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

Average Error: 5.6 → 1.7

Time: 8.9s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_1 := \left(k \cdot j\right) \cdot -27\\ t_2 := i \cdot \left(x \cdot -4\right)\\ t_3 := \left(\left(b \cdot c + \left(18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right) + \left(t \cdot a\right) \cdot -4\right)\right) + t_2\right) + t_1\\ t_4 := \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + t_2\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_4 \leq 3.5770137452836846 \cdot 10^{+278}:\\ \;\;\;\;t_4 + t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

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 (* (* k j) -27.0))
        (t_2 (* i (* x -4.0)))
        (t_3
         (+
          (+ (+ (* b c) (+ (* 18.0 (* y (* t (* x z)))) (* (* t a) -4.0))) t_2)
          t_1))
        (t_4
         (+
          (+ (+ (* (* (* (* x 18.0) y) z) t) (* t (* a -4.0))) (* b c))
          t_2)))
   (if (<= t_4 (- INFINITY))
     t_3
     (if (<= t_4 3.5770137452836846e+278) (+ t_4 t_1) t_3))))

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 = (k * j) * -27.0;
	double t_2 = i * (x * -4.0);
	double t_3 = (((b * c) + ((18.0 * (y * (t * (x * z)))) + ((t * a) * -4.0))) + t_2) + t_1;
	double t_4 = ((((((x * 18.0) * y) * z) * t) + (t * (a * -4.0))) + (b * c)) + t_2;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_4 <= 3.5770137452836846e+278) {
		tmp = t_4 + t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

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 = (k * j) * -27.0;
	double t_2 = i * (x * -4.0);
	double t_3 = (((b * c) + ((18.0 * (y * (t * (x * z)))) + ((t * a) * -4.0))) + t_2) + t_1;
	double t_4 = ((((((x * 18.0) * y) * z) * t) + (t * (a * -4.0))) + (b * c)) + t_2;
	double tmp;
	if (t_4 <= -Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else if (t_4 <= 3.5770137452836846e+278) {
		tmp = t_4 + t_1;
	} else {
		tmp = t_3;
	}
	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 = (k * j) * -27.0
	t_2 = i * (x * -4.0)
	t_3 = (((b * c) + ((18.0 * (y * (t * (x * z)))) + ((t * a) * -4.0))) + t_2) + t_1
	t_4 = ((((((x * 18.0) * y) * z) * t) + (t * (a * -4.0))) + (b * c)) + t_2
	tmp = 0
	if t_4 <= -math.inf:
		tmp = t_3
	elif t_4 <= 3.5770137452836846e+278:
		tmp = t_4 + t_1
	else:
		tmp = t_3
	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(Float64(k * j) * -27.0)
	t_2 = Float64(i * Float64(x * -4.0))
	t_3 = Float64(Float64(Float64(Float64(b * c) + Float64(Float64(18.0 * Float64(y * Float64(t * Float64(x * z)))) + Float64(Float64(t * a) * -4.0))) + t_2) + t_1)
	t_4 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 18.0) * y) * z) * t) + Float64(t * Float64(a * -4.0))) + Float64(b * c)) + t_2)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_4 <= 3.5770137452836846e+278)
		tmp = Float64(t_4 + t_1);
	else
		tmp = t_3;
	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 = (k * j) * -27.0;
	t_2 = i * (x * -4.0);
	t_3 = (((b * c) + ((18.0 * (y * (t * (x * z)))) + ((t * a) * -4.0))) + t_2) + t_1;
	t_4 = ((((((x * 18.0) * y) * z) * t) + (t * (a * -4.0))) + (b * c)) + t_2;
	tmp = 0.0;
	if (t_4 <= -Inf)
		tmp = t_3;
	elseif (t_4 <= 3.5770137452836846e+278)
		tmp = t_4 + t_1;
	else
		tmp = t_3;
	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[(N[(k * j), $MachinePrecision] * -27.0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(x * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(b * c), $MachinePrecision] + N[(N[(18.0 * N[(y * N[(t * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[(N[(N[(x * 18.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] + N[(t * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], t$95$3, If[LessEqual[t$95$4, 3.5770137452836846e+278], N[(t$95$4 + t$95$1), $MachinePrecision], t$95$3]]]]]]

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	1.6
Herbie	1.7

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t) (*.f64 (*.f64 a 4) t)) (*.f64 b c)) (*.f64 (*.f64 x 4) i)) < -inf.0 or 3.57701374528368462e278 < (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t) (*.f64 (*.f64 a 4) t)) (*.f64 b c)) (*.f64 (*.f64 x 4) i))

   1. Initial program 42.9

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

   2. Taylor expanded in x around 0 11.5

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

   3. Taylor expanded in j around 0 11.5

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

   4. Taylor expanded in a around 0 11.1

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

   if -inf.0 < (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t) (*.f64 (*.f64 a 4) t)) (*.f64 b c)) (*.f64 (*.f64 x 4) i)) < 3.57701374528368462e278

   1. Initial program 0.4

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

   2. Taylor expanded in j around 0 0.4

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

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right) \leq -\infty:\\ \;\;\;\;\left(\left(b \cdot c + \left(18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right) + \left(t \cdot a\right) \cdot -4\right)\right) + i \cdot \left(x \cdot -4\right)\right) + \left(k \cdot j\right) \cdot -27\\ \mathbf{elif}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right) \leq 3.5770137452836846 \cdot 10^{+278}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + \left(k \cdot j\right) \cdot -27\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right) + \left(t \cdot a\right) \cdot -4\right)\right) + i \cdot \left(x \cdot -4\right)\right) + \left(k \cdot j\right) \cdot -27\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(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)))