Average Error: 2.2 → 3.3
Time: 29.2s
Precision: 64
\[x + \left(y - x\right) \cdot \frac{z}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.83245230970383374 \cdot 10^{-70}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;x \le -2.07809484407539427 \cdot 10^{-240}:\\ \;\;\;\;x + \left(\frac{z \cdot y}{t} - \frac{x \cdot z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}\\ \end{array}\]
x + \left(y - x\right) \cdot \frac{z}{t}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.83245230970383374 \cdot 10^{-70}:\\
\;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\

\mathbf{elif}\;x \le -2.07809484407539427 \cdot 10^{-240}:\\
\;\;\;\;x + \left(\frac{z \cdot y}{t} - \frac{x \cdot z}{t}\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r490733 = x;
        double r490734 = y;
        double r490735 = r490734 - r490733;
        double r490736 = z;
        double r490737 = t;
        double r490738 = r490736 / r490737;
        double r490739 = r490735 * r490738;
        double r490740 = r490733 + r490739;
        return r490740;
}
↓
double f(double x, double y, double z, double t) {
        double r490741 = x;
        double r490742 = -2.8324523097038337e-70;
        bool r490743 = r490741 <= r490742;
        double r490744 = y;
        double r490745 = r490744 - r490741;
        double r490746 = z;
        double r490747 = t;
        double r490748 = r490746 / r490747;
        double r490749 = r490745 * r490748;
        double r490750 = r490741 + r490749;
        double r490751 = -2.0780948440753943e-240;
        bool r490752 = r490741 <= r490751;
        double r490753 = r490746 * r490744;
        double r490754 = r490753 / r490747;
        double r490755 = r490741 * r490746;
        double r490756 = r490755 / r490747;
        double r490757 = r490754 - r490756;
        double r490758 = r490741 + r490757;
        double r490759 = cbrt(r490747);
        double r490760 = r490759 * r490759;
        double r490761 = r490745 / r490760;
        double r490762 = r490746 / r490759;
        double r490763 = r490761 * r490762;
        double r490764 = r490741 + r490763;
        double r490765 = r490752 ? r490758 : r490764;
        double r490766 = r490743 ? r490750 : r490765;
        return r490766;
}

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

Original2.2
Target2.4
Herbie3.3
\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} \lt -1013646692435.887:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} \lt -0.0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -2.8324523097038337e-70

    1. Initial program 0.4

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

    if -2.8324523097038337e-70 < x < -2.0780948440753943e-240

    1. Initial program 4.4

      \[x + \left(y - x\right) \cdot \frac{z}{t}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt5.0

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

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

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

    5. Applied times-frac5.0

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

    6. Applied associate-*r*3.4

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

    7. Simplified3.4

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

    8. Taylor expanded around 0 3.5

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

    if -2.0780948440753943e-240 < x

    1. Initial program 2.4

      \[x + \left(y - x\right) \cdot \frac{z}{t}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt3.0

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

    4. Applied *-un-lft-identity3.0

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

    5. Applied times-frac3.0

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

    6. Applied associate-*r*4.7

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

    7. Simplified4.7

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

  4. Final simplification3.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.83245230970383374 \cdot 10^{-70}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;x \le -2.07809484407539427 \cdot 10^{-240}:\\ \;\;\;\;x + \left(\frac{z \cdot y}{t} - \frac{x \cdot z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))))