Average Error: 16.6 → 8.6
Time: 3.4s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.1300734185978326 \cdot 10^{79} \lor \neg \left(t \le 1.87955801658874242 \cdot 10^{143}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{a - t}, t - z, x + y\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.1300734185978326 \cdot 10^{79} \lor \neg \left(t \le 1.87955801658874242 \cdot 10^{143}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{a - t}, t - z, x + y\right)\\

\end{array}
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 VAR;
	if (((t <= -1.1300734185978326e+79) || !(t <= 1.8795580165887424e+143))) {
		VAR = fma((z / t), y, x);
	} else {
		VAR = fma((y * (1.0 / (a - t))), (t - z), (x + y));
	}
	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

Original16.6
Target8.6
Herbie8.6
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -1.1300734185978326e+79 or 1.8795580165887424e+143 < t

    1. Initial program 31.1

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

    2. Simplified22.8

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

    4. Applied div-inv22.8

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

    6. Applied fma-udef22.9

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

    7. Simplified22.9

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

    8. Taylor expanded around inf 17.6

      \[\leadsto \color{blue}{\frac{z \cdot y}{t} + x}\]

    9. Simplified12.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}\]

    if -1.1300734185978326e+79 < t < 1.8795580165887424e+143

    1. Initial program 9.1

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

    2. Simplified6.8

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

    4. Applied div-inv6.8

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \frac{1}{a - t}}, t - z, x + y\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.1300734185978326 \cdot 10^{79} \lor \neg \left(t \le 1.87955801658874242 \cdot 10^{143}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{a - t}, t - z, x + y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(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))))