Average Error: 16.2 → 9.1
Time: 24.6s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.31413808910975569189734655926999448624 \cdot 10^{-225} \lor \neg \left(a \le 5.913406836150693827223293523747761324681 \cdot 10^{-231}\right):\\ \;\;\;\;\left(x + y\right) - \frac{\frac{z - t}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;a \le -1.31413808910975569189734655926999448624 \cdot 10^{-225} \lor \neg \left(a \le 5.913406836150693827223293523747761324681 \cdot 10^{-231}\right):\\
\;\;\;\;\left(x + y\right) - \frac{\frac{z - t}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r378418 = x;
        double r378419 = y;
        double r378420 = r378418 + r378419;
        double r378421 = z;
        double r378422 = t;
        double r378423 = r378421 - r378422;
        double r378424 = r378423 * r378419;
        double r378425 = a;
        double r378426 = r378425 - r378422;
        double r378427 = r378424 / r378426;
        double r378428 = r378420 - r378427;
        return r378428;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r378429 = a;
        double r378430 = -1.3141380891097557e-225;
        bool r378431 = r378429 <= r378430;
        double r378432 = 5.913406836150694e-231;
        bool r378433 = r378429 <= r378432;
        double r378434 = !r378433;
        bool r378435 = r378431 || r378434;
        double r378436 = x;
        double r378437 = y;
        double r378438 = r378436 + r378437;
        double r378439 = z;
        double r378440 = t;
        double r378441 = r378439 - r378440;
        double r378442 = r378429 - r378440;
        double r378443 = cbrt(r378442);
        double r378444 = r378443 * r378443;
        double r378445 = cbrt(r378437);
        double r378446 = r378445 * r378445;
        double r378447 = r378444 / r378446;
        double r378448 = r378441 / r378447;
        double r378449 = r378443 / r378445;
        double r378450 = r378448 / r378449;
        double r378451 = r378438 - r378450;
        double r378452 = r378439 * r378437;
        double r378453 = r378452 / r378440;
        double r378454 = r378453 + r378436;
        double r378455 = r378435 ? r378451 : r378454;
        return r378455;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original16.2
Target8.3
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.366497088939072697550672266103566343531 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.475429344457723334351036314450840066235 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -1.3141380891097557e-225 or 5.913406836150694e-231 < a

    1. Initial program 15.5

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

    3. Applied associate-/l*10.4

      \[\leadsto \left(x + y\right) - \color{blue}{\frac{z - t}{\frac{a - t}{y}}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt10.6

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

    6. Applied add-cube-cbrt10.6

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

    7. Applied times-frac10.6

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

    8. Applied associate-/r*9.5

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

    if -1.3141380891097557e-225 < a < 5.913406836150694e-231

    1. Initial program 21.3

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

    2. Taylor expanded around inf 6.5

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.31413808910975569189734655926999448624 \cdot 10^{-225} \lor \neg \left(a \le 5.913406836150693827223293523747761324681 \cdot 10^{-231}\right):\\ \;\;\;\;\left(x + y\right) - \frac{\frac{z - t}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-07) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.4754293444577233e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y))))

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