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

Average Error: 6.1 → 1.6

Time: 11.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(x, y, z \cdot t\right)\\ \mathbf{if}\;i \leq -3.0819729376635466 \cdot 10^{-49}:\\ \;\;\;\;2 \cdot \left(t_1 - i \cdot \left(c \cdot \left(c \cdot b\right) + c \cdot a\right)\right)\\ \mathbf{elif}\;i \leq 2.578892516458991 \cdot 10^{+25}:\\ \;\;\;\;2 \cdot \left(t_1 - c \cdot \left(c \cdot \left(i \cdot b\right) + i \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 - i \cdot \left(c \cdot a + b \cdot {c}^{2}\right)\right)\\ \end{array} \]

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
 (let* ((t_1 (fma x y (* z t))))
   (if (<= i -3.0819729376635466e-49)
     (* 2.0 (- t_1 (* i (+ (* c (* c b)) (* c a)))))
     (if (<= i 2.578892516458991e+25)
       (* 2.0 (- t_1 (* c (+ (* c (* i b)) (* i a)))))
       (* 2.0 (- t_1 (* i (+ (* c a) (* b (pow c 2.0))))))))))

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 t_1 = fma(x, y, (z * t));
	double tmp;
	if (i <= -3.0819729376635466e-49) {
		tmp = 2.0 * (t_1 - (i * ((c * (c * b)) + (c * a))));
	} else if (i <= 2.578892516458991e+25) {
		tmp = 2.0 * (t_1 - (c * ((c * (i * b)) + (i * a))));
	} else {
		tmp = 2.0 * (t_1 - (i * ((c * a) + (b * pow(c, 2.0)))));
	}
	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	6.1
Target	2.0
Herbie	1.6

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

Derivation

1. Split input into 3 regimes

2. if i < -3.0819729376635466e-49

   1. Initial program 1.1

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

   2. Simplified 1.1

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

   3. Applied fma-udef_binary64 1.1

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

   4. Applied distribute-lft-in_binary64 1.1

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

   if -3.0819729376635466e-49 < i < 2.57889251645899121e25

   1. Initial program 9.7

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

   2. Simplified 9.7

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

   3. Taylor expanded in c around 0 8.8

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

   4. Simplified 1.9

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

   5. Applied fma-udef_binary64 1.9

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

   6. Applied distribute-rgt-in_binary64 1.9

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

   7. Simplified 1.4

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

   8. Simplified 1.4

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

   if 2.57889251645899121e25 < i

   1. Initial program 0.8

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

   2. Simplified 0.8

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

   3. Taylor expanded in i around 0 2.9

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

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -3.0819729376635466 \cdot 10^{-49}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(x, y, z \cdot t\right) - i \cdot \left(c \cdot \left(c \cdot b\right) + c \cdot a\right)\right)\\ \mathbf{elif}\;i \leq 2.578892516458991 \cdot 10^{+25}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(x, y, z \cdot t\right) - c \cdot \left(c \cdot \left(i \cdot b\right) + i \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(x, y, z \cdot t\right) - i \cdot \left(c \cdot a + b \cdot {c}^{2}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021340 
(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))))