Average Error: 7.9 → 5.1
Time: 20.7s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \le -5.588536951192011766760384954663213229832 \cdot 10^{216}:\\ \;\;\;\;\frac{y \cdot x}{a} \cdot 0.5 - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 220957729404425301813777018323200120979500:\\ \;\;\;\;\frac{y \cdot x - \left(t \cdot 9\right) \cdot z}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{a} \cdot 0.5 - \left(4.5 \cdot t\right) \cdot \frac{z}{a}\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
\mathbf{if}\;\left(z \cdot 9\right) \cdot t \le -5.588536951192011766760384954663213229832 \cdot 10^{216}:\\
\;\;\;\;\frac{y \cdot x}{a} \cdot 0.5 - 4.5 \cdot \frac{t}{\frac{a}{z}}\\

\mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 220957729404425301813777018323200120979500:\\
\;\;\;\;\frac{y \cdot x - \left(t \cdot 9\right) \cdot z}{a \cdot 2}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r39408554 = x;
        double r39408555 = y;
        double r39408556 = r39408554 * r39408555;
        double r39408557 = z;
        double r39408558 = 9.0;
        double r39408559 = r39408557 * r39408558;
        double r39408560 = t;
        double r39408561 = r39408559 * r39408560;
        double r39408562 = r39408556 - r39408561;
        double r39408563 = a;
        double r39408564 = 2.0;
        double r39408565 = r39408563 * r39408564;
        double r39408566 = r39408562 / r39408565;
        return r39408566;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r39408567 = z;
        double r39408568 = 9.0;
        double r39408569 = r39408567 * r39408568;
        double r39408570 = t;
        double r39408571 = r39408569 * r39408570;
        double r39408572 = -5.588536951192012e+216;
        bool r39408573 = r39408571 <= r39408572;
        double r39408574 = y;
        double r39408575 = x;
        double r39408576 = r39408574 * r39408575;
        double r39408577 = a;
        double r39408578 = r39408576 / r39408577;
        double r39408579 = 0.5;
        double r39408580 = r39408578 * r39408579;
        double r39408581 = 4.5;
        double r39408582 = r39408577 / r39408567;
        double r39408583 = r39408570 / r39408582;
        double r39408584 = r39408581 * r39408583;
        double r39408585 = r39408580 - r39408584;
        double r39408586 = 2.209577294044253e+41;
        bool r39408587 = r39408571 <= r39408586;
        double r39408588 = r39408570 * r39408568;
        double r39408589 = r39408588 * r39408567;
        double r39408590 = r39408576 - r39408589;
        double r39408591 = 2.0;
        double r39408592 = r39408577 * r39408591;
        double r39408593 = r39408590 / r39408592;
        double r39408594 = r39408581 * r39408570;
        double r39408595 = r39408567 / r39408577;
        double r39408596 = r39408594 * r39408595;
        double r39408597 = r39408580 - r39408596;
        double r39408598 = r39408587 ? r39408593 : r39408597;
        double r39408599 = r39408573 ? r39408585 : r39408598;
        return r39408599;
}

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

Original7.9
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 3 regimes

  2. if (* (* z 9.0) t) < -5.588536951192012e+216

    1. Initial program 33.8

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

    2. Taylor expanded around 0 33.1

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

    4. Applied associate-/l*5.5

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

    if -5.588536951192012e+216 < (* (* z 9.0) t) < 2.209577294044253e+41

    1. Initial program 4.1

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

    3. Applied associate-*l*4.2

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

    if 2.209577294044253e+41 < (* (* z 9.0) t)

    1. Initial program 15.4

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

    2. Taylor expanded around 0 15.1

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

    4. Applied *-un-lft-identity15.1

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{\color{blue}{1 \cdot a}}\]

    5. Applied times-frac9.1

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \color{blue}{\left(\frac{t}{1} \cdot \frac{z}{a}\right)}\]

    6. Applied associate-*r*9.1

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \color{blue}{\left(4.5 \cdot \frac{t}{1}\right) \cdot \frac{z}{a}}\]

    7. Simplified9.1

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

  4. Final simplification5.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \le -5.588536951192011766760384954663213229832 \cdot 10^{216}:\\ \;\;\;\;\frac{y \cdot x}{a} \cdot 0.5 - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 220957729404425301813777018323200120979500:\\ \;\;\;\;\frac{y \cdot x - \left(t \cdot 9\right) \cdot z}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{a} \cdot 0.5 - \left(4.5 \cdot t\right) \cdot \frac{z}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"

  :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 (* 9.0 t))) (* a 2.0)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))