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

Average Error: 6.0 → 3.8

Time: 12.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le 2.4693420887247912 \cdot 10^{-98}:\\ \;\;\;\;\mathsf{fma}\left(2, \mathsf{fma}\left(t, z, x \cdot y\right), \left(2 \cdot \left(\left(-\mathsf{fma}\left(c, b, a\right)\right) \cdot \left(c \cdot \left(\sqrt[3]{i} \cdot \sqrt[3]{i}\right)\right)\right)\right) \cdot \sqrt[3]{i}\right)\\ \mathbf{elif}\;c \le 8.5463068787004626 \cdot 10^{134}:\\ \;\;\;\;\mathsf{fma}\left(2, \mathsf{fma}\left(t, z, x \cdot y\right), \left(-2\right) \cdot \mathsf{fma}\left(i \cdot b, {c}^{2}, a \cdot \left(i \cdot c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \mathsf{fma}\left(t, z, x \cdot y\right), 2 \cdot \left(\left(-\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot i\right)\right)\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double VAR;
	if ((c <= 2.4693420887247912e-98)) {
		VAR = ((double) fma(2.0, ((double) fma(t, z, ((double) (x * y)))), ((double) (((double) (2.0 * ((double) (((double) -(((double) fma(c, b, a)))) * ((double) (c * ((double) (((double) cbrt(i)) * ((double) cbrt(i)))))))))) * ((double) cbrt(i))))));
	} else {
		double VAR_1;
		if ((c <= 8.546306878700463e+134)) {
			VAR_1 = ((double) fma(2.0, ((double) fma(t, z, ((double) (x * y)))), ((double) (((double) -(2.0)) * ((double) fma(((double) (i * b)), ((double) pow(c, 2.0)), ((double) (a * ((double) (i * c))))))))));
		} else {
			VAR_1 = ((double) fma(2.0, ((double) fma(t, z, ((double) (x * y)))), ((double) (2.0 * ((double) (((double) -(((double) (((double) fma(c, b, a)) * c)))) * i))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

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	6.0
Target	1.8
Herbie	3.8

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

Derivation

1. Split input into 3 regimes

2. if c < 2.4693420887247912e-98

   1. Initial program 4.0

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

   2. Simplified 1.4

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

   3. Taylor expanded around inf 1.4

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

   4. Simplified 1.4

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

   6. Applied add-cube-cbrt 1.7

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

   7. Applied associate-*r* 1.7

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

   8. Applied associate-*r* 1.9

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

   9. Applied associate-*r* 1.9

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

   if 2.4693420887247912e-98 < c < 8.546306878700463e+134

   1. Initial program 5.8

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

   2. Simplified 2.2

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

   3. Taylor expanded around inf 2.2

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

   4. Simplified 2.2

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

   6. Applied add-cube-cbrt 2.6

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

   7. Applied associate-*r* 2.6

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

   8. Applied associate-*r* 2.2

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

   9. Applied associate-*r* 2.2

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

   10. Taylor expanded around inf 4.4

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

   11. Simplified 2.6

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

   if 8.546306878700463e+134 < c

   1. Initial program 31.3

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

   2. Simplified 4.3

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

   3. Taylor expanded around inf 4.3

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

   4. Simplified 4.3

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

   6. Applied associate-*r* 31.3

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

   7. Simplified 31.3

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

4. Final simplification 3.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le 2.4693420887247912 \cdot 10^{-98}:\\ \;\;\;\;\mathsf{fma}\left(2, \mathsf{fma}\left(t, z, x \cdot y\right), \left(2 \cdot \left(\left(-\mathsf{fma}\left(c, b, a\right)\right) \cdot \left(c \cdot \left(\sqrt[3]{i} \cdot \sqrt[3]{i}\right)\right)\right)\right) \cdot \sqrt[3]{i}\right)\\ \mathbf{elif}\;c \le 8.5463068787004626 \cdot 10^{134}:\\ \;\;\;\;\mathsf{fma}\left(2, \mathsf{fma}\left(t, z, x \cdot y\right), \left(-2\right) \cdot \mathsf{fma}\left(i \cdot b, {c}^{2}, a \cdot \left(i \cdot c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \mathsf{fma}\left(t, z, x \cdot y\right), 2 \cdot \left(\left(-\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot i\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 +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))))