Average Error: 10.1 → 1.8
Time: 2.5min
Precision: binary64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.71856029181173682 \cdot 10^{-77} \lor \neg \left(z \le 1.73940474756971312 \cdot 10^{-69}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{1}{\frac{\frac{t}{z} - a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.71856029181173682 \cdot 10^{-77} \lor \neg \left(z \le 1.73940474756971312 \cdot 10^{-69}\right):\\
\;\;\;\;\frac{x}{t - a \cdot z} - \frac{1}{\frac{\frac{t}{z} - a}{y}}\\

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

\end{array}
double code(double x, double y, double z, double t, double a) {
	return (((double) (x - ((double) (y * z)))) / ((double) (t - ((double) (a * z)))));
}
↓
double code(double x, double y, double z, double t, double a) {
	double VAR;
	if (((z <= -2.718560291811737e-77) || !(z <= 1.739404747569713e-69))) {
		VAR = ((double) ((x / ((double) (t - ((double) (a * z))))) - (1.0 / (((double) ((t / z) - a)) / y))));
	} else {
		VAR = ((double) (((double) (x - ((double) (y * z)))) * (1.0 / ((double) (t - ((double) (a * z)))))));
	}
	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

Original10.1
Target1.6
Herbie1.8
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.71856029181173682e-77 or 1.73940474756971312e-69 < z

    1. Initial program 16.4

      \[\frac{x - y \cdot z}{t - a \cdot z}\]
    2. Using strategy rm

    3. Applied div-sub16.4

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

    4. Simplified2.5

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

    6. Applied clear-num2.8

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

    if -2.71856029181173682e-77 < z < 1.73940474756971312e-69

    1. Initial program 0.1

      \[\frac{x - y \cdot z}{t - a \cdot z}\]
    2. Using strategy rm

    3. Applied div-inv0.3

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

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.71856029181173682 \cdot 10^{-77} \lor \neg \left(z \le 1.73940474756971312 \cdot 10^{-69}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{1}{\frac{\frac{t}{z} - a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -32113435955957344.0) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

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