Average Error: 7.3 → 7.3
Time: 8.3s
Precision: binary64
\[\frac{x \cdot y - z \cdot t}{a} \]
↓
\[\frac{y \cdot x - t \cdot z}{a} \]
\frac{x \cdot y - z \cdot t}{a}
↓
\frac{y \cdot x - t \cdot z}{a}

(FPCore (x y z t a) :precision binary64 (/ (- (* x y) (* z t)) a))
↓
(FPCore (x y z t a) :precision binary64 (/ (- (* y x) (* t z)) a))
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) {
	return ((y * x) - (t * z)) / a;
}

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

Original7.3
Target5.8
Herbie7.3
\[\begin{array}{l} \mathbf{if}\;z < -2.468684968699548 \cdot 10^{+170}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{elif}\;z < 6.309831121978371 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \end{array} \]

Derivation

  1. Initial program 7.3

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

  3. Applied *-un-lft-identity_binary647.3

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

  4. Applied add-cube-cbrt_binary648.2

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

  5. Applied times-frac_binary648.2

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

  6. Simplified8.2

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

  7. Simplified8.2

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

  9. Applied div-inv_binary648.2

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

  10. Applied associate-*r*_binary648.2

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

  11. Simplified7.4

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

  13. Applied *-un-lft-identity_binary647.4

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

  14. Applied associate-*l*_binary647.4

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

  15. Simplified7.3

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

  16. Final simplification7.3

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

Reproduce

herbie shell --seed 2021207 
(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))