Average Error: 10.8 → 5.7
Time: 19.5s
Precision: binary64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -7.46567153473929 \cdot 10^{+293}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 2.5016904252996703 \cdot 10^{+299}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{y}}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -7.46567153473929 \cdot 10^{+293}:\\
\;\;\;\;\frac{y}{a}\\

\mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 2.5016904252996703 \cdot 10^{+299}:\\
\;\;\;\;\frac{x}{t - z \cdot a} - \frac{y \cdot z}{t - z \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{a}{y}}\\

\end{array}
(FPCore (x y z t a) :precision binary64 (/ (- x (* y z)) (- t (* a z))))
↓
(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- x (* y z)) (- t (* z a))) -7.46567153473929e+293)
   (/ y a)
   (if (<= (/ (- x (* y z)) (- t (* z a))) 2.5016904252996703e+299)
     (- (/ x (- t (* z a))) (/ (* y z) (- t (* z a))))
     (/ 1.0 (/ a y)))))
double code(double x, double y, double z, double t, double a) {
	return (x - (y * z)) / (t - (a * z));
}
↓
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x - (y * z)) / (t - (z * a))) <= -7.46567153473929e+293) {
		tmp = y / a;
	} else if (((x - (y * z)) / (t - (z * a))) <= 2.5016904252996703e+299) {
		tmp = (x / (t - (z * a))) - ((y * z) / (t - (z * a)));
	} else {
		tmp = 1.0 / (a / y);
	}
	return tmp;
}

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

Original10.8
Target1.7
Herbie5.7
\[\begin{array}{l} \mathbf{if}\;z < -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z < 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -7.46567153473928992e293

    1. Initial program 57.3

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

    2. Taylor expanded around inf 26.7

      \[\leadsto \color{blue}{\frac{y}{a}}\]

    if -7.46567153473928992e293 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 2.50169042529967034e299

    1. Initial program 4.3

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

    2. Taylor expanded around 0 4.3

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

    if 2.50169042529967034e299 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

    1. Initial program 62.2

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

    3. Applied clear-num_binary64_2190162.2

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

    4. Simplified62.2

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

    5. Taylor expanded around inf 12.4

      \[\leadsto \frac{1}{\color{blue}{\frac{a}{y}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification5.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -7.46567153473929 \cdot 10^{+293}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 2.5016904252996703 \cdot 10^{+299}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021023 
(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))))