Average Error: 10.8 → 1.0
Time: 2.5s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a - t} \le -6.77757148623794926 \cdot 10^{195}:\\ \;\;\;\;\left(\frac{z}{\frac{a - t}{y}} - \frac{t}{\frac{a - t}{y}}\right) + x\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a - t} \le 3.33962653386183323 \cdot 10^{197}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - t}, z - t, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a - t} \le -6.77757148623794926 \cdot 10^{195}:\\
\;\;\;\;\left(\frac{z}{\frac{a - t}{y}} - \frac{t}{\frac{a - t}{y}}\right) + x\\

\mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a - t} \le 3.33962653386183323 \cdot 10^{197}:\\
\;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a - t}\\

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

\end{array}
double code(double x, double y, double z, double t, double a) {
	return (x + ((y * (z - t)) / (a - t)));
}
↓
double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((((y * (z - t)) / (a - t)) <= -6.777571486237949e+195)) {
		VAR = (((z / ((a - t) / y)) - (t / ((a - t) / y))) + x);
	} else {
		double VAR_1;
		if ((((y * (z - t)) / (a - t)) <= 3.339626533861833e+197)) {
			VAR_1 = (x + ((y * (z - t)) / (a - t)));
		} else {
			VAR_1 = fma((y / (a - t)), (z - t), x);
		}
		VAR = VAR_1;
	}
	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

Original10.8
Target1.3
Herbie1.0
\[x + \frac{y}{\frac{a - t}{z - t}}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (* y (- z t)) (- a t)) < -6.777571486237949e+195

    1. Initial program 45.1

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

    2. Simplified4.7

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

    4. Applied clear-num4.8

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

    6. Applied fma-udef4.8

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

    7. Simplified4.2

      \[\leadsto \color{blue}{\frac{z - t}{\frac{a - t}{y}}} + x\]
    8. Using strategy rm

    9. Applied div-sub4.2

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

    if -6.777571486237949e+195 < (/ (* y (- z t)) (- a t)) < 3.339626533861833e+197

    1. Initial program 0.2

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

    if 3.339626533861833e+197 < (/ (* y (- z t)) (- a t))

    1. Initial program 47.4

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

    2. Simplified3.2

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a - t} \le -6.77757148623794926 \cdot 10^{195}:\\ \;\;\;\;\left(\frac{z}{\frac{a - t}{y}} - \frac{t}{\frac{a - t}{y}}\right) + x\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a - t} \le 3.33962653386183323 \cdot 10^{197}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - t}, z - t, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- a t) (- z t))))

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