Average Error: 8.0 → 7.9
Time: 8.0s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\frac{y}{\frac{a}{0.5 \cdot x}} - \left(4.5 \cdot \frac{z}{a}\right) \cdot t\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\frac{y}{\frac{a}{0.5 \cdot x}} - \left(4.5 \cdot \frac{z}{a}\right) \cdot t
double f(double x, double y, double z, double t, double a) {
        double r691052 = x;
        double r691053 = y;
        double r691054 = r691052 * r691053;
        double r691055 = z;
        double r691056 = 9.0;
        double r691057 = r691055 * r691056;
        double r691058 = t;
        double r691059 = r691057 * r691058;
        double r691060 = r691054 - r691059;
        double r691061 = a;
        double r691062 = 2.0;
        double r691063 = r691061 * r691062;
        double r691064 = r691060 / r691063;
        return r691064;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r691065 = y;
        double r691066 = a;
        double r691067 = 0.5;
        double r691068 = x;
        double r691069 = r691067 * r691068;
        double r691070 = r691066 / r691069;
        double r691071 = r691065 / r691070;
        double r691072 = 4.5;
        double r691073 = z;
        double r691074 = r691073 / r691066;
        double r691075 = r691072 * r691074;
        double r691076 = t;
        double r691077 = r691075 * r691076;
        double r691078 = r691071 - r691077;
        return r691078;
}

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

Original8.0
Target5.6
Herbie7.9
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709043451944897028999329376 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.144030707833976090627817222818061808815 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (* x y) (* (* z 9.0) t)) < -inf.0 or 3.635549497414519e+213 < (- (* x y) (* (* z 9.0) t))

    1. Initial program 41.9

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

    2. Taylor expanded around 0 41.5

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt41.6

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}} - 4.5 \cdot \frac{t \cdot z}{a}\]

    5. Applied times-frac23.2

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right)} - 4.5 \cdot \frac{t \cdot z}{a}\]
    6. Using strategy rm

    7. Applied associate-/l*1.3

      \[\leadsto 0.5 \cdot \left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right) - 4.5 \cdot \color{blue}{\frac{t}{\frac{a}{z}}}\]

    if -inf.0 < (- (* x y) (* (* z 9.0) t)) < 3.635549497414519e+213

    1. Initial program 0.8

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

    2. Taylor expanded around 0 0.8

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt0.8

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \color{blue}{\left(\left(\sqrt[3]{4.5} \cdot \sqrt[3]{4.5}\right) \cdot \sqrt[3]{4.5}\right)} \cdot \frac{t \cdot z}{a}\]

    5. Applied associate-*l*0.9

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \color{blue}{\left(\sqrt[3]{4.5} \cdot \sqrt[3]{4.5}\right) \cdot \left(\sqrt[3]{4.5} \cdot \frac{t \cdot z}{a}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification7.9

    \[\leadsto \frac{y}{\frac{a}{0.5 \cdot x}} - \left(4.5 \cdot \frac{z}{a}\right) \cdot t\]

Reproduce

herbie shell --seed 1978988140 
(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)))