Average Error: 7.7 → 1.4
Time: 17.0s
Precision: 64
\[\frac{x \cdot y - z \cdot t}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \le -1.35814929359811122012683914750974806748 \cdot 10^{213}:\\ \;\;\;\;\left(\frac{x}{a} \cdot y - \frac{t}{a} \cdot z\right) + \mathsf{fma}\left(-\frac{t}{a}, z, \frac{t}{a} \cdot z\right)\\ \mathbf{elif}\;x \cdot y - z \cdot t \le 1.304590199957326978034100659392854141042 \cdot 10^{142}:\\ \;\;\;\;\frac{1}{a} \cdot \left(x \cdot y - z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{a} \cdot y - \frac{t}{a} \cdot z\right) + \mathsf{fma}\left(-\frac{t}{a}, z, \frac{t}{a} \cdot z\right)\\ \end{array}\]
\frac{x \cdot y - z \cdot t}{a}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y - z \cdot t \le -1.35814929359811122012683914750974806748 \cdot 10^{213}:\\
\;\;\;\;\left(\frac{x}{a} \cdot y - \frac{t}{a} \cdot z\right) + \mathsf{fma}\left(-\frac{t}{a}, z, \frac{t}{a} \cdot z\right)\\

\mathbf{elif}\;x \cdot y - z \cdot t \le 1.304590199957326978034100659392854141042 \cdot 10^{142}:\\
\;\;\;\;\frac{1}{a} \cdot \left(x \cdot y - z \cdot t\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{x}{a} \cdot y - \frac{t}{a} \cdot z\right) + \mathsf{fma}\left(-\frac{t}{a}, z, \frac{t}{a} \cdot z\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r34936051 = x;
        double r34936052 = y;
        double r34936053 = r34936051 * r34936052;
        double r34936054 = z;
        double r34936055 = t;
        double r34936056 = r34936054 * r34936055;
        double r34936057 = r34936053 - r34936056;
        double r34936058 = a;
        double r34936059 = r34936057 / r34936058;
        return r34936059;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r34936060 = x;
        double r34936061 = y;
        double r34936062 = r34936060 * r34936061;
        double r34936063 = z;
        double r34936064 = t;
        double r34936065 = r34936063 * r34936064;
        double r34936066 = r34936062 - r34936065;
        double r34936067 = -1.3581492935981112e+213;
        bool r34936068 = r34936066 <= r34936067;
        double r34936069 = a;
        double r34936070 = r34936060 / r34936069;
        double r34936071 = r34936070 * r34936061;
        double r34936072 = r34936064 / r34936069;
        double r34936073 = r34936072 * r34936063;
        double r34936074 = r34936071 - r34936073;
        double r34936075 = -r34936072;
        double r34936076 = fma(r34936075, r34936063, r34936073);
        double r34936077 = r34936074 + r34936076;
        double r34936078 = 1.304590199957327e+142;
        bool r34936079 = r34936066 <= r34936078;
        double r34936080 = 1.0;
        double r34936081 = r34936080 / r34936069;
        double r34936082 = r34936081 * r34936066;
        double r34936083 = r34936079 ? r34936082 : r34936077;
        double r34936084 = r34936068 ? r34936077 : r34936083;
        return r34936084;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original7.7
Target5.9
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;z \lt -2.468684968699548224247694913169778644284 \cdot 10^{170}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{elif}\;z \lt 6.309831121978371209578784129518242708809 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (* x y) (* z t)) < -1.3581492935981112e+213 or 1.304590199957327e+142 < (- (* x y) (* z t))

    1. Initial program 24.4

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

    3. Applied div-sub24.4

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

    5. Applied *-un-lft-identity24.4

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

    6. Applied times-frac14.2

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

    7. Applied add-cube-cbrt14.5

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}} - \frac{z}{1} \cdot \frac{t}{a}\]

    8. Applied times-frac2.4

      \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}} - \frac{z}{1} \cdot \frac{t}{a}\]

    9. Applied prod-diff2.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{y}{\sqrt[3]{a}}, -\frac{t}{a} \cdot \frac{z}{1}\right) + \mathsf{fma}\left(-\frac{t}{a}, \frac{z}{1}, \frac{t}{a} \cdot \frac{z}{1}\right)}\]

    10. Taylor expanded around inf 24.9

      \[\leadsto \color{blue}{\left(\frac{x \cdot y}{a} - \frac{t \cdot z}{a}\right)} + \mathsf{fma}\left(-\frac{t}{a}, \frac{z}{1}, \frac{t}{a} \cdot \frac{z}{1}\right)\]

    11. Simplified2.2

      \[\leadsto \color{blue}{\left(\frac{x}{a} \cdot y - \frac{t}{a} \cdot z\right)} + \mathsf{fma}\left(-\frac{t}{a}, \frac{z}{1}, \frac{t}{a} \cdot \frac{z}{1}\right)\]

    if -1.3581492935981112e+213 < (- (* x y) (* z t)) < 1.304590199957327e+142

    1. Initial program 1.0

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

    3. Applied div-sub1.0

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

    5. Applied div-inv1.0

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

    6. Applied div-inv1.0

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

    7. Applied distribute-rgt-out--1.0

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \le -1.35814929359811122012683914750974806748 \cdot 10^{213}:\\ \;\;\;\;\left(\frac{x}{a} \cdot y - \frac{t}{a} \cdot z\right) + \mathsf{fma}\left(-\frac{t}{a}, z, \frac{t}{a} \cdot z\right)\\ \mathbf{elif}\;x \cdot y - z \cdot t \le 1.304590199957326978034100659392854141042 \cdot 10^{142}:\\ \;\;\;\;\frac{1}{a} \cdot \left(x \cdot y - z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{a} \cdot y - \frac{t}{a} \cdot z\right) + \mathsf{fma}\left(-\frac{t}{a}, z, \frac{t}{a} \cdot z\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"

  :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))