Average Error: 7.7 → 7.8
Time: 17.6s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\frac{1}{a} \cdot \frac{\left(x \cdot y - 9 \cdot \left(t \cdot z\right)\right) + \left(z \cdot 9\right) \cdot \left(\left(-t\right) + t\right)}{2}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\frac{1}{a} \cdot \frac{\left(x \cdot y - 9 \cdot \left(t \cdot z\right)\right) + \left(z \cdot 9\right) \cdot \left(\left(-t\right) + t\right)}{2}
double f(double x, double y, double z, double t, double a) {
        double r34432 = x;
        double r34433 = y;
        double r34434 = r34432 * r34433;
        double r34435 = z;
        double r34436 = 9.0;
        double r34437 = r34435 * r34436;
        double r34438 = t;
        double r34439 = r34437 * r34438;
        double r34440 = r34434 - r34439;
        double r34441 = a;
        double r34442 = 2.0;
        double r34443 = r34441 * r34442;
        double r34444 = r34440 / r34443;
        return r34444;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r34445 = 1.0;
        double r34446 = a;
        double r34447 = r34445 / r34446;
        double r34448 = x;
        double r34449 = y;
        double r34450 = r34448 * r34449;
        double r34451 = 9.0;
        double r34452 = t;
        double r34453 = z;
        double r34454 = r34452 * r34453;
        double r34455 = r34451 * r34454;
        double r34456 = r34450 - r34455;
        double r34457 = r34453 * r34451;
        double r34458 = -r34452;
        double r34459 = r34458 + r34452;
        double r34460 = r34457 * r34459;
        double r34461 = r34456 + r34460;
        double r34462 = 2.0;
        double r34463 = r34461 / r34462;
        double r34464 = r34447 * r34463;
        return r34464;
}

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

Original7.7
Target5.6
Herbie7.8
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709043451944897028999329376 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.144030707833976090627817222818061808815 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Initial program 7.7

    \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity7.7

    \[\leadsto \frac{\color{blue}{1 \cdot \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)}}{a \cdot 2}\]

  4. Applied times-frac7.8

    \[\leadsto \color{blue}{\frac{1}{a} \cdot \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{2}}\]
  5. Using strategy rm

  6. Applied prod-diff7.8

    \[\leadsto \frac{1}{a} \cdot \frac{\color{blue}{\mathsf{fma}\left(x, y, -t \cdot \left(z \cdot 9\right)\right) + \mathsf{fma}\left(-t, z \cdot 9, t \cdot \left(z \cdot 9\right)\right)}}{2}\]

  7. Simplified7.8

    \[\leadsto \frac{1}{a} \cdot \frac{\color{blue}{\left(x \cdot y - 9 \cdot \left(t \cdot z\right)\right)} + \mathsf{fma}\left(-t, z \cdot 9, t \cdot \left(z \cdot 9\right)\right)}{2}\]

  8. Simplified7.8

    \[\leadsto \frac{1}{a} \cdot \frac{\left(x \cdot y - 9 \cdot \left(t \cdot z\right)\right) + \color{blue}{\left(z \cdot 9\right) \cdot \left(\left(-t\right) + t\right)}}{2}\]

  9. Final simplification7.8

    \[\leadsto \frac{1}{a} \cdot \frac{\left(x \cdot y - 9 \cdot \left(t \cdot z\right)\right) + \left(z \cdot 9\right) \cdot \left(\left(-t\right) + t\right)}{2}\]

Reproduce

herbie shell --seed 2019315 +o rules:numerics
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :herbie-target
  (if (< a -2.090464557976709e86) (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z)))) (if (< a 2.14403070783397609e99) (/ (- (* x y) (* z (* 9 t))) (* a 2)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9) t)) (* a 2)))