Average Error: 10.5 → 2.8
Time: 10.3s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}
double f(double x, double y, double z, double t, double a) {
        double r844800 = x;
        double r844801 = y;
        double r844802 = z;
        double r844803 = r844801 * r844802;
        double r844804 = r844800 - r844803;
        double r844805 = t;
        double r844806 = a;
        double r844807 = r844806 * r844802;
        double r844808 = r844805 - r844807;
        double r844809 = r844804 / r844808;
        return r844809;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r844810 = x;
        double r844811 = t;
        double r844812 = a;
        double r844813 = z;
        double r844814 = r844812 * r844813;
        double r844815 = r844811 - r844814;
        double r844816 = r844810 / r844815;
        double r844817 = y;
        double r844818 = r844811 / r844813;
        double r844819 = r844818 - r844812;
        double r844820 = r844817 / r844819;
        double r844821 = r844816 - r844820;
        return r844821;
}

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
Herbie2.8
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

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. Taylor expanded around 0 2.9

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

  9. Applied pow12.9

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

  10. Applied pow12.9

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

  11. Applied pow-prod-down2.9

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

  12. Simplified2.8

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

  13. Final simplification2.8

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

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))))