Average Error: 10.6 → 10.7
Time: 4.2s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\frac{1}{t - a \cdot z} \cdot \left(x - y \cdot z\right)\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\frac{1}{t - a \cdot z} \cdot \left(x - y \cdot z\right)
double f(double x, double y, double z, double t, double a) {
        double r671821 = x;
        double r671822 = y;
        double r671823 = z;
        double r671824 = r671822 * r671823;
        double r671825 = r671821 - r671824;
        double r671826 = t;
        double r671827 = a;
        double r671828 = r671827 * r671823;
        double r671829 = r671826 - r671828;
        double r671830 = r671825 / r671829;
        return r671830;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r671831 = 1.0;
        double r671832 = t;
        double r671833 = a;
        double r671834 = z;
        double r671835 = r671833 * r671834;
        double r671836 = r671832 - r671835;
        double r671837 = r671831 / r671836;
        double r671838 = x;
        double r671839 = y;
        double r671840 = r671839 * r671834;
        double r671841 = r671838 - r671840;
        double r671842 = r671837 * r671841;
        return r671842;
}

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

Original10.6
Target1.6
Herbie10.7
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Initial program 10.6

    \[\frac{x - y \cdot z}{t - a \cdot z}\]
  2. Using strategy rm

  3. Applied clear-num10.9

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

  5. Applied div-inv10.9

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

  6. Applied add-cube-cbrt10.9

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\left(t - a \cdot z\right) \cdot \frac{1}{x - y \cdot z}}\]

  7. Applied times-frac10.7

    \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{t - a \cdot z} \cdot \frac{\sqrt[3]{1}}{\frac{1}{x - y \cdot z}}}\]

  8. Simplified10.7

    \[\leadsto \color{blue}{\frac{1}{t - a \cdot z}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{x - y \cdot z}}\]

  9. Simplified10.7

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

  10. Final simplification10.7

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

Reproduce

herbie shell --seed 2020081 +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))))