Average Error: 10.5 → 2.9
Time: 15.6s
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 r630636 = x;
        double r630637 = y;
        double r630638 = z;
        double r630639 = r630637 * r630638;
        double r630640 = r630636 - r630639;
        double r630641 = t;
        double r630642 = a;
        double r630643 = r630642 * r630638;
        double r630644 = r630641 - r630643;
        double r630645 = r630640 / r630644;
        return r630645;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r630646 = x;
        double r630647 = 1.0;
        double r630648 = t;
        double r630649 = a;
        double r630650 = z;
        double r630651 = r630649 * r630650;
        double r630652 = r630648 - r630651;
        double r630653 = r630647 / r630652;
        double r630654 = r630646 * r630653;
        double r630655 = y;
        double r630656 = r630648 / r630650;
        double r630657 = r630656 - r630649;
        double r630658 = r630655 / r630657;
        double r630659 = r630654 - r630658;
        return r630659;
}

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
Herbie2.9
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

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 pow18.2

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

  9. Applied pow18.2

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

  10. Applied pow-prod-down8.2

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

  11. Simplified2.8

    \[\leadsto \frac{x}{t - a \cdot z} - {\color{blue}{\left(\frac{y}{\frac{t}{z} - a}\right)}}^{1}\]
  12. Using strategy rm

  13. Applied div-inv2.9

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

  14. Final simplification2.9

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

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))))