);
        double r25709308 = cbrt(r25709297);
        double r25709309 = r25709301 / r25709308;
        double r25709310 = r25709309 / r25709308;
        double r25709311 = r25709307 * r25709310;
        double r25709312 = r25709311 * r25709307;
        double r25709313 = r25709307 / r25709308;
        double r25709314 = r25709312 * r25709313;
        double r25709315 = r25709305 ? r25709306 : r25709314;
        double r25709316 = r25709300 ? r25709303 : r25709315;
        return r25709316;
}

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

Target

Original14.8
Target1.6
Herbie2.4
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{y}{z} \cdot t}{t} \lt -1.206722051230450047215521150762600712224 \cdot 10^{245}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt -5.90752223693390632993316700759382836344 \cdot 10^{-275}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 5.658954423153415216825328199697215652986 \cdot 10^{-65}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 2.008718050240713347941382056648619307142 \cdot 10^{217}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ y z) < -3.636613989781371e+235

    1. Initial program 48.0

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

    2. Simplified0.4

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

    if -3.636613989781371e+235 < (/ y z) < -4.26886939917843e-310

    1. Initial program 9.6

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

    2. Simplified8.0

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

    4. Applied *-un-lft-identity8.0

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

    5. Applied times-frac0.2

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

    6. Simplified0.2

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

    if -4.26886939917843e-310 < (/ y z)

    1. Initial program 15.5

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

    2. Simplified5.5

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

    4. Applied add-cube-cbrt6.2

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\]

    5. Applied times-frac5.2

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

    7. Applied *-un-lft-identity5.2

      \[\leadsto \frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{\color{blue}{1 \cdot z}}}\]

    8. Applied cbrt-prod5.2

      \[\leadsto \frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\color{blue}{\sqrt[3]{1} \cdot \sqrt[3]{z}}}\]

    9. Applied add-cube-cbrt5.4

      \[\leadsto \frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\sqrt[3]{1} \cdot \sqrt[3]{z}}\]

    10. Applied times-frac5.4

      \[\leadsto \frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{1}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\right)}\]

    11. Applied associate-*r*4.0

      \[\leadsto \color{blue}{\left(\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{1}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}}\]

    12. Simplified4.0

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

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -3.636613989781371082537060282645406790623 \cdot 10^{235}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -4.268869399178432590236189660705002059275 \cdot 10^{-310}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{y} \cdot \frac{\frac{x}{\sqrt[3]{z}}}{\sqrt[3]{z}}\right) \cdot \sqrt[3]{y}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B"

  :herbie-target
  (if (< (/ (* (/ y z) t) t) -1.20672205123045e+245) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) -5.907522236933906e-275) (* x (/ y z)) (if (< (/ (* (/ y z) t) t) 5.658954423153415e-65) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) 2.0087180502407133e+217) (* x (/ y z)) (/ (* y x) z)))))

  (* x (/ (* (/ y z) t) t)))