Average Error: 10.8 → 1.7
Time: 8.9s
Precision: binary64
Cost: 1416
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -1.8309315438264195 \cdot 10^{-90} \lor \neg \left(z \leq 5.350417554604714 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{z \cdot y}{t - z \cdot a}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -1.8309315438264195 \cdot 10^{-90} \lor \neg \left(z \leq 5.350417554604714 \cdot 10^{-20}\right):\\
\;\;\;\;\frac{x}{t - z \cdot a} - \frac{y}{\frac{t}{z} - a}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{t - z \cdot a} - \frac{z \cdot y}{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 (or (<= z -1.8309315438264195e-90) (not (<= z 5.350417554604714e-20)))
   (- (/ x (- t (* z a))) (/ y (- (/ t z) a)))
   (- (/ x (- t (* z a))) (/ (* z y) (- 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 ((z <= -1.8309315438264195e-90) || !(z <= 5.350417554604714e-20)) {
		tmp = (x / (t - (z * a))) - (y / ((t / z) - a));
	} else {
		tmp = (x / (t - (z * a))) - ((z * y) / (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
Target1.8
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;z < -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z < 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Alternatives

Alternative 1
Error1.7
Cost1288
\[\begin{array}{l} \mathbf{if}\;z \leq -3.5402791921895114 \cdot 10^{-89} \lor \neg \left(z \leq 1.0806495670300669 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \end{array}\]
Alternative 2
Error5.8
Cost1346
\[\begin{array}{l} \mathbf{if}\;z \leq -2.849008921045424 \cdot 10^{+137}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 8.116864161191549 \cdot 10^{+104}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \end{array}\]
Alternative 3
Error18.2
Cost1475
\[\begin{array}{l} \mathbf{if}\;z \leq -2.5631144847457444 \cdot 10^{-85}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 1.484576833844367 \cdot 10^{-223}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 1.8686172713538164 \cdot 10^{+21}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \end{array}\]
Alternative 4
Error18.4
Cost1411
\[\begin{array}{l} \mathbf{if}\;z \leq -2.5631144847457444 \cdot 10^{-85}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 7.694522913990753 \cdot 10^{-224}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 126256121638203.44:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array}\]
Alternative 5
Error18.5
Cost776
\[\begin{array}{l} \mathbf{if}\;z \leq -2.5631144847457444 \cdot 10^{-85} \lor \neg \left(z \leq 3.047966088704956 \cdot 10^{+25}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array}\]
Alternative 6
Error22.1
Cost776
\[\begin{array}{l} \mathbf{if}\;z \leq -3.1272847213492848 \cdot 10^{+66} \lor \neg \left(z \leq 3.181744006878424 \cdot 10^{+52}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array}\]
Alternative 7
Error29.8
Cost520
\[\begin{array}{l} \mathbf{if}\;z \leq -2.1913474323769692 \cdot 10^{-91} \lor \neg \left(z \leq 2.847776845149297 \cdot 10^{+29}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array}\]
Alternative 8
Error41.8
Cost513
\[\begin{array}{l} \mathbf{if}\;t \leq 6.059723920873667 \cdot 10^{+226}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
Alternative 9
Error56.3
Cost64
\[0\]
Alternative 10
Error61.7
Cost64
\[1\]

Error

Derivation

  1. Split input into 2 regimes

  2. if z < -1.8309315438264195e-90 or 5.3504175546047138e-20 < z

    1. Initial program 18.6

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

    3. Applied div-sub_binary64_2020218.6

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

    4. Simplified18.6

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

    5. Simplified18.6

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

    7. Applied *-un-lft-identity_binary64_2019718.6

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

    8. Applied times-frac_binary64_2020311.9

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

    9. Simplified11.9

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

    10. Taylor expanded around 0 18.6

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

    11. Simplified2.9

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

    12. Simplified2.9

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

    if -1.8309315438264195e-90 < z < 5.3504175546047138e-20

    1. Initial program 0.1

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

    3. Applied div-sub_binary64_202020.1

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

    4. Simplified0.1

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

    5. Simplified0.1

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

    6. Simplified0.1

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

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8309315438264195 \cdot 10^{-90} \lor \neg \left(z \leq 5.350417554604714 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{z \cdot y}{t - z \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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))))