Average Error: 10.5 → 2.8
Time: 12.2s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\frac{x}{t - a \cdot z} + \left(-\frac{y}{\frac{t}{z} - a}\right)\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\frac{x}{t - a \cdot z} + \left(-\frac{y}{\frac{t}{z} - a}\right)
double f(double x, double y, double z, double t, double a) {
        double r859777 = x;
        double r859778 = y;
        double r859779 = z;
        double r859780 = r859778 * r859779;
        double r859781 = r859777 - r859780;
        double r859782 = t;
        double r859783 = a;
        double r859784 = r859783 * r859779;
        double r859785 = r859782 - r859784;
        double r859786 = r859781 / r859785;
        return r859786;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r859787 = x;
        double r859788 = t;
        double r859789 = a;
        double r859790 = z;
        double r859791 = r859789 * r859790;
        double r859792 = r859788 - r859791;
        double r859793 = r859787 / r859792;
        double r859794 = y;
        double r859795 = r859788 / r859790;
        double r859796 = r859795 - r859789;
        double r859797 = r859794 / r859796;
        double r859798 = -r859797;
        double r859799 = r859793 + r859798;
        return r859799;
}

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 sub-neg2.9

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

  10. Simplified2.8

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

  11. Final simplification2.8

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

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))))