Average Error: 1.4 → 0.5
Time: 5.7s
Precision: 64
\[x + y \cdot \frac{z - t}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -9.4186501671601072 \cdot 10^{28}:\\ \;\;\;\;\frac{1}{\frac{\frac{a - t}{z - t}}{y}} + x\\ \mathbf{elif}\;y \le 297107934094652.88:\\ \;\;\;\;\frac{1}{a - t} \cdot \left(y \cdot \left(z - t\right)\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a - t} - \frac{t}{a - t}, x\right)\\ \end{array}\]
x + y \cdot \frac{z - t}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;y \le -9.4186501671601072 \cdot 10^{28}:\\
\;\;\;\;\frac{1}{\frac{\frac{a - t}{z - t}}{y}} + x\\

\mathbf{elif}\;y \le 297107934094652.88:\\
\;\;\;\;\frac{1}{a - t} \cdot \left(y \cdot \left(z - t\right)\right) + x\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{z}{a - t} - \frac{t}{a - 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 <= -9.418650167160107e+28)) {
		VAR = ((1.0 / (((a - t) / (z - t)) / y)) + x);
	} else {
		double VAR_1;
		if ((y <= 297107934094652.9)) {
			VAR_1 = (((1.0 / (a - t)) * (y * (z - t))) + x);
		} else {
			VAR_1 = fma(y, ((z / (a - t)) - (t / (a - 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

Original1.4
Target0.4
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y \lt -8.50808486055124107 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -9.418650167160107e+28

    1. Initial program 0.4

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

    2. Simplified0.4

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

    4. Applied clear-num0.6

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

    6. Applied fma-udef0.6

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

    7. Simplified0.6

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

    9. Applied clear-num0.7

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

    if -9.418650167160107e+28 < y < 297107934094652.9

    1. Initial program 2.1

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

    2. Simplified2.1

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

    4. Applied clear-num2.2

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

    6. Applied fma-udef2.2

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

    7. Simplified1.9

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

    9. Applied div-inv1.9

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

    10. Applied *-un-lft-identity1.9

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

    11. Applied times-frac0.5

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

    12. Simplified0.5

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

    if 297107934094652.9 < y

    1. Initial program 0.6

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

    2. Simplified0.6

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

    4. Applied div-sub0.5

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -9.4186501671601072 \cdot 10^{28}:\\ \;\;\;\;\frac{1}{\frac{\frac{a - t}{z - t}}{y}} + x\\ \mathbf{elif}\;y \le 297107934094652.88:\\ \;\;\;\;\frac{1}{a - t} \cdot \left(y \cdot \left(z - t\right)\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a - t} - \frac{t}{a - t}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020092 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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