Average Error: 7.4 → 1.0
Time: 3.7s
Precision: 64
\[\frac{x \cdot y - z \cdot t}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t = -\infty \lor \neg \left(x \cdot y - z \cdot t \le 1.1925199982606428 \cdot 10^{305}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{y}{\sqrt[3]{a}}, -\frac{z}{\sqrt[3]{a}} \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) + \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\left(-\frac{z}{\sqrt[3]{a}}\right) + \frac{z}{\sqrt[3]{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{x \cdot y - z \cdot t}}\\ \end{array}\]
\frac{x \cdot y - z \cdot t}{a}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y - z \cdot t = -\infty \lor \neg \left(x \cdot y - z \cdot t \le 1.1925199982606428 \cdot 10^{305}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{y}{\sqrt[3]{a}}, -\frac{z}{\sqrt[3]{a}} \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) + \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\left(-\frac{z}{\sqrt[3]{a}}\right) + \frac{z}{\sqrt[3]{a}}\right)\\

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

\end{array}
double code(double x, double y, double z, double t, double a) {
	return (((x * y) - (z * t)) / a);
}
↓
double code(double x, double y, double z, double t, double a) {
	double temp;
	if (((((x * y) - (z * t)) <= -inf.0) || !(((x * y) - (z * t)) <= 1.1925199982606428e+305))) {
		temp = (fma((x / (cbrt(a) * cbrt(a))), (y / cbrt(a)), -((z / cbrt(a)) * (t / (cbrt(a) * cbrt(a))))) + ((t / (cbrt(a) * cbrt(a))) * (-(z / cbrt(a)) + (z / cbrt(a)))));
	} else {
		temp = (1.0 / (a / ((x * y) - (z * t))));
	}
	return temp;
}

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

Original7.4
Target6.2
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;z \lt -2.46868496869954822 \cdot 10^{170}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{elif}\;z \lt 6.30983112197837121 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (* x y) (* z t)) < -inf.0 or 1.1925199982606428e+305 < (- (* x y) (* z t))

    1. Initial program 63.3

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

    3. Applied div-sub63.3

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

    4. Simplified63.3

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

    6. Applied add-cube-cbrt63.3

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

    7. Applied times-frac35.2

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

    8. Applied add-cube-cbrt35.2

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

    9. Applied times-frac1.3

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

    10. Applied prod-diff1.3

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

    11. Simplified1.3

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

    if -inf.0 < (- (* x y) (* z t)) < 1.1925199982606428e+305

    1. Initial program 0.7

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

    3. Applied clear-num1.0

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t = -\infty \lor \neg \left(x \cdot y - z \cdot t \le 1.1925199982606428 \cdot 10^{305}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{y}{\sqrt[3]{a}}, -\frac{z}{\sqrt[3]{a}} \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) + \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\left(-\frac{z}{\sqrt[3]{a}}\right) + \frac{z}{\sqrt[3]{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{x \cdot y - z \cdot t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 +o rules:numerics
(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))