Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 6.6 → 1.8

Time: 3.8s

Precision: binary64

Math

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

↓

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

FPCore

(FPCore (x y z t) :precision binary64 (/ (* x 2.0) (- (* y z) (* t z))))

↓

(FPCore (x y z t)
 :precision binary64
 (*
  (/ x (* (cbrt (- y t)) (* (cbrt (- y t)) (* (cbrt z) (cbrt z)))))
  (/ (/ 2.0 (cbrt z)) (cbrt (- y t)))))

C

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

↓

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.6
Target	1.9
Herbie	1.8

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} < -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} < 1.0450278273301259 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

1. Initial program 6.6

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

2. Simplified 5.5

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

4. Applied associate-/r* 5.2

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

6. Applied add-cube-cbrt 5.8

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

7. Applied add-cube-cbrt 6.0

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

8. Applied *-un-lft-identity 6.0

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

9. Applied times-frac 6.0

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

10. Applied times-frac 6.0

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

11. Applied associate-*r* 1.7

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

12. Simplified 1.8

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

13. Final simplification 1.8

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

Reproduce

herbie shell --seed 2020198 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (if (< (/ (* x 2.0) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2.0) (if (< (/ (* x 2.0) (- (* y z) (* t z))) 1.0450278273301259e-269) (/ (* (/ x z) 2.0) (- y t)) (* (/ x (* (- y t) z)) 2.0)))

  (/ (* x 2.0) (- (* y z) (* t z))))