Average Error: 1.5 → 1.3
Time: 3.8s
Precision: 64
\[x + y \cdot \frac{z - t}{z - a}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -7.0231454598615249 \cdot 10^{-277}:\\ \;\;\;\;\frac{y}{\frac{z - a}{z - t}} + x\\ \mathbf{elif}\;z \le 2.80308765864311706 \cdot 10^{-161}:\\ \;\;\;\;\frac{y}{z - a} \cdot \left(z - t\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\\ \end{array}\]
x + y \cdot \frac{z - t}{z - a}
↓
\begin{array}{l}
\mathbf{if}\;z \le -7.0231454598615249 \cdot 10^{-277}:\\
\;\;\;\;\frac{y}{\frac{z - a}{z - t}} + x\\

\mathbf{elif}\;z \le 2.80308765864311706 \cdot 10^{-161}:\\
\;\;\;\;\frac{y}{z - a} \cdot \left(z - t\right) + x\\

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

\end{array}
double code(double x, double y, double z, double t, double a) {
	return (x + (y * ((z - t) / (z - a))));
}
↓
double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((z <= -7.023145459861525e-277)) {
		VAR = ((y / ((z - a) / (z - t))) + x);
	} else {
		double VAR_1;
		if ((z <= 2.803087658643117e-161)) {
			VAR_1 = (((y / (z - a)) * (z - t)) + x);
		} else {
			VAR_1 = fma(y, ((z - t) / (z - a)), 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

Original1.5
Target1.4
Herbie1.3
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -7.023145459861525e-277

    1. Initial program 1.4

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

    2. Simplified1.4

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

    4. Applied clear-num1.4

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

    6. Applied fma-udef1.4

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

    7. Simplified1.2

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

    if -7.023145459861525e-277 < z < 2.803087658643117e-161

    1. Initial program 4.5

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

    2. Simplified4.5

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

    4. Applied clear-num4.6

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

    6. Applied fma-udef4.6

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

    7. Simplified4.2

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

    9. Applied associate-/r/3.2

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

    if 2.803087658643117e-161 < z

    1. Initial program 0.7

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

    2. Simplified0.7

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.0231454598615249 \cdot 10^{-277}:\\ \;\;\;\;\frac{y}{\frac{z - a}{z - t}} + x\\ \mathbf{elif}\;z \le 2.80308765864311706 \cdot 10^{-161}:\\ \;\;\;\;\frac{y}{z - a} \cdot \left(z - t\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020103 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

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

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