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

Average Error: 5.3 → 1.6

Time: 5.7s

Precision: binary64

\[ \begin{array}{c}[y, z] = \mathsf{sort}([y, z])\\ [j, k] = \mathsf{sort}([j, k])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_1 := i \cdot \left(x \cdot -4\right)\\ t_2 := t \cdot \left(a \cdot -4\right)\\ \mathbf{if}\;y \leq -1.65 \cdot 10^{-146}:\\ \;\;\;\;\left(\left(\left(y \cdot \left(\left(z \cdot x\right) \cdot \left(18 \cdot t\right)\right) + t_2\right) + b \cdot c\right) + t_1\right) + \left(k \cdot j\right) \cdot -27\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(z \cdot \left(x \cdot \left(18 \cdot \left(y \cdot t\right)\right)\right) + t_2\right)\right) + t_1\right) + k \cdot \left(j \cdot -27\right)\\ \end{array} \]

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 (* i (* x -4.0))) (t_2 (* t (* a -4.0))))
   (if (<= y -1.65e-146)
     (+
      (+ (+ (+ (* y (* (* z x) (* 18.0 t))) t_2) (* b c)) t_1)
      (* (* k j) -27.0))
     (+
      (+ (+ (* b c) (+ (* z (* x (* 18.0 (* y t)))) t_2)) t_1)
      (* k (* j -27.0))))))

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 = i * (x * -4.0);
	double t_2 = t * (a * -4.0);
	double tmp;
	if (y <= -1.65e-146) {
		tmp = ((((y * ((z * x) * (18.0 * t))) + t_2) + (b * c)) + t_1) + ((k * j) * -27.0);
	} else {
		tmp = (((b * c) + ((z * (x * (18.0 * (y * t)))) + t_2)) + t_1) + (k * (j * -27.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j, k)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    code = (((((((x * 18.0d0) * y) * z) * t) - ((a * 4.0d0) * t)) + (b * c)) - ((x * 4.0d0) * i)) - ((j * 27.0d0) * k)
end function

↓

real(8) function code(x, y, z, t, a, b, c, i, j, k)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = i * (x * (-4.0d0))
    t_2 = t * (a * (-4.0d0))
    if (y <= (-1.65d-146)) then
        tmp = ((((y * ((z * x) * (18.0d0 * t))) + t_2) + (b * c)) + t_1) + ((k * j) * (-27.0d0))
    else
        tmp = (((b * c) + ((z * (x * (18.0d0 * (y * t)))) + t_2)) + t_1) + (k * (j * (-27.0d0)))
    end if
    code = tmp
end function

Java

public static 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);
}

↓

public static 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 = i * (x * -4.0);
	double t_2 = t * (a * -4.0);
	double tmp;
	if (y <= -1.65e-146) {
		tmp = ((((y * ((z * x) * (18.0 * t))) + t_2) + (b * c)) + t_1) + ((k * j) * -27.0);
	} else {
		tmp = (((b * c) + ((z * (x * (18.0 * (y * t)))) + t_2)) + t_1) + (k * (j * -27.0));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t, a, b, c, i, j, k):
	t_1 = i * (x * -4.0)
	t_2 = t * (a * -4.0)
	tmp = 0
	if y <= -1.65e-146:
		tmp = ((((y * ((z * x) * (18.0 * t))) + t_2) + (b * c)) + t_1) + ((k * j) * -27.0)
	else:
		tmp = (((b * c) + ((z * (x * (18.0 * (y * t)))) + t_2)) + t_1) + (k * (j * -27.0))
	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)
	t_1 = Float64(i * Float64(x * -4.0))
	t_2 = Float64(t * Float64(a * -4.0))
	tmp = 0.0
	if (y <= -1.65e-146)
		tmp = Float64(Float64(Float64(Float64(Float64(y * Float64(Float64(z * x) * Float64(18.0 * t))) + t_2) + Float64(b * c)) + t_1) + Float64(Float64(k * j) * -27.0));
	else
		tmp = Float64(Float64(Float64(Float64(b * c) + Float64(Float64(z * Float64(x * Float64(18.0 * Float64(y * t)))) + t_2)) + t_1) + Float64(k * Float64(j * -27.0)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t, a, b, c, i, j, k)
	t_1 = i * (x * -4.0);
	t_2 = t * (a * -4.0);
	tmp = 0.0;
	if (y <= -1.65e-146)
		tmp = ((((y * ((z * x) * (18.0 * t))) + t_2) + (b * c)) + t_1) + ((k * j) * -27.0);
	else
		tmp = (((b * c) + ((z * (x * (18.0 * (y * t)))) + t_2)) + t_1) + (k * (j * -27.0));
	end
	tmp_2 = 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_] := Block[{t$95$1 = N[(i * N[(x * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.65e-146], N[(N[(N[(N[(N[(y * N[(N[(z * x), $MachinePrecision] * N[(18.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(N[(k * j), $MachinePrecision] * -27.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b * c), $MachinePrecision] + N[(N[(z * N[(x * N[(18.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(k * N[(j * -27.0), $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	5.3
Target	1.7
Herbie	1.6

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

Derivation

1. Split input into 2 regimes

2. if y < -1.65e-146

   1. Initial program 7.6

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

   2. Taylor expanded in x around 0 2.0

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

   3. Simplified 2.1

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

   4. Taylor expanded in j around 0 2.0

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

   if -1.65e-146 < y

   1. Initial program 2.3

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

   2. Taylor expanded in x around 0 8.5

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

   3. Simplified 8.6

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

   4. Taylor expanded in y around 0 8.5

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

   5. Simplified 0.9

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(x \cdot \left(18 \cdot \left(t \cdot y\right)\right)\right)} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-146}:\\ \;\;\;\;\left(\left(\left(y \cdot \left(\left(z \cdot x\right) \cdot \left(18 \cdot t\right)\right) + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + \left(k \cdot j\right) \cdot -27\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(z \cdot \left(x \cdot \left(18 \cdot \left(y \cdot t\right)\right)\right) + t \cdot \left(a \cdot -4\right)\right)\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(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)))