Average Error: 1.3 → 0.6
Time: 4.6s
Precision: 64
\[x + y \cdot \frac{z - t}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.74547300710632633 \cdot 10^{-55} \lor \neg \left(y \le 1.4339579110756609 \cdot 10^{-126}\right):\\ \;\;\;\;\frac{y}{\frac{a - t}{z - t}} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a - t}{y \cdot \left(z - t\right)}} + x\\ \end{array}\]
x + y \cdot \frac{z - t}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.74547300710632633 \cdot 10^{-55} \lor \neg \left(y \le 1.4339579110756609 \cdot 10^{-126}\right):\\
\;\;\;\;\frac{y}{\frac{a - t}{z - t}} + x\\

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

\end{array}
double code(double x, double y, double z, double t, double a) {
	return ((double) (x + ((double) (y * ((double) (((double) (z - t)) / ((double) (a - t))))))));
}
↓
double code(double x, double y, double z, double t, double a) {
	double VAR;
	if (((y <= -1.7454730071063263e-55) || !(y <= 1.4339579110756609e-126))) {
		VAR = ((double) (((double) (y / ((double) (((double) (a - t)) / ((double) (z - t)))))) + x));
	} else {
		VAR = ((double) (((double) (1.0 / ((double) (((double) (a - t)) / ((double) (y * ((double) (z - t)))))))) + x));
	}
	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

Original1.3
Target0.4
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;y \lt -8.50808486055124107 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.7454730071063263e-55 or 1.4339579110756609e-126 < 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 clear-num0.7

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

    6. Applied fma-udef0.7

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

    7. Simplified0.6

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

    if -1.7454730071063263e-55 < y < 1.4339579110756609e-126

    1. Initial program 2.4

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

    2. Simplified2.4

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

    4. Applied clear-num2.4

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

    6. Applied fma-udef2.4

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

    7. Simplified2.0

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

    9. Applied clear-num2.1

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{a - t}{z - t}}{y}}} + x\]
    10. Using strategy rm

    11. Applied div-inv2.1

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

    12. Applied associate-/l*0.5

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

    13. Simplified0.5

      \[\leadsto \frac{1}{\frac{a - t}{\color{blue}{y \cdot \left(z - t\right)}}} + x\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.74547300710632633 \cdot 10^{-55} \lor \neg \left(y \le 1.4339579110756609 \cdot 10^{-126}\right):\\ \;\;\;\;\frac{y}{\frac{a - t}{z - t}} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a - t}{y \cdot \left(z - t\right)}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 +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)))))