Average Error: 10.9 → 1.0
Time: 2.8s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a - t} \le -1.1933061297129669 \cdot 10^{275}:\\ \;\;\;\;\frac{z - t}{\frac{a - t}{y}} + x\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a - t} \le 5.07564913742574227 \cdot 10^{-57}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - t}, z - t, x\right)\\ \end{array}\]
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 -1.1933061297129669 \cdot 10^{275}:\\
\;\;\;\;\frac{z - t}{\frac{a - t}{y}} + x\\

\mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a - t} \le 5.07564913742574227 \cdot 10^{-57}:\\
\;\;\;\;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)) <= -1.1933061297129669e+275)) {
		VAR = (((z - t) / ((a - t) / y)) + x);
	} else {
		double VAR_1;
		if ((((y * (z - t)) / (a - t)) <= 5.075649137425742e-57)) {
			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

Original10.9
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)) < -1.1933061297129669e+275

    1. Initial program 58.9

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

    2. Simplified1.8

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

    4. Applied clear-num1.9

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

    6. Applied fma-udef1.9

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

    7. Simplified1.6

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

    if -1.1933061297129669e+275 < (/ (* y (- z t)) (- a t)) < 5.075649137425742e-57

    1. Initial program 0.3

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

    if 5.075649137425742e-57 < (/ (* y (- z t)) (- a t))

    1. Initial program 20.1

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

    2. Simplified2.5

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

    4. Applied clear-num2.7

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

    6. Applied clear-num2.7

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

    7. Applied remove-double-div2.5

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a - t} \le -1.1933061297129669 \cdot 10^{275}:\\ \;\;\;\;\frac{z - t}{\frac{a - t}{y}} + x\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a - t} \le 5.07564913742574227 \cdot 10^{-57}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - t}, z - t, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +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))))