Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, A

Average Error: 6.5 → 1.9

Time: 10.7s

Precision: 64

Math

\[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)\]

↓

\[\mathsf{fma}\left(2, \mathsf{fma}\left(t, z, x \cdot y\right), 1 \cdot \left(2 \cdot \left(\left(-\mathsf{fma}\left(c, b, a\right)\right) \cdot \left(c \cdot i\right)\right)\right)\right)\]

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i)));
}

↓

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return fma(2.0, fma(t, z, (x * y)), (1.0 * (2.0 * (-fma(c, b, a) * (c * i)))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus i

Target

Original	6.5
Target	1.9
Herbie	1.9

\[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(a + b \cdot c\right) \cdot \left(c \cdot i\right)\right)\]

Derivation

1. Initial program 6.5

   \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)\]

2. Simplified 1.9

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, \mathsf{fma}\left(x, y, z \cdot t\right), 2 \cdot \left(\left(-\mathsf{fma}\left(c, b, a\right)\right) \cdot \left(c \cdot i\right)\right)\right)}\]

3. Taylor expanded around inf 1.9

   \[\leadsto \mathsf{fma}\left(2, \color{blue}{t \cdot z + x \cdot y}, 2 \cdot \left(\left(-\mathsf{fma}\left(c, b, a\right)\right) \cdot \left(c \cdot i\right)\right)\right)\]

4. Simplified 1.9

   \[\leadsto \mathsf{fma}\left(2, \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)}, 2 \cdot \left(\left(-\mathsf{fma}\left(c, b, a\right)\right) \cdot \left(c \cdot i\right)\right)\right)\]
5. Using strategy rm

6. Applied add-sqr-sqrt 2.2

   \[\leadsto \mathsf{fma}\left(2, \mathsf{fma}\left(t, z, x \cdot y\right), \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot \left(\left(-\mathsf{fma}\left(c, b, a\right)\right) \cdot \left(c \cdot i\right)\right)\right)\]

7. Applied associate-*l* 2.1

   \[\leadsto \mathsf{fma}\left(2, \mathsf{fma}\left(t, z, x \cdot y\right), \color{blue}{\sqrt{2} \cdot \left(\sqrt{2} \cdot \left(\left(-\mathsf{fma}\left(c, b, a\right)\right) \cdot \left(c \cdot i\right)\right)\right)}\right)\]
8. Using strategy rm

9. Applied add-sqr-sqrt 2.1

   \[\leadsto \mathsf{fma}\left(2, \mathsf{fma}\left(t, z, x \cdot y\right), \sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}} \cdot \left(\sqrt{2} \cdot \left(\left(-\mathsf{fma}\left(c, b, a\right)\right) \cdot \left(c \cdot i\right)\right)\right)\right)\]

10. Applied sqrt-prod 1.9

    \[\leadsto \mathsf{fma}\left(2, \mathsf{fma}\left(t, z, x \cdot y\right), \color{blue}{\left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}\right)} \cdot \left(\sqrt{2} \cdot \left(\left(-\mathsf{fma}\left(c, b, a\right)\right) \cdot \left(c \cdot i\right)\right)\right)\right)\]

11. Applied associate-*l* 1.9

    \[\leadsto \mathsf{fma}\left(2, \mathsf{fma}\left(t, z, x \cdot y\right), \color{blue}{\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{2} \cdot \left(\left(-\mathsf{fma}\left(c, b, a\right)\right) \cdot \left(c \cdot i\right)\right)\right)\right)}\right)\]
12. Using strategy rm

13. Applied *-un-lft-identity 1.9

    \[\leadsto \mathsf{fma}\left(2, \mathsf{fma}\left(t, z, x \cdot y\right), \color{blue}{\left(1 \cdot \sqrt{\sqrt{2}}\right)} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{2} \cdot \left(\left(-\mathsf{fma}\left(c, b, a\right)\right) \cdot \left(c \cdot i\right)\right)\right)\right)\right)\]

14. Applied associate-*l* 1.9

    \[\leadsto \mathsf{fma}\left(2, \mathsf{fma}\left(t, z, x \cdot y\right), \color{blue}{1 \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{2} \cdot \left(\left(-\mathsf{fma}\left(c, b, a\right)\right) \cdot \left(c \cdot i\right)\right)\right)\right)\right)}\right)\]

15. Simplified 1.9

    \[\leadsto \mathsf{fma}\left(2, \mathsf{fma}\left(t, z, x \cdot y\right), 1 \cdot \color{blue}{\left(2 \cdot \left(\left(-\mathsf{fma}\left(c, b, a\right)\right) \cdot \left(c \cdot i\right)\right)\right)}\right)\]

16. Final simplification 1.9

    \[\leadsto \mathsf{fma}\left(2, \mathsf{fma}\left(t, z, x \cdot y\right), 1 \cdot \left(2 \cdot \left(\left(-\mathsf{fma}\left(c, b, a\right)\right) \cdot \left(c \cdot i\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x y z t a b c i)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (* 2 (- (+ (* x y) (* z t)) (* (+ a (* b c)) (* c i))))

  (* 2 (- (+ (* x y) (* z t)) (* (* (+ a (* b c)) c) i))))