Average Error: 16.8 → 5.9
Time: 7.8s
Precision: binary64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{y \cdot \left(z - t\right)}{a - t} \leq -7.115427134883366 \cdot 10^{-232} \lor \neg \left(\left(x + y\right) - \frac{y \cdot \left(z - t\right)}{a - t} \leq 3.506851936346399 \cdot 10^{-253}\right):\\ \;\;\;\;x + \left(y - \frac{z - t}{\frac{a - t}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{y \cdot z}{t}\right) - \frac{y \cdot a}{t}\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;\left(x + y\right) - \frac{y \cdot \left(z - t\right)}{a - t} \leq -7.115427134883366 \cdot 10^{-232} \lor \neg \left(\left(x + y\right) - \frac{y \cdot \left(z - t\right)}{a - t} \leq 3.506851936346399 \cdot 10^{-253}\right):\\
\;\;\;\;x + \left(y - \frac{z - t}{\frac{a - t}{y}}\right)\\

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

\end{array}
(FPCore (x y z t a) :precision binary64 (- (+ x y) (/ (* (- z t) y) (- a t))))
↓
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- (+ x y) (/ (* y (- z t)) (- a t))) -7.115427134883366e-232)
         (not
          (<= (- (+ x y) (/ (* y (- z t)) (- a t))) 3.506851936346399e-253)))
   (+ x (- y (/ (- z t) (/ (- a t) y))))
   (- (+ x (/ (* y z) t)) (/ (* y a) t))))
double code(double x, double y, double z, double t, double a) {
	return (x + y) - (((z - t) * y) / (a - t));
}
↓
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((((x + y) - ((y * (z - t)) / (a - t))) <= -7.115427134883366e-232) || !(((x + y) - ((y * (z - t)) / (a - t))) <= 3.506851936346399e-253)) {
		tmp = x + (y - ((z - t) / ((a - t) / y)));
	} else {
		tmp = (x + ((y * z) / t)) - ((y * a) / t);
	}
	return tmp;
}

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

Original16.8
Target8.8
Herbie5.9
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} < -1.3664970889390727 \cdot 10^{-07}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} < 1.4754293444577233 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -7.11542713488336607e-232 or 3.506851936346399e-253 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

    1. Initial program 13.0

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

    3. Applied add-cube-cbrt_binary6413.2

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

    4. Applied times-frac_binary647.8

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

    5. Applied cancel-sign-sub-inv_binary647.8

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

    6. Simplified7.8

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

    8. Applied associate-+l+_binary647.3

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

    9. Simplified6.0

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

    if -7.11542713488336607e-232 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 3.506851936346399e-253

    1. Initial program 55.5

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

    2. Taylor expanded around inf 4.1

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

    3. Simplified4.1

      \[\leadsto \color{blue}{\left(\frac{y \cdot z}{t} + x\right) - \frac{y \cdot a}{t}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{y \cdot \left(z - t\right)}{a - t} \leq -7.115427134883366 \cdot 10^{-232} \lor \neg \left(\left(x + y\right) - \frac{y \cdot \left(z - t\right)}{a - t} \leq 3.506851936346399 \cdot 10^{-253}\right):\\ \;\;\;\;x + \left(y - \frac{z - t}{\frac{a - t}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{y \cdot z}{t}\right) - \frac{y \cdot a}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2021173 
(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) (/ 1.0 (- 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) (/ 1.0 (- a t))) y))))

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