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

Average Error: 6.5 → 1.7

Time: 21.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -2.926840426293390090061766031766601179458 \cdot 10^{-61}:\\ \;\;\;\;2 \cdot \left(\left(y \cdot x + z \cdot t\right) - i \cdot \left(c \cdot \left(a + c \cdot b\right)\right)\right)\\ \mathbf{elif}\;i \le 7.568213044503462274218583496034052027993 \cdot 10^{-168}:\\ \;\;\;\;2 \cdot \left(\left(\left(z \cdot t - \left(a \cdot i\right) \cdot c\right) + y \cdot x\right) - \left(\left(c \cdot i\right) \cdot c\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot x + z \cdot t\right) - \sqrt[3]{\left(c \cdot i\right) \cdot \left(a + c \cdot b\right)} \cdot \left(\sqrt[3]{\left(c \cdot i\right) \cdot \left(a + c \cdot b\right)} \cdot \sqrt[3]{\left(c \cdot i\right) \cdot \left(a + c \cdot b\right)}\right)\right) \cdot 2\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r589351 = 2.0;
        double r589352 = x;
        double r589353 = y;
        double r589354 = r589352 * r589353;
        double r589355 = z;
        double r589356 = t;
        double r589357 = r589355 * r589356;
        double r589358 = r589354 + r589357;
        double r589359 = a;
        double r589360 = b;
        double r589361 = c;
        double r589362 = r589360 * r589361;
        double r589363 = r589359 + r589362;
        double r589364 = r589363 * r589361;
        double r589365 = i;
        double r589366 = r589364 * r589365;
        double r589367 = r589358 - r589366;
        double r589368 = r589351 * r589367;
        return r589368;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r589369 = i;
        double r589370 = -2.92684042629339e-61;
        bool r589371 = r589369 <= r589370;
        double r589372 = 2.0;
        double r589373 = y;
        double r589374 = x;
        double r589375 = r589373 * r589374;
        double r589376 = z;
        double r589377 = t;
        double r589378 = r589376 * r589377;
        double r589379 = r589375 + r589378;
        double r589380 = c;
        double r589381 = a;
        double r589382 = b;
        double r589383 = r589380 * r589382;
        double r589384 = r589381 + r589383;
        double r589385 = r589380 * r589384;
        double r589386 = r589369 * r589385;
        double r589387 = r589379 - r589386;
        double r589388 = r589372 * r589387;
        double r589389 = 7.568213044503462e-168;
        bool r589390 = r589369 <= r589389;
        double r589391 = r589381 * r589369;
        double r589392 = r589391 * r589380;
        double r589393 = r589378 - r589392;
        double r589394 = r589393 + r589375;
        double r589395 = r589380 * r589369;
        double r589396 = r589395 * r589380;
        double r589397 = r589396 * r589382;
        double r589398 = r589394 - r589397;
        double r589399 = r589372 * r589398;
        double r589400 = r589395 * r589384;
        double r589401 = cbrt(r589400);
        double r589402 = r589401 * r589401;
        double r589403 = r589401 * r589402;
        double r589404 = r589379 - r589403;
        double r589405 = r589404 * r589372;
        double r589406 = r589390 ? r589399 : r589405;
        double r589407 = r589371 ? r589388 : r589406;
        return r589407;
}

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	2.1
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 3 regimes

2. if i < -2.92684042629339e-61

   1. Initial program 1.3

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

   2. Simplified 1.3

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

   if -2.92684042629339e-61 < i < 7.568213044503462e-168

   1. Initial program 12.5

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

   2. Simplified 12.5

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

   4. Applied *-un-lft-identity 12.5

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

   5. Applied associate-*l* 12.5

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

   6. Simplified 3.3

      \[\leadsto 2 \cdot \left(\left(y \cdot x + z \cdot t\right) - 1 \cdot \color{blue}{\left(\left(i \cdot c\right) \cdot \left(a + b \cdot c\right)\right)}\right)\]
   7. Using strategy rm

   8. Applied distribute-lft-in 3.3

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

   9. Simplified 2.7

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

   10. Simplified 2.1

       \[\leadsto 2 \cdot \left(\left(y \cdot x + z \cdot t\right) - 1 \cdot \left(\left(a \cdot i\right) \cdot c + \color{blue}{b \cdot \left(c \cdot \left(c \cdot i\right)\right)}\right)\right)\]
   11. Using strategy rm

   12. Applied distribute-rgt-in 2.1

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

   13. Applied associate--r+ 2.1

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

   14. Simplified 2.1

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

   if 7.568213044503462e-168 < i

   1. Initial program 3.3

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

   2. Simplified 3.3

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

   4. Applied add-cube-cbrt 3.6

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

   5. Simplified 4.5

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

   6. Simplified 1.7

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

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -2.926840426293390090061766031766601179458 \cdot 10^{-61}:\\ \;\;\;\;2 \cdot \left(\left(y \cdot x + z \cdot t\right) - i \cdot \left(c \cdot \left(a + c \cdot b\right)\right)\right)\\ \mathbf{elif}\;i \le 7.568213044503462274218583496034052027993 \cdot 10^{-168}:\\ \;\;\;\;2 \cdot \left(\left(\left(z \cdot t - \left(a \cdot i\right) \cdot c\right) + y \cdot x\right) - \left(\left(c \cdot i\right) \cdot c\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot x + z \cdot t\right) - \sqrt[3]{\left(c \cdot i\right) \cdot \left(a + c \cdot b\right)} \cdot \left(\sqrt[3]{\left(c \cdot i\right) \cdot \left(a + c \cdot b\right)} \cdot \sqrt[3]{\left(c \cdot i\right) \cdot \left(a + c \cdot b\right)}\right)\right) \cdot 2\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x y z t a b c i)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, A"

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

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