Average Error: 1.9 → 2.0
Time: 29.0s
Precision: 64
\[x + \left(y - x\right) \cdot \frac{z}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.038679537701190426521563963363797141776 \cdot 10^{-68}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{elif}\;t \le -2.343259671890845919694745995838773941708 \cdot 10^{-154}:\\ \;\;\;\;\frac{y}{\frac{t}{z}} - \left(\frac{x}{\frac{t}{z}} - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \frac{\sqrt[3]{y - x}}{\frac{t}{\sqrt[3]{z}}} + x\\ \end{array}\]
x + \left(y - x\right) \cdot \frac{z}{t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.038679537701190426521563963363797141776 \cdot 10^{-68}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\

\mathbf{elif}\;t \le -2.343259671890845919694745995838773941708 \cdot 10^{-154}:\\
\;\;\;\;\frac{y}{\frac{t}{z}} - \left(\frac{x}{\frac{t}{z}} - x\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r443250 = x;
        double r443251 = y;
        double r443252 = r443251 - r443250;
        double r443253 = z;
        double r443254 = t;
        double r443255 = r443253 / r443254;
        double r443256 = r443252 * r443255;
        double r443257 = r443250 + r443256;
        return r443257;
}
↓
double f(double x, double y, double z, double t) {
        double r443258 = t;
        double r443259 = -1.0386795377011904e-68;
        bool r443260 = r443258 <= r443259;
        double r443261 = y;
        double r443262 = x;
        double r443263 = r443261 - r443262;
        double r443264 = r443263 / r443258;
        double r443265 = z;
        double r443266 = fma(r443264, r443265, r443262);
        double r443267 = -2.343259671890846e-154;
        bool r443268 = r443258 <= r443267;
        double r443269 = r443258 / r443265;
        double r443270 = r443261 / r443269;
        double r443271 = r443262 / r443269;
        double r443272 = r443271 - r443262;
        double r443273 = r443270 - r443272;
        double r443274 = cbrt(r443263);
        double r443275 = r443274 * r443274;
        double r443276 = cbrt(r443265);
        double r443277 = r443276 * r443276;
        double r443278 = r443275 * r443277;
        double r443279 = r443258 / r443276;
        double r443280 = r443274 / r443279;
        double r443281 = r443278 * r443280;
        double r443282 = r443281 + r443262;
        double r443283 = r443268 ? r443273 : r443282;
        double r443284 = r443260 ? r443266 : r443283;
        return r443284;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original1.9
Target2.2
Herbie2.0
\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} \lt -1013646692435.88671875:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} \lt -0.0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -1.0386795377011904e-68

    1. Initial program 1.3

      \[x + \left(y - x\right) \cdot \frac{z}{t}\]

    2. Simplified1.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)}\]
    3. Using strategy rm

    4. Applied fma-udef1.3

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

    5. Simplified1.3

      \[\leadsto \color{blue}{\frac{y - x}{\frac{t}{z}}} + x\]
    6. Using strategy rm

    7. Applied *-un-lft-identity1.3

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

    8. Applied *-un-lft-identity1.3

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

    9. Applied distribute-lft-out1.3

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

    10. Simplified1.6

      \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)}\]

    if -1.0386795377011904e-68 < t < -2.343259671890846e-154

    1. Initial program 3.2

      \[x + \left(y - x\right) \cdot \frac{z}{t}\]

    2. Simplified3.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)}\]
    3. Using strategy rm

    4. Applied fma-udef3.2

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

    5. Simplified3.1

      \[\leadsto \color{blue}{\frac{y - x}{\frac{t}{z}}} + x\]
    6. Using strategy rm

    7. Applied div-sub3.1

      \[\leadsto \color{blue}{\left(\frac{y}{\frac{t}{z}} - \frac{x}{\frac{t}{z}}\right)} + x\]

    8. Applied associate-+l-3.1

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

    if -2.343259671890846e-154 < t

    1. Initial program 2.2

      \[x + \left(y - x\right) \cdot \frac{z}{t}\]

    2. Simplified2.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)}\]
    3. Using strategy rm

    4. Applied fma-udef2.2

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

    5. Simplified2.1

      \[\leadsto \color{blue}{\frac{y - x}{\frac{t}{z}}} + x\]
    6. Using strategy rm

    7. Applied add-cube-cbrt2.7

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

    8. Applied *-un-lft-identity2.7

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

    9. Applied times-frac2.7

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

    10. Applied add-cube-cbrt2.8

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

    11. Applied times-frac2.1

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

    12. Simplified2.1

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

  4. Final simplification2.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.038679537701190426521563963363797141776 \cdot 10^{-68}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{elif}\;t \le -2.343259671890845919694745995838773941708 \cdot 10^{-154}:\\ \;\;\;\;\frac{y}{\frac{t}{z}} - \left(\frac{x}{\frac{t}{z}} - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \frac{\sqrt[3]{y - x}}{\frac{t}{\sqrt[3]{z}}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4"
  :precision binary64

  :herbie-target
  (if (< (* (- y x) (/ z t)) -1013646692435.887) (+ x (/ (- y x) (/ t z))) (if (< (* (- y x) (/ z t)) -0.0) (+ x (/ (* (- y x) z) t)) (+ x (/ (- y x) (/ t z)))))

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