Average Error: 15.7 → 7.0
Time: 6.3s
Precision: binary64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -9.459372584668667 \cdot 10^{-116} \lor \neg \left(y \leq 7.60177063837458 \cdot 10^{-20}\right):\\ \;\;\;\;x + \left(y + y \cdot \left(\left(z - t\right) \cdot \frac{1}{t - a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y + \frac{y \cdot \left(z - t\right)}{t - a}\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -9.459372584668667 \cdot 10^{-116} \lor \neg \left(y \leq 7.60177063837458 \cdot 10^{-20}\right):\\
\;\;\;\;x + \left(y + y \cdot \left(\left(z - t\right) \cdot \frac{1}{t - a}\right)\right)\\

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

\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 (<= y -9.459372584668667e-116) (not (<= y 7.60177063837458e-20)))
   (+ x (+ y (* y (* (- z t) (/ 1.0 (- t a))))))
   (+ x (+ y (/ (* y (- z t)) (- t a))))))
double code(double x, double y, double z, double t, double a) {
	return ((double) (((double) (x + y)) - (((double) (((double) (z - t)) * y)) / ((double) (a - t)))));
}
↓
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((y <= -9.459372584668667e-116) || !(y <= 7.60177063837458e-20))) {
		tmp = ((double) (x + ((double) (y + ((double) (y * ((double) (((double) (z - t)) * (1.0 / ((double) (t - a)))))))))));
	} else {
		tmp = ((double) (x + ((double) (y + (((double) (y * ((double) (z - t)))) / ((double) (t - a)))))));
	}
	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

Original15.7
Target8.4
Herbie7.0
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} < -1.3664970889390727 \cdot 10^{-07}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} < 1.4754293444577233 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -9.4593725846686666e-116 or 7.6017706383745804e-20 < y

    1. Initial program Error: 25.3 bits

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

    2. SimplifiedError: 10.5 bits

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

    4. Applied div-invError: 11.2 bits

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

    if -9.4593725846686666e-116 < y < 7.6017706383745804e-20

    1. Initial program Error: 3.6 bits

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

    2. SimplifiedError: 3.6 bits

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

    4. Applied associate-*r/Error: 1.7 bits

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

  4. Final simplificationError: 7.0 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.459372584668667 \cdot 10^{-116} \lor \neg \left(y \leq 7.60177063837458 \cdot 10^{-20}\right):\\ \;\;\;\;x + \left(y + y \cdot \left(\left(z - t\right) \cdot \frac{1}{t - a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y + \frac{y \cdot \left(z - t\right)}{t - a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(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))))