Average Error: 10.8 → 3.8
Time: 8.8s
Precision: binary64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -1.4282238240949 \cdot 10^{-312}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{z}{\sqrt[3]{t - z \cdot a} \cdot \sqrt[3]{t - z \cdot a}} \cdot \frac{y}{\sqrt[3]{t - z \cdot a}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0 \lor \neg \left(\frac{x - y \cdot z}{t - z \cdot a} \leq 3.932088919627915 \cdot 10^{+305}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -1.4282238240949 \cdot 10^{-312}:\\
\;\;\;\;\frac{x}{t - z \cdot a} - \frac{z}{\sqrt[3]{t - z \cdot a} \cdot \sqrt[3]{t - z \cdot a}} \cdot \frac{y}{\sqrt[3]{t - z \cdot a}}\\

\mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0 \lor \neg \left(\frac{x - y \cdot z}{t - z \cdot a} \leq 3.932088919627915 \cdot 10^{+305}\right):\\
\;\;\;\;\frac{y - \frac{x}{z}}{a}\\

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

\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))) -1.4282238240949e-312)
   (-
    (/ x (- t (* z a)))
    (*
     (/ z (* (cbrt (- t (* z a))) (cbrt (- t (* z a)))))
     (/ y (cbrt (- t (* z a))))))
   (if (or (<= (/ (- x (* y z)) (- t (* z a))) 0.0)
           (not (<= (/ (- x (* y z)) (- t (* z a))) 3.932088919627915e+305)))
     (/ (- y (/ x z)) a)
     (/ (- x (* y z)) (- t (* z a))))))
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))) <= -1.4282238240949e-312) {
		tmp = (x / (t - (z * a))) - ((z / (cbrt(t - (z * a)) * cbrt(t - (z * a)))) * (y / cbrt(t - (z * a))));
	} else if ((((x - (y * z)) / (t - (z * a))) <= 0.0) || !(((x - (y * z)) / (t - (z * a))) <= 3.932088919627915e+305)) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (y * z)) / (t - (z * a));
	}
	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
Target1.9
Herbie3.8
\[\begin{array}{l} \mathbf{if}\;z < -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z < 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.42822382409e-312

    1. Initial program 4.8

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

    3. Applied div-sub_binary64_208844.8

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

    4. Simplified4.8

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

    5. Simplified4.8

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

    7. Applied add-cube-cbrt_binary64_209145.2

      \[\leadsto \frac{x}{t - z \cdot a} - \frac{z \cdot y}{\color{blue}{\left(\sqrt[3]{t - z \cdot a} \cdot \sqrt[3]{t - z \cdot a}\right) \cdot \sqrt[3]{t - z \cdot a}}}\]

    8. Applied times-frac_binary64_208852.0

      \[\leadsto \frac{x}{t - z \cdot a} - \color{blue}{\frac{z}{\sqrt[3]{t - z \cdot a} \cdot \sqrt[3]{t - z \cdot a}} \cdot \frac{y}{\sqrt[3]{t - z \cdot a}}}\]

    if -1.42822382409e-312 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0 or 3.9320889196279146e305 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

    1. Initial program 39.8

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

    3. Applied div-sub_binary64_2088439.8

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

    4. Simplified39.8

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

    5. Simplified39.8

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

    6. Taylor expanded around 0 20.3

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

    7. Simplified13.2

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

    if 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 3.9320889196279146e305

    1. Initial program 0.2

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

  4. Final simplification3.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -1.4282238240949 \cdot 10^{-312}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{z}{\sqrt[3]{t - z \cdot a} \cdot \sqrt[3]{t - z \cdot a}} \cdot \frac{y}{\sqrt[3]{t - z \cdot a}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0 \lor \neg \left(\frac{x - y \cdot z}{t - z \cdot a} \leq 3.932088919627915 \cdot 10^{+305}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2021019 
(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))))