Average Error: 10.5 → 2.9
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 r713683 = x;
        double r713684 = y;
        double r713685 = z;
        double r713686 = r713684 * r713685;
        double r713687 = r713683 - r713686;
        double r713688 = t;
        double r713689 = a;
        double r713690 = r713689 * r713685;
        double r713691 = r713688 - r713690;
        double r713692 = r713687 / r713691;
        return r713692;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r713693 = x;
        double r713694 = 1.0;
        double r713695 = t;
        double r713696 = a;
        double r713697 = z;
        double r713698 = r713696 * r713697;
        double r713699 = r713695 - r713698;
        double r713700 = r713694 / r713699;
        double r713701 = r713693 * r713700;
        double r713702 = y;
        double r713703 = r713695 / r713697;
        double r713704 = r713703 - r713696;
        double r713705 = r713702 / r713704;
        double r713706 = r713701 - r713705;
        return r713706;
}

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 *-un-lft-identity10.5

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

  5. Applied div-sub10.5

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

  6. Simplified10.5

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

  7. Simplified8.1

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

  9. Applied div-sub8.1

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

  10. Simplified2.8

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

  12. Applied div-inv2.9

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

  13. Final simplification2.9

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

Reproduce

herbie shell --seed 2020047 
(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))))