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

Average Error: 6.6 → 1.7

Time: 29.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq -1558263795518608400 \lor \neg \left(i \leq 3.860982310378207 \cdot 10^{+95}\right):\\ \;\;\;\;2 \cdot \left(\left(x \cdot y + z \cdot t\right) - i \cdot \left(c \cdot \left(a + b \cdot c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - c \cdot \left(i \cdot \mathsf{fma}\left(c, b, a\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (* 2.0 (- (+ (* x y) (* z t)) (* (* (+ a (* b c)) c) i))))

↓

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= i -1558263795518608400.0) (not (<= i 3.860982310378207e+95)))
   (* 2.0 (- (+ (* x y) (* z t)) (* i (* c (+ a (* b c))))))
   (* 2.0 (- (fma t z (* x y)) (* c (* i (fma c b a)))))))

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) {
	double tmp;
	if ((i <= -1558263795518608400.0) || !(i <= 3.860982310378207e+95)) {
		tmp = 2.0 * (((x * y) + (z * t)) - (i * (c * (a + (b * c)))));
	} else {
		tmp = 2.0 * (fma(t, z, (x * y)) - (c * (i * fma(c, b, a))));
	}
	return tmp;
}

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	6.6
Target	1.9
Herbie	1.7

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

Derivation

1. Split input into 2 regimes

2. if i < -1558263795518608400 or 3.86098231037820712e95 < i

   1. Initial program 0.8

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

   if -1558263795518608400 < i < 3.86098231037820712e95

   1. Initial program 9.0

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

   2. Simplified 9.0

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

   3. Taylor expanded in x around 0 9.2

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

   4. Simplified 2.0

      \[\leadsto 2 \cdot \color{blue}{\left(\mathsf{fma}\left(t, z, y \cdot x\right) - c \cdot \left(i \cdot \mathsf{fma}\left(c, b, a\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1558263795518608400 \lor \neg \left(i \leq 3.860982310378207 \cdot 10^{+95}\right):\\ \;\;\;\;2 \cdot \left(\left(x \cdot y + z \cdot t\right) - i \cdot \left(c \cdot \left(a + b \cdot c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - c \cdot \left(i \cdot \mathsf{fma}\left(c, b, a\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021291 
(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
  (* 2.0 (- (+ (* x y) (* z t)) (* (+ a (* b c)) (* c i))))

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