Average Error: 7.7 → 4.2
Time: 15.8s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot 2 \le -3.077577436686377797275553991166215453874 \cdot 10^{-99}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \cdot 2 \le 6474202012076836110009892864:\\ \;\;\;\;\frac{0.5 \cdot \left(x \cdot y\right) - 4.5 \cdot \left(t \cdot z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right) - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
\mathbf{if}\;a \cdot 2 \le -3.077577436686377797275553991166215453874 \cdot 10^{-99}:\\
\;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\

\mathbf{elif}\;a \cdot 2 \le 6474202012076836110009892864:\\
\;\;\;\;\frac{0.5 \cdot \left(x \cdot y\right) - 4.5 \cdot \left(t \cdot z\right)}{a}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right) - 4.5 \cdot \frac{t}{\frac{a}{z}}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r472095 = x;
        double r472096 = y;
        double r472097 = r472095 * r472096;
        double r472098 = z;
        double r472099 = 9.0;
        double r472100 = r472098 * r472099;
        double r472101 = t;
        double r472102 = r472100 * r472101;
        double r472103 = r472097 - r472102;
        double r472104 = a;
        double r472105 = 2.0;
        double r472106 = r472104 * r472105;
        double r472107 = r472103 / r472106;
        return r472107;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r472108 = a;
        double r472109 = 2.0;
        double r472110 = r472108 * r472109;
        double r472111 = -3.077577436686378e-99;
        bool r472112 = r472110 <= r472111;
        double r472113 = 0.5;
        double r472114 = x;
        double r472115 = y;
        double r472116 = r472108 / r472115;
        double r472117 = r472114 / r472116;
        double r472118 = r472113 * r472117;
        double r472119 = 4.5;
        double r472120 = t;
        double r472121 = z;
        double r472122 = r472108 / r472121;
        double r472123 = r472120 / r472122;
        double r472124 = r472119 * r472123;
        double r472125 = r472118 - r472124;
        double r472126 = 6.474202012076836e+27;
        bool r472127 = r472110 <= r472126;
        double r472128 = r472114 * r472115;
        double r472129 = r472113 * r472128;
        double r472130 = r472120 * r472121;
        double r472131 = r472119 * r472130;
        double r472132 = r472129 - r472131;
        double r472133 = r472132 / r472108;
        double r472134 = cbrt(r472108);
        double r472135 = r472134 * r472134;
        double r472136 = r472114 / r472135;
        double r472137 = r472115 / r472134;
        double r472138 = r472136 * r472137;
        double r472139 = r472113 * r472138;
        double r472140 = r472139 - r472124;
        double r472141 = r472127 ? r472133 : r472140;
        double r472142 = r472112 ? r472125 : r472141;
        return r472142;
}

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
Target5.2
Herbie4.2
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709043451944897028999329376 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.144030707833976090627817222818061808815 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* a 2.0) < -3.077577436686378e-99

    1. Initial program 9.0

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

    2. Taylor expanded around 0 8.9

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

    4. Applied associate-/l*7.4

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

    6. Applied associate-/l*5.2

      \[\leadsto 0.5 \cdot \color{blue}{\frac{x}{\frac{a}{y}}} - 4.5 \cdot \frac{t}{\frac{a}{z}}\]

    if -3.077577436686378e-99 < (* a 2.0) < 6.474202012076836e+27

    1. Initial program 1.5

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

    2. Taylor expanded around 0 1.6

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

    4. Applied associate-*r/1.6

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

    5. Applied associate-*r/1.5

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

    6. Applied sub-div1.5

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

    if 6.474202012076836e+27 < (* a 2.0)

    1. Initial program 12.3

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

    2. Taylor expanded around 0 12.1

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

    4. Applied associate-/l*9.6

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

    6. Applied add-cube-cbrt10.0

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

    7. Applied times-frac5.6

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

  4. Final simplification4.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 2 \le -3.077577436686377797275553991166215453874 \cdot 10^{-99}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \cdot 2 \le 6474202012076836110009892864:\\ \;\;\;\;\frac{0.5 \cdot \left(x \cdot y\right) - 4.5 \cdot \left(t \cdot z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right) - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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)))