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

Average Error: 6.2 → 1.1

Time: 10.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t_1 \leq -3.2895597596916134 \cdot 10^{+300}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(t, z, y \cdot x\right) - c \cdot \left(c \cdot \left(b \cdot i\right) + a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_2 := \mathsf{fma}\left(x, y, t \cdot z\right)\\ \mathbf{if}\;t_1 \leq 4.316811525763463 \cdot 10^{+296}:\\ \;\;\;\;2 \cdot \left(t_2 - i \cdot \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_3 := \sqrt{c \cdot \left(i \cdot \mathsf{fma}\left(c, b, a\right)\right)}\\ 2 \cdot \left(t_2 - t_3 \cdot t_3\right) \end{array}\\ \end{array}\\ \end{array} \]

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 (* (* c (+ a (* b c))) i)))
   (if (<= t_1 -3.2895597596916134e+300)
     (* 2.0 (- (fma t z (* y x)) (* c (+ (* c (* b i)) (* a i)))))
     (let* ((t_2 (fma x y (* t z))))
       (if (<= t_1 4.316811525763463e+296)
         (* 2.0 (- t_2 (* i (* c (fma b c a)))))
         (let* ((t_3 (sqrt (* c (* i (fma c b a))))))
           (* 2.0 (- t_2 (* t_3 t_3)))))))))

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 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_1 <= -3.2895597596916134e+300) {
		tmp = 2.0 * (fma(t, z, (y * x)) - (c * ((c * (b * i)) + (a * i))));
	} else {
		double t_2 = fma(x, y, (t * z));
		double tmp_1;
		if (t_1 <= 4.316811525763463e+296) {
			tmp_1 = 2.0 * (t_2 - (i * (c * fma(b, c, a))));
		} else {
			double t_3 = sqrt(c * (i * fma(c, b, a)));
			tmp_1 = 2.0 * (t_2 - (t_3 * t_3));
		}
		tmp = tmp_1;
	}
	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	6.2
Target	1.9
Herbie	1.1

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -3.28955975969161342e300

   1. Initial program 60.2

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

   2. Simplified 60.2

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

   3. Taylor expanded in x around 0 34.7

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

   4. Simplified 15.6

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

   5. Taylor expanded in c around 0 5.5

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

   if -3.28955975969161342e300 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 4.31681152576346306e296

   1. Initial program 0.4

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

   2. Simplified 0.3

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

   if 4.31681152576346306e296 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 57.3

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

   2. Simplified 57.3

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

   3. Applied add-sqr-sqrt_binary64 57.3

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

   4. Simplified 59.2

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

   5. Simplified 10.0

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -3.2895597596916134 \cdot 10^{+300}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(t, z, y \cdot x\right) - c \cdot \left(c \cdot \left(b \cdot i\right) + a \cdot i\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 4.316811525763463 \cdot 10^{+296}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(x, y, t \cdot z\right) - i \cdot \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(x, y, t \cdot z\right) - \sqrt{c \cdot \left(i \cdot \mathsf{fma}\left(c, b, a\right)\right)} \cdot \sqrt{c \cdot \left(i \cdot \mathsf{fma}\left(c, b, a\right)\right)}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021313 
(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))))