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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -1.693067188060901 \cdot 10^{-126} \lor \neg \left(a \le 2.11759892460565624 \cdot 10^{-87}\right):\\ \;\;\;\;\left(x + y\right) - \left(\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;a \le -1.693067188060901 \cdot 10^{-126} \lor \neg \left(a \le 2.11759892460565624 \cdot 10^{-87}\right):\\
\;\;\;\;\left(x + y\right) - \left(\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t}}}\\

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

\end{array}

double f(double x, double y, double z, double t, double a) {
        double r637618 = x;
        double r637619 = y;
        double r637620 = r637618 + r637619;
        double r637621 = z;
        double r637622 = t;
        double r637623 = r637621 - r637622;
        double r637624 = r637623 * r637619;
        double r637625 = a;
        double r637626 = r637625 - r637622;
        double r637627 = r637624 / r637626;
        double r637628 = r637620 - r637627;
        return r637628;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r637629 = a;
        double r637630 = -1.693067188060901e-126;
        bool r637631 = r637629 <= r637630;
        double r637632 = 2.1175989246056562e-87;
        bool r637633 = r637629 <= r637632;
        double r637634 = !r637633;
        bool r637635 = r637631 || r637634;
        double r637636 = x;
        double r637637 = y;
        double r637638 = r637636 + r637637;
        double r637639 = z;
        double r637640 = t;
        double r637641 = r637639 - r637640;
        double r637642 = r637629 - r637640;
        double r637643 = cbrt(r637642);
        double r637644 = r637643 * r637643;
        double r637645 = r637641 / r637644;
        double r637646 = cbrt(r637637);
        double r637647 = r637646 * r637646;
        double r637648 = cbrt(r637643);
        double r637649 = r637648 * r637648;
        double r637650 = r637647 / r637649;
        double r637651 = r637645 * r637650;
        double r637652 = r637646 / r637648;
        double r637653 = r637651 * r637652;
        double r637654 = r637638 - r637653;
        double r637655 = r637639 * r637637;
        double r637656 = r637655 / r637640;
        double r637657 = r637656 + r637636;
        double r637658 = r637635 ? r637654 : r637657;
        return r637658;
}

Original	16.5
Target	8.1
Herbie	8.8

Derivation

Split input into 2 regimes
if a < -1.693067188060901e-126 or 2.1175989246056562e-87 < a
1. Initial program 14.9
  \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
2. Using strategy rm
3. Applied add-cube-cbrt15.0
  \[\leadsto \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{\color{blue}{\left(\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}\right) \cdot \sqrt[3]{a - t}}}\]
4. Applied times-frac7.9
  \[\leadsto \left(x + y\right) - \color{blue}{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}}\]
5. Using strategy rm
6. Applied add-cube-cbrt8.0
  \[\leadsto \left(x + y\right) - \frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\color{blue}{\left(\sqrt[3]{\sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}\right) \cdot \sqrt[3]{\sqrt[3]{a - t}}}}\]
7. Applied add-cube-cbrt8.0
  \[\leadsto \left(x + y\right) - \frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\left(\sqrt[3]{\sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}\right) \cdot \sqrt[3]{\sqrt[3]{a - t}}}\]
8. Applied times-frac8.0
  \[\leadsto \left(x + y\right) - \frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t}}}\right)}\]
9. Applied associate-*r*7.8
  \[\leadsto \left(x + y\right) - \color{blue}{\left(\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t}}}}\]
if -1.693067188060901e-126 < a < 2.1175989246056562e-87
1. Initial program 20.1
  \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
2. Taylor expanded around inf 11.0
  \[\leadsto \color{blue}{\frac{z \cdot y}{t} + x}\]
Recombined 2 regimes into one program.
Final simplification8.8
\[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.693067188060901 \cdot 10^{-126} \lor \neg \left(a \le 2.11759892460565624 \cdot 10^{-87}\right):\\ \;\;\;\;\left(x + y\right) - \left(\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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))))

Error

Try it out

Target

Derivation

`if a < -1.693067188060901e-126 or 2.1175989246056562e-87 < a`

`if -1.693067188060901e-126 < a < 2.1175989246056562e-87`

Reproduce

In
Out