Average Error: 10.9 → 1.6
Time: 3.6s
Precision: binary64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -6.120866528703716 \cdot 10^{-47} \lor \neg \left(z \leq 1.15914428873438 \cdot 10^{-119}\right):\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -6.120866528703716 \cdot 10^{-47} \lor \neg \left(z \leq 1.15914428873438 \cdot 10^{-119}\right):\\
\;\;\;\;\frac{x}{t - z \cdot a} - \frac{y}{\frac{t}{z} - a}\\

\mathbf{else}:\\
\;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\

\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 <= -6.120866528703716e-47) || !(z <= 1.15914428873438e-119))) {
		VAR = ((double) ((x / ((double) (t - ((double) (z * a))))) - (y / ((double) ((t / z) - a)))));
	} else {
		VAR = (((double) (x - ((double) (z * y)))) / ((double) (t - ((double) (z * a)))));
	}
	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

Original10.9
Target1.6
Herbie1.6
\[\begin{array}{l} \mathbf{if}\;z < -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z < 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -6.12086652870371594e-47 or 1.15914428873438e-119 < z

    1. Initial program 17.1

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

    3. Applied div-sub17.1

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

    4. Simplified17.1

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

    5. Simplified10.7

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

    7. Applied sub-neg10.7

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

    8. Simplified2.5

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

    if -6.12086652870371594e-47 < z < 1.15914428873438e-119

    1. Initial program 0.1

      \[\frac{x - y \cdot z}{t - a \cdot z}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.120866528703716 \cdot 10^{-47} \lor \neg \left(z \leq 1.15914428873438 \cdot 10^{-119}\right):\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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))))