Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, E

Average Error: 5.6 → 2.4

Time: 20.8s

Precision: binary64

\[[y, z] = \mathsf{sort}([y, z]) \[j, k] = \mathsf{sort}([j, k]) \\]

Math

\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(t, a, x \cdot i\right)\\ \mathbf{if}\;x \leq -2.946096318727387 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(-4, t_1, \mathsf{fma}\left(k, j \cdot -27, \mathsf{fma}\left(c, b, 18 \cdot \left(x \cdot \left(\left(t \cdot y\right) \cdot z\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 3.5401247757472833 \cdot 10^{-130}:\\ \;\;\;\;\mathsf{fma}\left(-4, t_1, \mathsf{fma}\left(k, j \cdot -27, \mathsf{fma}\left(c, b, \frac{y \cdot \left(x \cdot t\right)}{\frac{-1}{z}} \cdot -18\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(18, y \cdot \left(t \cdot z\right), -4 \cdot i\right), \mathsf{fma}\left(a, -4 \cdot t, \mathsf{fma}\left(-27, k \cdot j, c \cdot b\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i j k)
 :precision binary64
 (-
  (-
   (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c))
   (* (* x 4.0) i))
  (* (* j 27.0) k)))

↓

(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (fma t a (* x i))))
   (if (<= x -2.946096318727387e-20)
     (fma -4.0 t_1 (fma k (* j -27.0) (fma c b (* 18.0 (* x (* (* t y) z))))))
     (if (<= x 3.5401247757472833e-130)
       (fma
        -4.0
        t_1
        (fma k (* j -27.0) (fma c b (* (/ (* y (* x t)) (/ -1.0 z)) -18.0))))
       (fma
        x
        (fma 18.0 (* y (* t z)) (* -4.0 i))
        (fma a (* -4.0 t) (fma -27.0 (* k j) (* c b))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	return (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
}

↓

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = fma(t, a, (x * i));
	double tmp;
	if (x <= -2.946096318727387e-20) {
		tmp = fma(-4.0, t_1, fma(k, (j * -27.0), fma(c, b, (18.0 * (x * ((t * y) * z))))));
	} else if (x <= 3.5401247757472833e-130) {
		tmp = fma(-4.0, t_1, fma(k, (j * -27.0), fma(c, b, (((y * (x * t)) / (-1.0 / z)) * -18.0))));
	} else {
		tmp = fma(x, fma(18.0, (y * (t * z)), (-4.0 * i)), fma(a, (-4.0 * t), fma(-27.0, (k * j), (c * b))));
	}
	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

Bits error versus j

Bits error versus k

Target

Original	5.6
Target	1.5
Herbie	2.4

\[\begin{array}{l} \mathbf{if}\;t < -1.6210815397541398 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t < 165.68027943805222:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if x < -2.9460963187273869e-20

   1. Initial program 11.7

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

   2. Simplified 1.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(18, y \cdot \left(z \cdot t\right), i \cdot -4\right), \mathsf{fma}\left(a, t \cdot -4, \mathsf{fma}\left(-27, j \cdot k, b \cdot c\right)\right)\right)} \]

   3. Taylor expanded in x around 0 10.0

      \[\leadsto \color{blue}{\left(c \cdot b + 18 \cdot \left(y \cdot \left(t \cdot \left(z \cdot x\right)\right)\right)\right) - \left(4 \cdot \left(i \cdot x\right) + \left(4 \cdot \left(a \cdot t\right) + 27 \cdot \left(k \cdot j\right)\right)\right)} \]

   4. Simplified 10.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \mathsf{fma}\left(t, a, x \cdot i\right), \mathsf{fma}\left(y, 18 \cdot \left(t \cdot \left(z \cdot x\right)\right), \mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right)\right)\right)} \]

   5. Taylor expanded in y around 0 10.0

      \[\leadsto \mathsf{fma}\left(-4, \mathsf{fma}\left(t, a, x \cdot i\right), \color{blue}{\left(c \cdot b + 18 \cdot \left(y \cdot \left(t \cdot \left(z \cdot x\right)\right)\right)\right) - 27 \cdot \left(k \cdot j\right)}\right) \]

   6. Simplified 10.2

      \[\leadsto \mathsf{fma}\left(-4, \mathsf{fma}\left(t, a, x \cdot i\right), \color{blue}{\mathsf{fma}\left(k, j \cdot -27, \mathsf{fma}\left(c, b, 18 \cdot \left(y \cdot \left(t \cdot \left(z \cdot x\right)\right)\right)\right)\right)}\right) \]

   7. Applied egg-rr 10.3

      \[\leadsto \mathsf{fma}\left(-4, \mathsf{fma}\left(t, a, x \cdot i\right), \mathsf{fma}\left(k, j \cdot -27, \mathsf{fma}\left(c, b, 18 \cdot \color{blue}{{\left(\sqrt[3]{y \cdot \left(t \cdot \left(z \cdot x\right)\right)}\right)}^{3}}\right)\right)\right) \]

   8. Applied egg-rr 1.9

      \[\leadsto \mathsf{fma}\left(-4, \mathsf{fma}\left(t, a, x \cdot i\right), \mathsf{fma}\left(k, j \cdot -27, \mathsf{fma}\left(c, b, 18 \cdot \color{blue}{\left(\left(\left(y \cdot t\right) \cdot z\right) \cdot x\right)}\right)\right)\right) \]

   if -2.9460963187273869e-20 < x < 3.5401247757472833e-130

   1. Initial program 1.2

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

   2. Simplified 8.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(18, y \cdot \left(z \cdot t\right), i \cdot -4\right), \mathsf{fma}\left(a, t \cdot -4, \mathsf{fma}\left(-27, j \cdot k, b \cdot c\right)\right)\right)} \]

   3. Taylor expanded in x around 0 1.1

      \[\leadsto \color{blue}{\left(c \cdot b + 18 \cdot \left(y \cdot \left(t \cdot \left(z \cdot x\right)\right)\right)\right) - \left(4 \cdot \left(i \cdot x\right) + \left(4 \cdot \left(a \cdot t\right) + 27 \cdot \left(k \cdot j\right)\right)\right)} \]

   4. Simplified 1.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \mathsf{fma}\left(t, a, x \cdot i\right), \mathsf{fma}\left(y, 18 \cdot \left(t \cdot \left(z \cdot x\right)\right), \mathsf{fma}\left(c, b, \left(k \cdot j\right) \cdot -27\right)\right)\right)} \]

   5. Taylor expanded in y around 0 1.1

      \[\leadsto \mathsf{fma}\left(-4, \mathsf{fma}\left(t, a, x \cdot i\right), \color{blue}{\left(c \cdot b + 18 \cdot \left(y \cdot \left(t \cdot \left(z \cdot x\right)\right)\right)\right) - 27 \cdot \left(k \cdot j\right)}\right) \]

   6. Simplified 1.2

      \[\leadsto \mathsf{fma}\left(-4, \mathsf{fma}\left(t, a, x \cdot i\right), \color{blue}{\mathsf{fma}\left(k, j \cdot -27, \mathsf{fma}\left(c, b, 18 \cdot \left(y \cdot \left(t \cdot \left(z \cdot x\right)\right)\right)\right)\right)}\right) \]

   7. Applied egg-rr 1.3

      \[\leadsto \mathsf{fma}\left(-4, \mathsf{fma}\left(t, a, x \cdot i\right), \mathsf{fma}\left(k, j \cdot -27, \mathsf{fma}\left(c, b, 18 \cdot \color{blue}{{\left(\sqrt[3]{y \cdot \left(t \cdot \left(z \cdot x\right)\right)}\right)}^{3}}\right)\right)\right) \]

   8. Taylor expanded in z around -inf 53.5

      \[\leadsto \mathsf{fma}\left(-4, \mathsf{fma}\left(t, a, x \cdot i\right), \mathsf{fma}\left(k, j \cdot -27, \mathsf{fma}\left(c, b, 18 \cdot \color{blue}{\left(-1 \cdot {\left(e^{0.3333333333333333 \cdot \left(\log \left(y \cdot \left(t \cdot x\right)\right) - \log \left(\frac{-1}{z}\right)\right)}\right)}^{3}\right)}\right)\right)\right) \]

   9. Simplified 1.6

      \[\leadsto \mathsf{fma}\left(-4, \mathsf{fma}\left(t, a, x \cdot i\right), \mathsf{fma}\left(k, j \cdot -27, \mathsf{fma}\left(c, b, 18 \cdot \color{blue}{\left(-\frac{y \cdot \left(t \cdot x\right)}{\frac{-1}{z}}\right)}\right)\right)\right) \]

   if 3.5401247757472833e-130 < x

   1. Initial program 8.1

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

   2. Simplified 3.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(18, y \cdot \left(z \cdot t\right), i \cdot -4\right), \mathsf{fma}\left(a, t \cdot -4, \mathsf{fma}\left(-27, j \cdot k, b \cdot c\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.946096318727387 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(-4, \mathsf{fma}\left(t, a, x \cdot i\right), \mathsf{fma}\left(k, j \cdot -27, \mathsf{fma}\left(c, b, 18 \cdot \left(x \cdot \left(\left(t \cdot y\right) \cdot z\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 3.5401247757472833 \cdot 10^{-130}:\\ \;\;\;\;\mathsf{fma}\left(-4, \mathsf{fma}\left(t, a, x \cdot i\right), \mathsf{fma}\left(k, j \cdot -27, \mathsf{fma}\left(c, b, \frac{y \cdot \left(x \cdot t\right)}{\frac{-1}{z}} \cdot -18\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(18, y \cdot \left(t \cdot z\right), -4 \cdot i\right), \mathsf{fma}\left(a, -4 \cdot t, \mathsf{fma}\left(-27, k \cdot j, c \cdot b\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022130 
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, E"
  :precision binary64

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b))) (if (< t 165.68027943805222) (+ (- (* (* 18.0 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4.0)) (- (* c b) (* 27.0 (* k j)))) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))