Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 6.9 → 1.3

Time: 5.0s

Precision: binary64

Math

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

↓

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

FPCore

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

↓

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

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 ((x * (((cbrt(sqrt(2.0)) * cbrt(sqrt(2.0))) / cbrt(y - t)) / cbrt(z))) * ((cbrt(sqrt(2.0)) / cbrt(y - t)) / cbrt(z))) * ((sqrt(2.0) / cbrt(y - t)) / cbrt(z));
}

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	1.3

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

Derivation

1. Initial program 6.9

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

2. Simplified 5.6

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

4. Applied add-cube-cbrt_binary64 6.3

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

5. Applied add-cube-cbrt_binary64 6.4

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

6. Applied add-sqr-sqrt_binary64 6.5

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

7. Applied times-frac_binary64 6.5

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

8. Applied times-frac_binary64 6.5

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

9. Applied associate-*r*_binary64 1.9

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

11. Applied add-cube-cbrt_binary64 1.9

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

12. Applied times-frac_binary64 1.9

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

13. Applied times-frac_binary64 1.9

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

14. Applied associate-*r*_binary64 1.3

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

15. Final simplification 1.3

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

Reproduce

herbie shell --seed 2020232 
(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))))