Average Error: 16.1 → 6.9
Time: 8.6s
Precision: binary64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -3.12213425314999685 \cdot 10^{-145} \lor \neg \left(a \le 3.70465193948904425 \cdot 10^{-112}\right):\\ \;\;\;\;x + y \cdot \left(1 - \left(z - t\right) \cdot \frac{1}{a - t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;a \le -3.12213425314999685 \cdot 10^{-145} \lor \neg \left(a \le 3.70465193948904425 \cdot 10^{-112}\right):\\
\;\;\;\;x + y \cdot \left(1 - \left(z - t\right) \cdot \frac{1}{a - t}\right)\\

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

\end{array}
double code(double x, double y, double z, double t, double a) {
	return ((double) (((double) (x + y)) - ((double) (((double) (((double) (z - t)) * y)) / ((double) (a - t))))));
}
↓
double code(double x, double y, double z, double t, double a) {
	double VAR;
	if (((a <= -3.122134253149997e-145) || !(a <= 3.7046519394890442e-112))) {
		VAR = ((double) (x + ((double) (y * ((double) (1.0 - ((double) (((double) (z - t)) * ((double) (1.0 / ((double) (a - t))))))))))));
	} else {
		VAR = ((double) (x + ((double) (y * ((double) (z / t))))));
	}
	return VAR;
}

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

Original16.1
Target8.2
Herbie6.9
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -3.12213425314999685e-145 or 3.70465193948904425e-112 < a

    1. Initial program 14.8

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

    3. Applied associate-/l*9.0

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

    5. Applied associate--l+7.4

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

    7. Applied associate-/r/5.8

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

    8. Applied *-un-lft-identity5.8

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

    9. Applied distribute-rgt-out--5.8

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

    11. Applied div-inv6.1

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

    if -3.12213425314999685e-145 < a < 3.70465193948904425e-112

    1. Initial program 19.6

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

    3. Applied associate-/l*19.0

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

    5. Applied associate--l+13.1

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

    7. Applied associate-/r/11.7

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

    8. Applied *-un-lft-identity11.7

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

    9. Applied distribute-rgt-out--11.7

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

    10. Taylor expanded around inf 8.9

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -3.12213425314999685 \cdot 10^{-145} \lor \neg \left(a \le 3.70465193948904425 \cdot 10^{-112}\right):\\ \;\;\;\;x + y \cdot \left(1 - \left(z - t\right) \cdot \frac{1}{a - t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020147 
(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))))