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

Average Error: 5.5 → 1.1

Time: 9.0s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_1 := -4 \cdot \left(x \cdot i\right)\\ t_2 := \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\\ t_3 := -27 \cdot \left(k \cdot j\right) + -4 \cdot \left(t \cdot a\right)\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;b \cdot c + \left(t_1 + \left(18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right) + t_3\right)\right)\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+301}:\\ \;\;\;\;t_2 + k \cdot \left(-27 \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c + \left(t_1 + \left(t_3 + 18 \cdot \left(y \cdot \left(x \cdot \left(z \cdot t\right)\right)\right)\right)\right)\\ \end{array} \]

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
 (let* ((t_1 (* -4.0 (* x i)))
        (t_2
         (+
          (+ (+ (* (* (* (* x 18.0) y) z) t) (* t (* a -4.0))) (* b c))
          (* i (* x -4.0))))
        (t_3 (+ (* -27.0 (* k j)) (* -4.0 (* t a)))))
   (if (<= t_2 (- INFINITY))
     (+ (* b c) (+ t_1 (+ (* 18.0 (* y (* t (* x z)))) t_3)))
     (if (<= t_2 5e+301)
       (+ t_2 (* k (* -27.0 j)))
       (+ (* b c) (+ t_1 (+ t_3 (* 18.0 (* y (* x (* z t)))))))))))

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 = -4.0 * (x * i);
	double t_2 = ((((((x * 18.0) * y) * z) * t) + (t * (a * -4.0))) + (b * c)) + (i * (x * -4.0));
	double t_3 = (-27.0 * (k * j)) + (-4.0 * (t * a));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (b * c) + (t_1 + ((18.0 * (y * (t * (x * z)))) + t_3));
	} else if (t_2 <= 5e+301) {
		tmp = t_2 + (k * (-27.0 * j));
	} else {
		tmp = (b * c) + (t_1 + (t_3 + (18.0 * (y * (x * (z * t))))));
	}
	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 = -4.0 * (x * i);
	double t_2 = ((((((x * 18.0) * y) * z) * t) + (t * (a * -4.0))) + (b * c)) + (i * (x * -4.0));
	double t_3 = (-27.0 * (k * j)) + (-4.0 * (t * a));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (b * c) + (t_1 + ((18.0 * (y * (t * (x * z)))) + t_3));
	} else if (t_2 <= 5e+301) {
		tmp = t_2 + (k * (-27.0 * j));
	} else {
		tmp = (b * c) + (t_1 + (t_3 + (18.0 * (y * (x * (z * t))))));
	}
	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 = -4.0 * (x * i)
	t_2 = ((((((x * 18.0) * y) * z) * t) + (t * (a * -4.0))) + (b * c)) + (i * (x * -4.0))
	t_3 = (-27.0 * (k * j)) + (-4.0 * (t * a))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (b * c) + (t_1 + ((18.0 * (y * (t * (x * z)))) + t_3))
	elif t_2 <= 5e+301:
		tmp = t_2 + (k * (-27.0 * j))
	else:
		tmp = (b * c) + (t_1 + (t_3 + (18.0 * (y * (x * (z * t))))))
	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(-4.0 * Float64(x * i))
	t_2 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 18.0) * y) * z) * t) + Float64(t * Float64(a * -4.0))) + Float64(b * c)) + Float64(i * Float64(x * -4.0)))
	t_3 = Float64(Float64(-27.0 * Float64(k * j)) + Float64(-4.0 * Float64(t * a)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(b * c) + Float64(t_1 + Float64(Float64(18.0 * Float64(y * Float64(t * Float64(x * z)))) + t_3)));
	elseif (t_2 <= 5e+301)
		tmp = Float64(t_2 + Float64(k * Float64(-27.0 * j)));
	else
		tmp = Float64(Float64(b * c) + Float64(t_1 + Float64(t_3 + Float64(18.0 * Float64(y * Float64(x * Float64(z * t)))))));
	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 = -4.0 * (x * i);
	t_2 = ((((((x * 18.0) * y) * z) * t) + (t * (a * -4.0))) + (b * c)) + (i * (x * -4.0));
	t_3 = (-27.0 * (k * j)) + (-4.0 * (t * a));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (b * c) + (t_1 + ((18.0 * (y * (t * (x * z)))) + t_3));
	elseif (t_2 <= 5e+301)
		tmp = t_2 + (k * (-27.0 * j));
	else
		tmp = (b * c) + (t_1 + (t_3 + (18.0 * (y * (x * (z * t))))));
	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[(-4.0 * N[(x * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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] + N[(i * N[(x * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-27.0 * N[(k * j), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(b * c), $MachinePrecision] + N[(t$95$1 + N[(N[(18.0 * N[(y * N[(t * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+301], N[(t$95$2 + N[(k * N[(-27.0 * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * c), $MachinePrecision] + N[(t$95$1 + N[(t$95$3 + N[(18.0 * N[(y * N[(x * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	5.5
Target	1.5
Herbie	1.1

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

Derivation

1. Split input into 3 regimes

2. if (-.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

   1. Initial program 64.0

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

   2. Simplified 37.7

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

   3. Taylor expanded in y around inf 11.9

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

   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)) < 5.0000000000000004e301

   1. Initial program 0.4

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

   if 5.0000000000000004e301 < (-.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 52.0

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

   2. Simplified 34.0

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

   3. Taylor expanded in y around inf 12.1

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

   4. Applied egg-rr 5.1

      \[\leadsto c \cdot b + \left(-4 \cdot \left(i \cdot x\right) + \left(18 \cdot \color{blue}{{\left(\left(x \cdot \left(t \cdot z\right)\right) \cdot y\right)}^{1}} + \left(-27 \cdot \left(k \cdot j\right) + -4 \cdot \left(a \cdot t\right)\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right) \leq -\infty:\\ \;\;\;\;b \cdot c + \left(-4 \cdot \left(x \cdot i\right) + \left(18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right) + \left(-27 \cdot \left(k \cdot j\right) + -4 \cdot \left(t \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right) \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(-27 \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot c + \left(-4 \cdot \left(x \cdot i\right) + \left(\left(-27 \cdot \left(k \cdot j\right) + -4 \cdot \left(t \cdot a\right)\right) + 18 \cdot \left(y \cdot \left(x \cdot \left(z \cdot t\right)\right)\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(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)))