Average Error: 16.4 → 7.5
Time: 3.2s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -9.0201884235280197 \cdot 10^{203} \lor \neg \left(t \le 1.5081531538129489 \cdot 10^{118}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a - t}, y, x + y\right) - \frac{z}{\frac{a - t}{y}}\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -9.0201884235280197 \cdot 10^{203} \lor \neg \left(t \le 1.5081531538129489 \cdot 10^{118}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\

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

\end{array}
double code(double x, double y, double z, double t, double a) {
	return ((x + y) - (((z - t) * y) / (a - t)));
}
↓
double code(double x, double y, double z, double t, double a) {
	double VAR;
	if (((t <= -9.02018842352802e+203) || !(t <= 1.508153153812949e+118))) {
		VAR = fma((z / t), y, x);
	} else {
		VAR = (fma((t / (a - t)), y, (x + y)) - (z / ((a - t) / y)));
	}
	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

Original16.4
Target8.6
Herbie7.5
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -9.02018842352802e+203 or 1.508153153812949e+118 < t

    1. Initial program 31.9

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

    2. Simplified23.1

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

    4. Applied clear-num23.2

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

    6. Applied fma-udef23.3

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

    7. Simplified23.2

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

    8. Taylor expanded around inf 16.5

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

    9. Simplified11.2

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

    if -9.02018842352802e+203 < t < 1.508153153812949e+118

    1. Initial program 11.2

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

    2. Simplified8.2

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

    4. Applied clear-num8.3

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

    6. Applied fma-udef8.3

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

    7. Simplified8.1

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

    9. Applied +-commutative8.1

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

    11. Applied div-sub8.1

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

    12. Applied associate-+r-6.4

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

    13. Simplified6.2

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

  4. Final simplification7.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -9.0201884235280197 \cdot 10^{203} \lor \neg \left(t \le 1.5081531538129489 \cdot 10^{118}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a - t}, y, x + y\right) - \frac{z}{\frac{a - t}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-07) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.4754293444577233e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y))))

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