Average Error: 10.5 → 3.0
Time: 11.8s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[x \cdot \frac{1}{t - a \cdot z} - y \cdot \frac{1}{\frac{t}{z} - a}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
x \cdot \frac{1}{t - a \cdot z} - y \cdot \frac{1}{\frac{t}{z} - a}
double f(double x, double y, double z, double t, double a) {
        double r794856 = x;
        double r794857 = y;
        double r794858 = z;
        double r794859 = r794857 * r794858;
        double r794860 = r794856 - r794859;
        double r794861 = t;
        double r794862 = a;
        double r794863 = r794862 * r794858;
        double r794864 = r794861 - r794863;
        double r794865 = r794860 / r794864;
        return r794865;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r794866 = x;
        double r794867 = 1.0;
        double r794868 = t;
        double r794869 = a;
        double r794870 = z;
        double r794871 = r794869 * r794870;
        double r794872 = r794868 - r794871;
        double r794873 = r794867 / r794872;
        double r794874 = r794866 * r794873;
        double r794875 = y;
        double r794876 = r794868 / r794870;
        double r794877 = r794876 - r794869;
        double r794878 = r794867 / r794877;
        double r794879 = r794875 * r794878;
        double r794880 = r794874 - r794879;
        return r794880;
}

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
Herbie3.0
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

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 div-inv3.0

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

  10. Final simplification3.0

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

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))))