Average Error: 16.5 → 11.9
Time: 6.2s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.74606125404172808 \cdot 10^{-145} \lor \neg \left(t \le 2.82207153459146274 \cdot 10^{-113}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - t}, t - z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t - z\right) \cdot y}{a - t} + \left(x + y\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -4.74606125404172808 \cdot 10^{-145} \lor \neg \left(t \le 2.82207153459146274 \cdot 10^{-113}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a - t}, t - z, x + y\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(t - z\right) \cdot y}{a - t} + \left(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 temp;
	if (((t <= -4.746061254041728e-145) || !(t <= 2.822071534591463e-113))) {
		temp = fma((y / (a - t)), (t - z), (x + y));
	} else {
		temp = ((((t - z) * y) / (a - t)) + (x + y));
	}
	return temp;
}

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

Original16.5
Target8.1
Herbie11.9
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -4.746061254041728e-145 or 2.822071534591463e-113 < t

    1. Initial program 20.8

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

    2. Simplified14.5

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

    if -4.746061254041728e-145 < t < 2.822071534591463e-113

    1. Initial program 4.7

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

    2. Simplified3.8

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

    4. Applied add-cube-cbrt4.1

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

    5. Applied associate-/l*4.1

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

    7. Applied fma-udef4.1

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

    8. Simplified4.7

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

  4. Final simplification11.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.74606125404172808 \cdot 10^{-145} \lor \neg \left(t \le 2.82207153459146274 \cdot 10^{-113}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - t}, t - z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t - z\right) \cdot y}{a - t} + \left(x + y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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))))