Average Error: 7.4 → 2.4
Time: 4.8s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\frac{y \cdot \left(\frac{1}{t \cdot z - x} \cdot z\right) + x}{\left(x + 1\right) \cdot 1} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\frac{y \cdot \left(\frac{1}{t \cdot z - x} \cdot z\right) + x}{\left(x + 1\right) \cdot 1} - \frac{\frac{x}{t \cdot z - x}}{x + 1}
double f(double x, double y, double z, double t) {
        double r630194 = x;
        double r630195 = y;
        double r630196 = z;
        double r630197 = r630195 * r630196;
        double r630198 = r630197 - r630194;
        double r630199 = t;
        double r630200 = r630199 * r630196;
        double r630201 = r630200 - r630194;
        double r630202 = r630198 / r630201;
        double r630203 = r630194 + r630202;
        double r630204 = 1.0;
        double r630205 = r630194 + r630204;
        double r630206 = r630203 / r630205;
        return r630206;
}
↓
double f(double x, double y, double z, double t) {
        double r630207 = y;
        double r630208 = 1.0;
        double r630209 = t;
        double r630210 = z;
        double r630211 = r630209 * r630210;
        double r630212 = x;
        double r630213 = r630211 - r630212;
        double r630214 = r630208 / r630213;
        double r630215 = r630214 * r630210;
        double r630216 = r630207 * r630215;
        double r630217 = r630216 + r630212;
        double r630218 = 1.0;
        double r630219 = r630212 + r630218;
        double r630220 = r630219 * r630208;
        double r630221 = r630217 / r630220;
        double r630222 = r630212 / r630213;
        double r630223 = r630222 / r630219;
        double r630224 = r630221 - r630223;
        return r630224;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.4
Target0.4
Herbie2.4
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Initial program 7.4

    \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
  2. Using strategy rm

  3. Applied div-sub7.4

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

  4. Applied associate-+r-7.4

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

  5. Applied div-sub7.4

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

  6. Simplified4.6

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1}} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\]
  7. Using strategy rm

  8. Applied fma-udef4.6

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

  10. Applied div-inv4.6

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

  11. Applied associate-*l*2.4

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

  12. Final simplification2.4

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

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1)))