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

Percentage Accurate: 95.2% → 98.8%

Time: 21.8s

Precision: binary64

Cost: 13764

• Unsound rule application detected in e-graph. Results may not be sound.

• 26.9% of points produce a very large (infinite) output. You may want to add a precondition.

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

Math

\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

↓

\[\begin{array}{l} \mathbf{if}\;z \leq 1.4 \cdot 10^{-287}:\\ \;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - 9 \cdot \left(y \cdot \left(z \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 2, \mathsf{fma}\left(t, \left(z \cdot y\right) \cdot -9, b \cdot \left(a \cdot 27\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))

↓

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 1.4e-287)
   (fma a (* 27.0 b) (- (* x 2.0) (* 9.0 (* y (* z t)))))
   (fma x 2.0 (fma t (* (* z y) -9.0) (* b (* a 27.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b);
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 1.4e-287) {
		tmp = fma(a, (27.0 * b), ((x * 2.0) - (9.0 * (y * (z * t)))));
	} else {
		tmp = fma(x, 2.0, fma(t, ((z * y) * -9.0), (b * (a * 27.0))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(x * 2.0) - Float64(Float64(Float64(y * 9.0) * z) * t)) + Float64(Float64(a * 27.0) * b))
end

↓

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 1.4e-287)
		tmp = fma(a, Float64(27.0 * b), Float64(Float64(x * 2.0) - Float64(9.0 * Float64(y * Float64(z * t)))));
	else
		tmp = fma(x, 2.0, fma(t, Float64(Float64(z * y) * -9.0), Float64(b * Float64(a * 27.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x * 2.0), $MachinePrecision] - N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 1.4e-287], N[(a * N[(27.0 * b), $MachinePrecision] + N[(N[(x * 2.0), $MachinePrecision] - N[(9.0 * N[(y * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * 2.0 + N[(t * N[(N[(z * y), $MachinePrecision] * -9.0), $MachinePrecision] + N[(b * N[(a * 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	95.2%
Target	94.7%
Herbie	98.8%

\[\begin{array}{l} \mathbf{if}\;y < 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if z < 1.4000000000000001e-287

   1. Initial program 93.8%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Simplified 96.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - 9 \cdot \left(y \cdot \left(z \cdot t\right)\right)\right)} \]
      Step-by-step derivation

      [Start] 93.8%	\[ \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
      +-commutative [=>] 93.8%	\[ \color{blue}{\left(a \cdot 27\right) \cdot b + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} \]
      associate-*l* [=>] 93.8%	\[ \color{blue}{a \cdot \left(27 \cdot b\right)} + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \]
      fma-def [=>] 95.4%	\[ \color{blue}{\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} \]
      associate-*l* [=>] 96.4%	\[ \mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - \color{blue}{\left(y \cdot 9\right) \cdot \left(z \cdot t\right)}\right) \]
      *-commutative [=>] 96.4%	\[ \mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - \color{blue}{\left(9 \cdot y\right)} \cdot \left(z \cdot t\right)\right) \]
      associate-*l* [=>] 96.4%	\[ \mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - \color{blue}{9 \cdot \left(y \cdot \left(z \cdot t\right)\right)}\right) \]

   if 1.4000000000000001e-287 < z

   1. Initial program 95.3%

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Simplified 96.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(t, \left(y \cdot z\right) \cdot -9, \left(a \cdot 27\right) \cdot b\right)\right)} \]
      Step-by-step derivation

      [Start] 95.3%	\[ \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
      associate-+l- [=>] 95.3%	\[ \color{blue}{x \cdot 2 - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)} \]
      fma-neg [=>] 95.3%	\[ \color{blue}{\mathsf{fma}\left(x, 2, -\left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)\right)} \]
      neg-sub0 [=>] 95.3%	\[ \mathsf{fma}\left(x, 2, \color{blue}{0 - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - \left(a \cdot 27\right) \cdot b\right)}\right) \]
      associate-+l- [<=] 95.3%	\[ \mathsf{fma}\left(x, 2, \color{blue}{\left(0 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b}\right) \]
      neg-sub0 [<=] 95.3%	\[ \mathsf{fma}\left(x, 2, \color{blue}{\left(-\left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)} + \left(a \cdot 27\right) \cdot b\right) \]
      *-commutative [=>] 95.3%	\[ \mathsf{fma}\left(x, 2, \left(-\color{blue}{t \cdot \left(\left(y \cdot 9\right) \cdot z\right)}\right) + \left(a \cdot 27\right) \cdot b\right) \]
      distribute-rgt-neg-in [=>] 95.3%	\[ \mathsf{fma}\left(x, 2, \color{blue}{t \cdot \left(-\left(y \cdot 9\right) \cdot z\right)} + \left(a \cdot 27\right) \cdot b\right) \]
      fma-def [=>] 96.1%	\[ \mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(t, -\left(y \cdot 9\right) \cdot z, \left(a \cdot 27\right) \cdot b\right)}\right) \]
      *-commutative [=>] 96.1%	\[ \mathsf{fma}\left(x, 2, \mathsf{fma}\left(t, -\color{blue}{z \cdot \left(y \cdot 9\right)}, \left(a \cdot 27\right) \cdot b\right)\right) \]
      associate-*r* [=>] 96.1%	\[ \mathsf{fma}\left(x, 2, \mathsf{fma}\left(t, -\color{blue}{\left(z \cdot y\right) \cdot 9}, \left(a \cdot 27\right) \cdot b\right)\right) \]
      distribute-rgt-neg-in [=>] 96.1%	\[ \mathsf{fma}\left(x, 2, \mathsf{fma}\left(t, \color{blue}{\left(z \cdot y\right) \cdot \left(-9\right)}, \left(a \cdot 27\right) \cdot b\right)\right) \]
      *-commutative [=>] 96.1%	\[ \mathsf{fma}\left(x, 2, \mathsf{fma}\left(t, \color{blue}{\left(y \cdot z\right)} \cdot \left(-9\right), \left(a \cdot 27\right) \cdot b\right)\right) \]
      metadata-eval [=>] 96.1%	\[ \mathsf{fma}\left(x, 2, \mathsf{fma}\left(t, \left(y \cdot z\right) \cdot \color{blue}{-9}, \left(a \cdot 27\right) \cdot b\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.4 \cdot 10^{-287}:\\ \;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - 9 \cdot \left(y \cdot \left(z \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 2, \mathsf{fma}\left(t, \left(z \cdot y\right) \cdot -9, b \cdot \left(a \cdot 27\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.8%
Cost	13764

\[\begin{array}{l} \mathbf{if}\;z \leq 1.4 \cdot 10^{-287}:\\ \;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - 9 \cdot \left(y \cdot \left(z \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 2, \mathsf{fma}\left(t, \left(z \cdot y\right) \cdot -9, b \cdot \left(a \cdot 27\right)\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	98.5%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;z \leq 10^{-287}:\\ \;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - 9 \cdot \left(y \cdot \left(z \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 - \left(a \cdot \left(b \cdot -27\right) + 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	81.2%
Cost	1609

\[\begin{array}{l} t_1 := b \cdot \left(a \cdot 27\right)\\ \mathbf{if}\;t_1 \leq -1.5 \cdot 10^{-65} \lor \neg \left(t_1 \leq 4 \cdot 10^{+132}\right):\\ \;\;\;\;27 \cdot \left(a \cdot b\right) - 9 \cdot \left(y \cdot \left(z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 - 9 \cdot \left(z \cdot \left(y \cdot t\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	47.9%
Cost	1372

\[\begin{array}{l} t_1 := -9 \cdot \left(y \cdot \left(z \cdot t\right)\right)\\ \mathbf{if}\;z \leq -3.7 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-256}:\\ \;\;\;\;x \cdot 2\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-290}:\\ \;\;\;\;27 \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-216}:\\ \;\;\;\;x \cdot 2\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-135}:\\ \;\;\;\;a \cdot \left(27 \cdot b\right)\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+31}:\\ \;\;\;\;x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-9 \cdot \left(z \cdot \left(y \cdot t\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	96.6%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;z \leq 6.5 \cdot 10^{+39}:\\ \;\;\;\;b \cdot \left(a \cdot 27\right) + \left(x \cdot 2 - y \cdot \left(z \cdot \left(9 \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 - z \cdot \left(9 \cdot \left(y \cdot t\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	98.1%
Cost	1220

\[\begin{array}{l} t_1 := b \cdot \left(a \cdot 27\right)\\ \mathbf{if}\;z \leq 10^{-287}:\\ \;\;\;\;t_1 + \left(x \cdot 2 - y \cdot \left(z \cdot \left(9 \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(x \cdot 2 - t \cdot \left(9 \cdot \left(z \cdot y\right)\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	98.1%
Cost	1220

\[\begin{array}{l} t_1 := b \cdot \left(a \cdot 27\right)\\ \mathbf{if}\;z \leq 10^{-287}:\\ \;\;\;\;t_1 + \left(x \cdot 2 - y \cdot \left(z \cdot \left(9 \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(x \cdot 2 - t \cdot \left(z \cdot \left(9 \cdot y\right)\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	98.1%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-287}:\\ \;\;\;\;b \cdot \left(a \cdot 27\right) + \left(x \cdot 2 - y \cdot \left(z \cdot \left(9 \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 - \left(a \cdot \left(b \cdot -27\right) + 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	98.0%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{-287}:\\ \;\;\;\;x \cdot 2 - \left(\left(z \cdot t\right) \cdot \left(9 \cdot y\right) - a \cdot \left(27 \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 - \left(a \cdot \left(b \cdot -27\right) + 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\\ \end{array} \]

Alternative 10
Accuracy	74.5%
Cost	1104

\[\begin{array}{l} t_1 := x \cdot 2 - a \cdot \left(b \cdot -27\right)\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{+47}:\\ \;\;\;\;y \cdot \left(\left(z \cdot t\right) \cdot -9\right)\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{-65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-25}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \left(t \cdot -9\right)\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+32}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-9 \cdot \left(z \cdot \left(y \cdot t\right)\right)\\ \end{array} \]

Alternative 11
Accuracy	77.3%
Cost	1100

\[\begin{array}{l} t_1 := x \cdot 2 - 9 \cdot \left(y \cdot \left(z \cdot t\right)\right)\\ \mathbf{if}\;z \leq -2.65 \cdot 10^{-28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-136}:\\ \;\;\;\;x \cdot 2 - a \cdot \left(b \cdot -27\right)\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-9 \cdot \left(z \cdot \left(y \cdot t\right)\right)\\ \end{array} \]

Alternative 12
Accuracy	78.2%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-25} \lor \neg \left(z \leq 1.22 \cdot 10^{-135}\right):\\ \;\;\;\;x \cdot 2 - 9 \cdot \left(z \cdot \left(y \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 - a \cdot \left(b \cdot -27\right)\\ \end{array} \]

Alternative 13
Accuracy	78.2%
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{-23}:\\ \;\;\;\;x \cdot 2 - 9 \cdot \left(z \cdot \left(y \cdot t\right)\right)\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-134}:\\ \;\;\;\;x \cdot 2 - a \cdot \left(b \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 - z \cdot \left(9 \cdot \left(y \cdot t\right)\right)\\ \end{array} \]

Alternative 14
Accuracy	46.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-177} \lor \neg \left(t \leq 3.4 \cdot 10^{+59}\right):\\ \;\;\;\;-9 \cdot \left(y \cdot \left(z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2\\ \end{array} \]

Alternative 15
Accuracy	47.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-171}:\\ \;\;\;\;-9 \cdot \left(y \cdot \left(z \cdot t\right)\right)\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+59}:\\ \;\;\;\;x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(z \cdot -9\right)\right)\\ \end{array} \]

Alternative 16
Accuracy	47.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-171}:\\ \;\;\;\;y \cdot \left(\left(z \cdot t\right) \cdot -9\right)\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+59}:\\ \;\;\;\;x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(z \cdot -9\right)\right)\\ \end{array} \]

Alternative 17
Accuracy	47.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-171}:\\ \;\;\;\;y \cdot \left(\left(z \cdot t\right) \cdot -9\right)\\ \mathbf{elif}\;t \leq 2.75 \cdot 10^{+60}:\\ \;\;\;\;x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \left(t \cdot -9\right)\\ \end{array} \]

Alternative 18
Accuracy	46.9%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-22}:\\ \;\;\;\;x \cdot 2\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;27 \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2\\ \end{array} \]

Alternative 19
Accuracy	46.9%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-22}:\\ \;\;\;\;x \cdot 2\\ \mathbf{elif}\;x \leq 1.8:\\ \;\;\;\;a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2\\ \end{array} \]

Alternative 20
Accuracy	30.2%
Cost	192

\[x \cdot 2 \]

Reproduce

herbie shell --seed 2023263 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< y 7.590524218811189e-161) (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* a (* 27.0 b))) (+ (- (* x 2.0) (* 9.0 (* y (* t z)))) (* (* a 27.0) b)))

  (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))