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

Average Error: 5.6 → 1.8

Time: 12.1s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -4.921965120671141 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(i, x \cdot -4, \mathsf{fma}\left(t, \mathsf{fma}\left(x, 18 \cdot \left(y \cdot z\right), -4 \cdot a\right), \left(k \cdot j\right) \cdot -27\right)\right)\right)\\ \mathbf{elif}\;t \leq 1.607584982652585 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(18, y \cdot \left(t \cdot z\right), i \cdot -4\right), \mathsf{fma}\left(a, t \cdot -4, \mathsf{fma}\left(-27, k \cdot j, c \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(t, \mathsf{fma}\left(y, 18 \cdot \left(x \cdot z\right), -4 \cdot a\right), \mathsf{fma}\left(k \cdot j, -27, -4 \cdot \left(i \cdot x\right)\right)\right)\right)\\ \end{array} \]

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
 (if (<= t -4.921965120671141e+36)
   (fma
    c
    b
    (fma
     i
     (* x -4.0)
     (fma t (fma x (* 18.0 (* y z)) (* -4.0 a)) (* (* k j) -27.0))))
   (if (<= t 1.607584982652585e-82)
     (fma
      x
      (fma 18.0 (* y (* t z)) (* i -4.0))
      (fma a (* t -4.0) (fma -27.0 (* k j) (* c b))))
     (fma
      c
      b
      (fma
       t
       (fma y (* 18.0 (* x z)) (* -4.0 a))
       (fma (* k j) -27.0 (* -4.0 (* i x))))))))

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 tmp;
	if (t <= -4.921965120671141e+36) {
		tmp = fma(c, b, fma(i, (x * -4.0), fma(t, fma(x, (18.0 * (y * z)), (-4.0 * a)), ((k * j) * -27.0))));
	} else if (t <= 1.607584982652585e-82) {
		tmp = fma(x, fma(18.0, (y * (t * z)), (i * -4.0)), fma(a, (t * -4.0), fma(-27.0, (k * j), (c * b))));
	} else {
		tmp = fma(c, b, fma(t, fma(y, (18.0 * (x * z)), (-4.0 * a)), fma((k * j), -27.0, (-4.0 * (i * x)))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j, k)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 18.0) * y) * z) * t) - Float64(Float64(a * 4.0) * t)) + Float64(b * c)) - Float64(Float64(x * 4.0) * i)) - Float64(Float64(j * 27.0) * k))
end

↓

function code(x, y, z, t, a, b, c, i, j, k)
	tmp = 0.0
	if (t <= -4.921965120671141e+36)
		tmp = fma(c, b, fma(i, Float64(x * -4.0), fma(t, fma(x, Float64(18.0 * Float64(y * z)), Float64(-4.0 * a)), Float64(Float64(k * j) * -27.0))));
	elseif (t <= 1.607584982652585e-82)
		tmp = fma(x, fma(18.0, Float64(y * Float64(t * z)), Float64(i * -4.0)), fma(a, Float64(t * -4.0), fma(-27.0, Float64(k * j), Float64(c * b))));
	else
		tmp = fma(c, b, fma(t, fma(y, Float64(18.0 * Float64(x * z)), Float64(-4.0 * a)), fma(Float64(k * j), -27.0, Float64(-4.0 * Float64(i * x)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := N[(N[(N[(N[(N[(N[(N[(N[(x * 18.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] - N[(N[(x * 4.0), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] - N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := If[LessEqual[t, -4.921965120671141e+36], N[(c * b + N[(i * N[(x * -4.0), $MachinePrecision] + N[(t * N[(x * N[(18.0 * N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * a), $MachinePrecision]), $MachinePrecision] + N[(N[(k * j), $MachinePrecision] * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.607584982652585e-82], N[(x * N[(18.0 * N[(y * N[(t * z), $MachinePrecision]), $MachinePrecision] + N[(i * -4.0), $MachinePrecision]), $MachinePrecision] + N[(a * N[(t * -4.0), $MachinePrecision] + N[(-27.0 * N[(k * j), $MachinePrecision] + N[(c * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * b + N[(t * N[(y * N[(18.0 * N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * a), $MachinePrecision]), $MachinePrecision] + N[(N[(k * j), $MachinePrecision] * -27.0 + N[(-4.0 * N[(i * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

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	1.8
Herbie	1.8

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

Derivation

1. Split input into 3 regimes

2. if t < -4.92196512067114091e36

   1. Initial program 1.9

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

   2. Simplified 13.5

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

   3. Taylor expanded in x around 0 5.5

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

   4. Simplified 1.3

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

   5. Taylor expanded in t around 0 5.4

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

   6. Simplified 1.3

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

   if -4.92196512067114091e36 < t < 1.6075849826525849e-82

   1. Initial program 8.0

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

   2. Simplified 1.6

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

   3. Applied *-un-lft-identity_binary64 1.6

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

   4. Applied associate-*l*_binary64 1.6

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

   5. Simplified 1.6

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

   if 1.6075849826525849e-82 < t

   1. Initial program 2.8

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

   2. Simplified 9.0

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

   3. Taylor expanded in x around 0 5.2

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

   4. Simplified 2.5

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

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.921965120671141 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(i, x \cdot -4, \mathsf{fma}\left(t, \mathsf{fma}\left(x, 18 \cdot \left(y \cdot z\right), -4 \cdot a\right), \left(k \cdot j\right) \cdot -27\right)\right)\right)\\ \mathbf{elif}\;t \leq 1.607584982652585 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(18, y \cdot \left(t \cdot z\right), i \cdot -4\right), \mathsf{fma}\left(a, t \cdot -4, \mathsf{fma}\left(-27, k \cdot j, c \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(t, \mathsf{fma}\left(y, 18 \cdot \left(x \cdot z\right), -4 \cdot a\right), \mathsf{fma}\left(k \cdot j, -27, -4 \cdot \left(i \cdot x\right)\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(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)))