Average Error: 10.8 → 0.7
Time: 3.2s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} \le -6.848064445188812 \cdot 10^{179}:\\ \;\;\;\;\mathsf{fma}\left(\left(z - t\right) \cdot \frac{1}{z - a}, y, x\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \le 3.25382333353132745 \cdot 10^{246}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} \le -6.848064445188812 \cdot 10^{179}:\\
\;\;\;\;\mathsf{fma}\left(\left(z - t\right) \cdot \frac{1}{z - a}, y, x\right)\\

\mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \le 3.25382333353132745 \cdot 10^{246}:\\
\;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\

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

\end{array}
double code(double x, double y, double z, double t, double a) {
	return ((double) (x + ((double) (((double) (y * ((double) (z - t)))) / ((double) (z - a))))));
}
↓
double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((((double) (((double) (y * ((double) (z - t)))) / ((double) (z - a)))) <= -6.848064445188812e+179)) {
		VAR = ((double) fma(((double) (((double) (z - t)) * ((double) (1.0 / ((double) (z - a)))))), y, x));
	} else {
		double VAR_1;
		if ((((double) (((double) (y * ((double) (z - t)))) / ((double) (z - a)))) <= 3.2538233335313274e+246)) {
			VAR_1 = ((double) (x + ((double) (((double) (y * ((double) (z - t)))) / ((double) (z - a))))));
		} else {
			VAR_1 = ((double) fma(((double) (((double) (z - t)) / ((double) (z - a)))), y, 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
Target1.2
Herbie0.7
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (* y (- z t)) (- z a)) < -6.848064445188812e+179

    1. Initial program 44.9

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

    2. Simplified3.9

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

    4. Applied clear-num4.0

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

    6. Applied fma-udef4.0

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

    7. Simplified3.6

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

    9. Applied associate-/r/3.0

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

    10. Applied fma-def3.0

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

    12. Applied div-inv3.1

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

    if -6.848064445188812e+179 < (/ (* y (- z t)) (- z a)) < 3.2538233335313274e+246

    1. Initial program 0.3

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

    if 3.2538233335313274e+246 < (/ (* y (- z t)) (- z a))

    1. Initial program 55.0

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

    2. Simplified2.3

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

    4. Applied clear-num2.5

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

    6. Applied fma-udef2.5

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

    7. Simplified2.1

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

    9. Applied associate-/r/1.5

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

    10. Applied fma-def1.5

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} \le -6.848064445188812 \cdot 10^{179}:\\ \;\;\;\;\mathsf{fma}\left(\left(z - t\right) \cdot \frac{1}{z - a}, y, x\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \le 3.25382333353132745 \cdot 10^{246}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64

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

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