Average Error: 7.8 → 5.1
Time: 4.6s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -4.019204525666223946553497990031878832169 \cdot 10^{163} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 3.585567167112418833993175021914148886831 \cdot 10^{159}\right):\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2} - \frac{9 \cdot t}{2} \cdot \frac{z - \left(\left(-z\right) + z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}}\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -4.019204525666223946553497990031878832169 \cdot 10^{163} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 3.585567167112418833993175021914148886831 \cdot 10^{159}\right):\\
\;\;\;\;\frac{x \cdot y}{a \cdot 2} - \frac{9 \cdot t}{2} \cdot \frac{z - \left(\left(-z\right) + z\right)}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r773027 = x;
        double r773028 = y;
        double r773029 = r773027 * r773028;
        double r773030 = z;
        double r773031 = 9.0;
        double r773032 = r773030 * r773031;
        double r773033 = t;
        double r773034 = r773032 * r773033;
        double r773035 = r773029 - r773034;
        double r773036 = a;
        double r773037 = 2.0;
        double r773038 = r773036 * r773037;
        double r773039 = r773035 / r773038;
        return r773039;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r773040 = x;
        double r773041 = y;
        double r773042 = r773040 * r773041;
        double r773043 = z;
        double r773044 = 9.0;
        double r773045 = r773043 * r773044;
        double r773046 = t;
        double r773047 = r773045 * r773046;
        double r773048 = r773042 - r773047;
        double r773049 = -4.019204525666224e+163;
        bool r773050 = r773048 <= r773049;
        double r773051 = 3.585567167112419e+159;
        bool r773052 = r773048 <= r773051;
        double r773053 = !r773052;
        bool r773054 = r773050 || r773053;
        double r773055 = a;
        double r773056 = 2.0;
        double r773057 = r773055 * r773056;
        double r773058 = r773042 / r773057;
        double r773059 = r773044 * r773046;
        double r773060 = r773059 / r773056;
        double r773061 = -r773043;
        double r773062 = r773061 + r773043;
        double r773063 = r773043 - r773062;
        double r773064 = r773063 / r773055;
        double r773065 = r773060 * r773064;
        double r773066 = r773058 - r773065;
        double r773067 = 1.0;
        double r773068 = r773057 / r773048;
        double r773069 = r773067 / r773068;
        double r773070 = r773054 ? r773066 : r773069;
        return r773070;
}

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

Original7.8
Target5.4
Herbie5.1
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709043451944897028999329376 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.144030707833976090627817222818061808815 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (* x y) (* (* z 9.0) t)) < -4.019204525666224e+163 or 3.585567167112419e+159 < (- (* x y) (* (* z 9.0) t))

    1. Initial program 22.8

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

    3. Applied associate-*l*22.7

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

    5. Applied prod-diff22.7

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

    6. Simplified22.7

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

    7. Simplified22.7

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

    9. Applied associate-+l-22.7

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

    10. Applied div-sub22.7

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

    11. Simplified13.5

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

    if -4.019204525666224e+163 < (- (* x y) (* (* z 9.0) t)) < 3.585567167112419e+159

    1. Initial program 1.0

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

    3. Applied clear-num1.3

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -4.019204525666223946553497990031878832169 \cdot 10^{163} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 3.585567167112418833993175021914148886831 \cdot 10^{159}\right):\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2} - \frac{9 \cdot t}{2} \cdot \frac{z - \left(\left(-z\right) + z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{x \cdot y - \left(z \cdot 9\right) \cdot t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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 -2.090464557976709e+86) (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z)))) (if (< a 2.144030707833976e+99) (/ (- (* x y) (* z (* 9 t))) (* a 2)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

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