Average Error: 10.8 → 1.8
Time: 6.0s
Precision: binary64
\[\frac{x - y \cdot z}{t - a \cdot z} \]
↓
\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq -8.0439677691137 \cdot 10^{-315} \lor \neg \left(t_1 \leq 0\right) \land t_1 \leq 2.0216730270562908 \cdot 10^{+285}\right):\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
t_1 := \frac{x - y \cdot z}{t - z \cdot a}\\
\mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq -8.0439677691137 \cdot 10^{-315} \lor \neg \left(t_1 \leq 0\right) \land t_1 \leq 2.0216730270562908 \cdot 10^{+285}\right):\\
\;\;\;\;\frac{-y}{\frac{t}{z} - a}\\

\mathbf{else}:\\
\;\;\;\;t_1\\


\end{array}

(FPCore (x y z t a) :precision binary64 (/ (- x (* y z)) (- t (* a z))))
↓
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- x (* y z)) (- t (* z a)))))
   (if (or (<= t_1 (- INFINITY))
           (not
            (or (<= t_1 -8.0439677691137e-315)
                (and (not (<= t_1 0.0)) (<= t_1 2.0216730270562908e+285)))))
     (/ (- y) (- (/ t z) a))
     t_1)))
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 t_1 = (x - (y * z)) / (t - (z * a));
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !((t_1 <= -8.0439677691137e-315) || (!(t_1 <= 0.0) && (t_1 <= 2.0216730270562908e+285)))) {
		tmp = -y / ((t / z) - a);
	} else {
		tmp = t_1;
	}
	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
Herbie1.8
\[\begin{array}{l} \mathbf{if}\;z < -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z < 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0 or -8.0439677691e-315 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0 or 2.02167302705629082e285 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

    1. Initial program 41.1

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

    3. Applied div-inv_binary6441.1

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

    4. Simplified41.1

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

    5. Taylor expanded around 0 42.2

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

    6. Simplified6.5

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

    if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -8.0439677691e-315 or 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 2.02167302705629082e285

    1. Initial program 0.2

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

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty \lor \neg \left(\frac{x - y \cdot z}{t - z \cdot a} \leq -8.0439677691137 \cdot 10^{-315} \lor \neg \left(\frac{x - y \cdot z}{t - z \cdot a} \leq 0\right) \land \frac{x - y \cdot z}{t - z \cdot a} \leq 2.0216730270562908 \cdot 10^{+285}\right):\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \end{array} \]

Reproduce

herbie shell --seed 2021190 
(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))))