Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 6.9 → 1.8

Time: 6.2s

Precision: 64

Math

\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

↓

\[\frac{\frac{\sqrt[3]{x}}{\frac{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}}{\sqrt[3]{x}}}}{z} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt[3]{y - t}}{2}}\]

C

double code(double x, double y, double z, double t) {
	return ((x * 2.0) / ((y * z) - (t * z)));
}

↓

double code(double x, double y, double z, double t) {
	return (((cbrt(x) / ((cbrt((y - t)) * cbrt((y - t))) / cbrt(x))) / z) * (cbrt(x) / (cbrt((y - t)) / 2.0)));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.9
Target	2.3
Herbie	1.8

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.04502782733012586 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

1. Initial program 6.9

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

2. Simplified 5.8

   \[\leadsto \color{blue}{\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}}\]
3. Using strategy rm

4. Applied *-un-lft-identity 5.8

   \[\leadsto \frac{x}{\frac{z \cdot \left(y - t\right)}{\color{blue}{1 \cdot 2}}}\]

5. Applied times-frac 5.8

   \[\leadsto \frac{x}{\color{blue}{\frac{z}{1} \cdot \frac{y - t}{2}}}\]

6. Applied *-un-lft-identity 5.8

   \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{z}{1} \cdot \frac{y - t}{2}}\]

7. Applied times-frac 5.6

   \[\leadsto \color{blue}{\frac{1}{\frac{z}{1}} \cdot \frac{x}{\frac{y - t}{2}}}\]

8. Simplified 5.6

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot \frac{x}{\frac{y - t}{2}}\]
9. Using strategy rm

10. Applied *-un-lft-identity 5.6

    \[\leadsto \frac{1}{z} \cdot \frac{x}{\frac{y - t}{\color{blue}{1 \cdot 2}}}\]

11. Applied add-cube-cbrt 6.2

    \[\leadsto \frac{1}{z} \cdot \frac{x}{\frac{\color{blue}{\left(\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}\right) \cdot \sqrt[3]{y - t}}}{1 \cdot 2}}\]

12. Applied times-frac 6.2

    \[\leadsto \frac{1}{z} \cdot \frac{x}{\color{blue}{\frac{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}}{1} \cdot \frac{\sqrt[3]{y - t}}{2}}}\]

13. Applied add-cube-cbrt 6.4

    \[\leadsto \frac{1}{z} \cdot \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\frac{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}}{1} \cdot \frac{\sqrt[3]{y - t}}{2}}\]

14. Applied times-frac 6.4

    \[\leadsto \frac{1}{z} \cdot \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}}{1}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt[3]{y - t}}{2}}\right)}\]

15. Applied associate-*r* 1.8

    \[\leadsto \color{blue}{\left(\frac{1}{z} \cdot \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}}{1}}\right) \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt[3]{y - t}}{2}}}\]

16. Simplified 1.8

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{x}}{\frac{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}}{\sqrt[3]{x}}}}{z}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt[3]{y - t}}{2}}\]

17. Final simplification 1.8

    \[\leadsto \frac{\frac{\sqrt[3]{x}}{\frac{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}}{\sqrt[3]{x}}}}{z} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt[3]{y - t}}{2}}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (if (< (/ (* x 2) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2) (if (< (/ (* x 2) (- (* y z) (* t z))) 1.0450278273301259e-269) (/ (* (/ x z) 2) (- y t)) (* (/ x (* (- y t) z)) 2)))

  (/ (* x 2) (- (* y z) (* t z))))