Average Error: 10.6 → 3.1
Time: 3.4s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[x \cdot \frac{1}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
x \cdot \frac{1}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}
double f(double x, double y, double z, double t, double a) {
        double r820444 = x;
        double r820445 = y;
        double r820446 = z;
        double r820447 = r820445 * r820446;
        double r820448 = r820444 - r820447;
        double r820449 = t;
        double r820450 = a;
        double r820451 = r820450 * r820446;
        double r820452 = r820449 - r820451;
        double r820453 = r820448 / r820452;
        return r820453;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r820454 = x;
        double r820455 = 1.0;
        double r820456 = t;
        double r820457 = a;
        double r820458 = z;
        double r820459 = r820457 * r820458;
        double r820460 = r820456 - r820459;
        double r820461 = r820455 / r820460;
        double r820462 = r820454 * r820461;
        double r820463 = y;
        double r820464 = r820456 / r820458;
        double r820465 = r820464 - r820457;
        double r820466 = r820463 / r820465;
        double r820467 = r820462 - r820466;
        return r820467;
}

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

Original10.6
Target1.7
Herbie3.1
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Initial program 10.6

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

  3. Applied div-sub10.6

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

  5. Applied associate-/l*8.2

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

  7. Applied div-sub8.2

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

  8. Simplified3.1

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

  10. Applied div-inv3.1

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

  11. Final simplification3.1

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

Reproduce

herbie shell --seed 2020062 
(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))))