Average Error: 10.2 → 5.4
Time: 9.1s
Precision: binary64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -1.4327903729396 \cdot 10^{-322}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0 \lor \neg \left(\frac{x - y \cdot z}{t - z \cdot a} \leq 1.799119497863352 \cdot 10^{+294}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y \cdot z}{t - z \cdot a}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\
\;\;\;\;\frac{y}{a}\\

\mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -1.4327903729396 \cdot 10^{-322}:\\
\;\;\;\;\frac{x - y \cdot z}{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 1.799119497863352 \cdot 10^{+294}\right):\\
\;\;\;\;\frac{y}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{t - z \cdot a} - \frac{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))) (- INFINITY))
   (/ y a)
   (if (<= (/ (- x (* y z)) (- t (* z a))) -1.4327903729396e-322)
     (/ (- x (* y z)) (- t (* z a)))
     (if (or (<= (/ (- x (* y z)) (- t (* z a))) 0.0)
             (not (<= (/ (- x (* y z)) (- t (* z a))) 1.799119497863352e+294)))
       (/ y a)
       (- (/ x (- t (* z a))) (/ (* 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))) <= -((double) INFINITY)) {
		tmp = y / a;
	} else if (((x - (y * z)) / (t - (z * a))) <= -1.4327903729396e-322) {
		tmp = (x - (y * z)) / (t - (z * a));
	} else if ((((x - (y * z)) / (t - (z * a))) <= 0.0) || !(((x - (y * z)) / (t - (z * a))) <= 1.799119497863352e+294)) {
		tmp = y / a;
	} else {
		tmp = (x / (t - (z * a))) - ((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

Original10.2
Target1.8
Herbie5.4
\[\begin{array}{l} \mathbf{if}\;z < -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z < 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0 or -1.43279e-322 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0 or 1.79911949786335196e294 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

    1. Initial program 40.8

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

    2. Taylor expanded around inf 21.2

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

    if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.43279e-322

    1. Initial program 0.2

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

    if -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1.79911949786335196e294

    1. Initial program 0.2

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

    3. Applied div-sub_binary64_174740.2

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

  4. Final simplification5.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -1.4327903729396 \cdot 10^{-322}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0 \lor \neg \left(\frac{x - y \cdot z}{t - z \cdot a} \leq 1.799119497863352 \cdot 10^{+294}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y \cdot z}{t - z \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2021043 
(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))))