Average Error: 10.5 → 3.0
Time: 9.4s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[x \cdot \frac{1}{t - a \cdot z} - y \cdot \frac{1}{\frac{t}{z} - a}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
x \cdot \frac{1}{t - a \cdot z} - y \cdot \frac{1}{\frac{t}{z} - a}
double f(double x, double y, double z, double t, double a) {
        double r746540 = x;
        double r746541 = y;
        double r746542 = z;
        double r746543 = r746541 * r746542;
        double r746544 = r746540 - r746543;
        double r746545 = t;
        double r746546 = a;
        double r746547 = r746546 * r746542;
        double r746548 = r746545 - r746547;
        double r746549 = r746544 / r746548;
        return r746549;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r746550 = x;
        double r746551 = 1.0;
        double r746552 = t;
        double r746553 = a;
        double r746554 = z;
        double r746555 = r746553 * r746554;
        double r746556 = r746552 - r746555;
        double r746557 = r746551 / r746556;
        double r746558 = r746550 * r746557;
        double r746559 = y;
        double r746560 = r746552 / r746554;
        double r746561 = r746560 - r746553;
        double r746562 = r746551 / r746561;
        double r746563 = r746559 * r746562;
        double r746564 = r746558 - r746563;
        return r746564;
}

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

Original10.5
Target1.6
Herbie3.0
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Initial program 10.5

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

  3. Applied div-sub10.5

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

  4. Simplified8.1

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

  6. Applied clear-num8.2

    \[\leadsto \frac{x}{t - a \cdot z} - y \cdot \color{blue}{\frac{1}{\frac{t - a \cdot z}{z}}}\]
  7. Using strategy rm

  8. Applied div-sub8.2

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

  9. Simplified2.9

    \[\leadsto \frac{x}{t - a \cdot z} - y \cdot \frac{1}{\frac{t}{z} - \color{blue}{a \cdot 1}}\]
  10. Using strategy rm

  11. Applied div-inv3.0

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

  12. Final simplification3.0

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -32113435955957344) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

  (/ (- x (* y z)) (- t (* a z))))