Average Error: 7.7 → 1.7
Time: 9.0s
Precision: 64
\[\frac{x \cdot y - z \cdot t}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \le -6.6779270858917678 \cdot 10^{224} \lor \neg \left(x \cdot y - z \cdot t \le 5.60997412174555331 \cdot 10^{132}\right):\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{x \cdot y - z \cdot t}}\\ \end{array}\]
\frac{x \cdot y - z \cdot t}{a}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y - z \cdot t \le -6.6779270858917678 \cdot 10^{224} \lor \neg \left(x \cdot y - z \cdot t \le 5.60997412174555331 \cdot 10^{132}\right):\\
\;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{a}{x \cdot y - z \cdot t}}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r879682 = x;
        double r879683 = y;
        double r879684 = r879682 * r879683;
        double r879685 = z;
        double r879686 = t;
        double r879687 = r879685 * r879686;
        double r879688 = r879684 - r879687;
        double r879689 = a;
        double r879690 = r879688 / r879689;
        return r879690;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r879691 = x;
        double r879692 = y;
        double r879693 = r879691 * r879692;
        double r879694 = z;
        double r879695 = t;
        double r879696 = r879694 * r879695;
        double r879697 = r879693 - r879696;
        double r879698 = -6.677927085891768e+224;
        bool r879699 = r879697 <= r879698;
        double r879700 = 5.609974121745553e+132;
        bool r879701 = r879697 <= r879700;
        double r879702 = !r879701;
        bool r879703 = r879699 || r879702;
        double r879704 = a;
        double r879705 = r879692 / r879704;
        double r879706 = r879691 * r879705;
        double r879707 = r879695 / r879704;
        double r879708 = r879694 * r879707;
        double r879709 = r879706 - r879708;
        double r879710 = 1.0;
        double r879711 = r879704 / r879697;
        double r879712 = r879710 / r879711;
        double r879713 = r879703 ? r879709 : r879712;
        return r879713;
}

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
Target6.2
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;z \lt -2.46868496869954822 \cdot 10^{170}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{elif}\;z \lt 6.30983112197837121 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (* x y) (* z t)) < -6.677927085891768e+224 or 5.609974121745553e+132 < (- (* x y) (* z t))

    1. Initial program 23.7

      \[\frac{x \cdot y - z \cdot t}{a}\]
    2. Using strategy rm

    3. Applied div-sub23.7

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

    5. Applied *-un-lft-identity23.7

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

    6. Applied times-frac14.0

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

    7. Simplified14.0

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

    9. Applied *-un-lft-identity14.0

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

    10. Applied times-frac2.7

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

    11. Simplified2.7

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

    if -6.677927085891768e+224 < (- (* x y) (* z t)) < 5.609974121745553e+132

    1. Initial program 0.9

      \[\frac{x \cdot y - z \cdot t}{a}\]
    2. Using strategy rm

    3. Applied clear-num1.3

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

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \le -6.6779270858917678 \cdot 10^{224} \lor \neg \left(x \cdot y - z \cdot t \le 5.60997412174555331 \cdot 10^{132}\right):\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{x \cdot y - z \cdot t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"
  :precision binary64

  :herbie-target
  (if (< z -2.468684968699548e+170) (- (* (/ y a) x) (* (/ t a) z)) (if (< z 6.309831121978371e-71) (/ (- (* x y) (* z t)) a) (- (* (/ y a) x) (* (/ t a) z))))

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