Diagrams.Solve.Polynomial:quartForm from diagrams-solve-0.1, C

Percentage Accurate: 97.8% → 98.9%

Time: 10.9s

Precision: binary64

Cost: 7876

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

• 33.8% of points produce a very large (infinite) output. You may want to add a precondition.

Math

\[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

↓

\[\begin{array}{l} t_1 := \left(\frac{z \cdot t}{16} + x \cdot y\right) - \frac{b \cdot a}{4}\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;c + t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot \left(t \cdot 0.0625\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))

↓

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (- (+ (/ (* z t) 16.0) (* x y)) (/ (* b a) 4.0))))
   (if (<= t_1 INFINITY) (+ c t_1) (fma y x (* z (* t 0.0625))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

↓

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (((z * t) / 16.0) + (x * y)) - ((b * a) / 4.0);
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = c + t_1;
	} else {
		tmp = fma(y, x, (z * (t * 0.0625)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0)) - Float64(Float64(a * b) / 4.0)) + c)
end

↓

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(Float64(z * t) / 16.0) + Float64(x * y)) - Float64(Float64(b * a) / 4.0))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = Float64(c + t_1);
	else
		tmp = fma(y, x, Float64(z * Float64(t * 0.0625)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] - N[(N[(b * a), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], N[(c + t$95$1), $MachinePrecision], N[(y * x + N[(z * N[(t * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) 16)) (/.f64 (*.f64 a b) 4)) < +inf.0

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   if +inf.0 < (-.f64 (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) 16)) (/.f64 (*.f64 a b) 4))

   1. Initial program 0.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Simplified 50.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \mathsf{fma}\left(b, -0.25 \cdot a, \mathsf{fma}\left(z, \frac{t}{16}, c\right)\right)\right)} \]
      Step-by-step derivation

      [Start] 0.0%	\[ \left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
      associate-+l- [=>] 0.0%	\[ \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(\frac{a \cdot b}{4} - c\right)} \]
      +-commutative [=>] 0.0%	\[ \color{blue}{\left(\frac{z \cdot t}{16} + x \cdot y\right)} - \left(\frac{a \cdot b}{4} - c\right) \]
      associate--l+ [=>] 0.0%	\[ \color{blue}{\frac{z \cdot t}{16} + \left(x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right)} \]
      +-commutative [=>] 0.0%	\[ \color{blue}{\left(x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right) + \frac{z \cdot t}{16}} \]
      associate-+l- [=>] 0.0%	\[ \color{blue}{x \cdot y - \left(\left(\frac{a \cdot b}{4} - c\right) - \frac{z \cdot t}{16}\right)} \]
      fma-neg [=>] 33.3%	\[ \color{blue}{\mathsf{fma}\left(x, y, -\left(\left(\frac{a \cdot b}{4} - c\right) - \frac{z \cdot t}{16}\right)\right)} \]
      neg-sub0 [=>] 33.3%	\[ \mathsf{fma}\left(x, y, \color{blue}{0 - \left(\left(\frac{a \cdot b}{4} - c\right) - \frac{z \cdot t}{16}\right)}\right) \]
      associate--l- [=>] 33.3%	\[ \mathsf{fma}\left(x, y, 0 - \color{blue}{\left(\frac{a \cdot b}{4} - \left(c + \frac{z \cdot t}{16}\right)\right)}\right) \]
      associate-+l- [<=] 33.3%	\[ \mathsf{fma}\left(x, y, \color{blue}{\left(0 - \frac{a \cdot b}{4}\right) + \left(c + \frac{z \cdot t}{16}\right)}\right) \]
      neg-sub0 [<=] 33.3%	\[ \mathsf{fma}\left(x, y, \color{blue}{\left(-\frac{a \cdot b}{4}\right)} + \left(c + \frac{z \cdot t}{16}\right)\right) \]
      *-commutative [=>] 33.3%	\[ \mathsf{fma}\left(x, y, \left(-\frac{\color{blue}{b \cdot a}}{4}\right) + \left(c + \frac{z \cdot t}{16}\right)\right) \]
      associate-*r/ [<=] 33.3%	\[ \mathsf{fma}\left(x, y, \left(-\color{blue}{b \cdot \frac{a}{4}}\right) + \left(c + \frac{z \cdot t}{16}\right)\right) \]
      distribute-rgt-neg-in [=>] 33.3%	\[ \mathsf{fma}\left(x, y, \color{blue}{b \cdot \left(-\frac{a}{4}\right)} + \left(c + \frac{z \cdot t}{16}\right)\right) \]
      fma-def [=>] 50.0%	\[ \mathsf{fma}\left(x, y, \color{blue}{\mathsf{fma}\left(b, -\frac{a}{4}, c + \frac{z \cdot t}{16}\right)}\right) \]
      distribute-frac-neg [<=] 50.0%	\[ \mathsf{fma}\left(x, y, \mathsf{fma}\left(b, \color{blue}{\frac{-a}{4}}, c + \frac{z \cdot t}{16}\right)\right) \]
      neg-mul-1 [=>] 50.0%	\[ \mathsf{fma}\left(x, y, \mathsf{fma}\left(b, \frac{\color{blue}{-1 \cdot a}}{4}, c + \frac{z \cdot t}{16}\right)\right) \]
      associate-/l* [=>] 50.0%	\[ \mathsf{fma}\left(x, y, \mathsf{fma}\left(b, \color{blue}{\frac{-1}{\frac{4}{a}}}, c + \frac{z \cdot t}{16}\right)\right) \]
      associate-/r/ [=>] 50.0%	\[ \mathsf{fma}\left(x, y, \mathsf{fma}\left(b, \color{blue}{\frac{-1}{4} \cdot a}, c + \frac{z \cdot t}{16}\right)\right) \]
      metadata-eval [=>] 50.0%	\[ \mathsf{fma}\left(x, y, \mathsf{fma}\left(b, \color{blue}{-0.25} \cdot a, c + \frac{z \cdot t}{16}\right)\right) \]

   3. Taylor expanded in b around 0 66.7%

      \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{c + 0.0625 \cdot \left(t \cdot z\right)}\right) \]

   4. Taylor expanded in c around 0 33.3%

      \[\leadsto \color{blue}{y \cdot x + 0.0625 \cdot \left(t \cdot z\right)} \]

   5. Simplified 66.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \left(0.0625 \cdot t\right) \cdot z\right)} \]
      Step-by-step derivation

      [Start] 33.3%	\[ y \cdot x + 0.0625 \cdot \left(t \cdot z\right) \]
      fma-def [=>] 66.7%	\[ \color{blue}{\mathsf{fma}\left(y, x, 0.0625 \cdot \left(t \cdot z\right)\right)} \]
      associate-*r* [=>] 66.7%	\[ \mathsf{fma}\left(y, x, \color{blue}{\left(0.0625 \cdot t\right) \cdot z}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{z \cdot t}{16} + x \cdot y\right) - \frac{b \cdot a}{4} \leq \infty:\\ \;\;\;\;c + \left(\left(\frac{z \cdot t}{16} + x \cdot y\right) - \frac{b \cdot a}{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot \left(t \cdot 0.0625\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.9%
Cost	7876

\[\begin{array}{l} t_1 := \left(\frac{z \cdot t}{16} + x \cdot y\right) - \frac{b \cdot a}{4}\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;c + t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot \left(t \cdot 0.0625\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	98.8%
Cost	19904

\[\mathsf{fma}\left(x, y, \mathsf{fma}\left(b, -0.25 \cdot a, \mathsf{fma}\left(z, \frac{t}{16}, c\right)\right)\right) \]

Alternative 3
Accuracy	98.9%
Cost	13632

\[\mathsf{fma}\left(t, \frac{z}{16}, \mathsf{fma}\left(x, y, c - b \cdot \frac{a}{4}\right)\right) \]

Alternative 4
Accuracy	98.1%
Cost	7360

\[\mathsf{fma}\left(x, y, \frac{z}{\frac{16}{t}}\right) + \left(c - \frac{a}{\frac{4}{b}}\right) \]

Alternative 5
Accuracy	44.4%
Cost	2660

\[\begin{array}{l} t_1 := z \cdot \left(t \cdot 0.0625\right)\\ t_2 := -0.25 \cdot \left(b \cdot a\right)\\ \mathbf{if}\;b \cdot a \leq -5.5 \cdot 10^{+79}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \cdot a \leq -520000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \cdot a \leq -3.7 \cdot 10^{-50}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;b \cdot a \leq -9.6 \cdot 10^{-104}:\\ \;\;\;\;c\\ \mathbf{elif}\;b \cdot a \leq -2.1 \cdot 10^{-191}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \cdot a \leq 1.7 \cdot 10^{-189}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;b \cdot a \leq 9.5 \cdot 10^{-140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \cdot a \leq 2.1 \cdot 10^{-94}:\\ \;\;\;\;c\\ \mathbf{elif}\;b \cdot a \leq 2.2 \cdot 10^{+28}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Accuracy	62.8%
Cost	2140

\[\begin{array}{l} t_1 := c + x \cdot y\\ t_2 := -0.25 \cdot \left(b \cdot a\right)\\ t_3 := c + 0.0625 \cdot \left(z \cdot t\right)\\ \mathbf{if}\;b \cdot a \leq -1.66 \cdot 10^{+214}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \cdot a \leq -1.8 \cdot 10^{+174}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \cdot a \leq -2.6 \cdot 10^{+80}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \cdot a \leq -3 \cdot 10^{+16}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \cdot a \leq -3.4 \cdot 10^{-50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \cdot a \leq -3.2 \cdot 10^{-192}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \cdot a \leq 1.95 \cdot 10^{+202}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Accuracy	68.3%
Cost	2136

\[\begin{array}{l} t_1 := c + x \cdot y\\ t_2 := 0.0625 \cdot \left(z \cdot t\right)\\ t_3 := x \cdot y + t_2\\ t_4 := x \cdot y - \left(b \cdot a\right) \cdot 0.25\\ \mathbf{if}\;b \cdot a \leq -2 \cdot 10^{+80}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;b \cdot a \leq -2 \cdot 10^{+18}:\\ \;\;\;\;c + t_2\\ \mathbf{elif}\;b \cdot a \leq -5 \cdot 10^{-103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \cdot a \leq 5 \cdot 10^{-140}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \cdot a \leq 2 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \cdot a \leq 5 \cdot 10^{+22}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 8
Accuracy	85.5%
Cost	1876

\[\begin{array}{l} t_1 := c + \left(x \cdot y + 0.0625 \cdot \left(z \cdot t\right)\right)\\ t_2 := x \cdot y - \left(b \cdot a\right) \cdot 0.25\\ \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{+225}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \cdot a \leq 5 \cdot 10^{+22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \cdot a \leq 2 \cdot 10^{+118}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \cdot a \leq 5 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \cdot a \leq 2 \cdot 10^{+185}:\\ \;\;\;\;c + b \cdot \left(-0.25 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Accuracy	85.2%
Cost	1876

\[\begin{array}{l} t_1 := \left(b \cdot a\right) \cdot 0.25\\ t_2 := x \cdot y - t_1\\ t_3 := 0.0625 \cdot \left(z \cdot t\right)\\ t_4 := c + \left(x \cdot y + t_3\right)\\ \mathbf{if}\;b \cdot a \leq -5 \cdot 10^{+225}:\\ \;\;\;\;t_3 - t_1\\ \mathbf{elif}\;b \cdot a \leq 5 \cdot 10^{+22}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;b \cdot a \leq 2 \cdot 10^{+118}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \cdot a \leq 5 \cdot 10^{+154}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;b \cdot a \leq 2 \cdot 10^{+185}:\\ \;\;\;\;c + b \cdot \left(-0.25 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Accuracy	65.6%
Cost	1748

\[\begin{array}{l} t_1 := c + 0.0625 \cdot \left(z \cdot t\right)\\ t_2 := c + b \cdot \left(-0.25 \cdot a\right)\\ t_3 := c + x \cdot y\\ \mathbf{if}\;b \cdot a \leq -2 \cdot 10^{+90}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \cdot a \leq -2 \cdot 10^{+18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \cdot a \leq -4 \cdot 10^{-50}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \cdot a \leq -4 \cdot 10^{-189}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \cdot a \leq 10^{+26}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Accuracy	88.3%
Cost	1746

\[\begin{array}{l} \mathbf{if}\;b \cdot a \leq -2 \cdot 10^{+108} \lor \neg \left(b \cdot a \leq -2 \cdot 10^{+18} \lor \neg \left(b \cdot a \leq -2 \cdot 10^{-81}\right) \land b \cdot a \leq 50000000000000\right):\\ \;\;\;\;x \cdot y + \left(c - \frac{a}{\frac{4}{b}}\right)\\ \mathbf{else}:\\ \;\;\;\;c + \left(x \cdot y + 0.0625 \cdot \left(z \cdot t\right)\right)\\ \end{array} \]

Alternative 12
Accuracy	88.3%
Cost	1746

\[\begin{array}{l} \mathbf{if}\;b \cdot a \leq -2 \cdot 10^{+108} \lor \neg \left(b \cdot a \leq -2 \cdot 10^{+18} \lor \neg \left(b \cdot a \leq -2 \cdot 10^{-81}\right) \land b \cdot a \leq 50000000000000\right):\\ \;\;\;\;\left(c + x \cdot y\right) - \left(b \cdot a\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;c + \left(x \cdot y + 0.0625 \cdot \left(z \cdot t\right)\right)\\ \end{array} \]

Alternative 13
Accuracy	62.1%
Cost	1360

\[\begin{array}{l} t_1 := c + x \cdot y\\ t_2 := -0.25 \cdot \left(b \cdot a\right)\\ \mathbf{if}\;b \cdot a \leq -1.55 \cdot 10^{+214}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \cdot a \leq -2.8 \cdot 10^{-109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \cdot a \leq -8 \cdot 10^{-190}:\\ \;\;\;\;z \cdot \left(t \cdot 0.0625\right)\\ \mathbf{elif}\;b \cdot a \leq 5.8 \cdot 10^{+200}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 14
Accuracy	60.6%
Cost	1236

\[\begin{array}{l} t_1 := 0.0625 \cdot \left(z \cdot t\right)\\ t_2 := c + b \cdot \left(-0.25 \cdot a\right)\\ t_3 := x \cdot y + t_1\\ \mathbf{if}\;a \leq -1.3 \cdot 10^{+87}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.75 \cdot 10^{-28}:\\ \;\;\;\;c + x \cdot y\\ \mathbf{elif}\;a \leq -3.8 \cdot 10^{-166}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -4.9 \cdot 10^{-233}:\\ \;\;\;\;c + t_1\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{-107}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 15
Accuracy	37.1%
Cost	1113

\[\begin{array}{l} t_1 := z \cdot \left(t \cdot 0.0625\right)\\ \mathbf{if}\;t \leq -3.15 \cdot 10^{-15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-95}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;t \leq 7500:\\ \;\;\;\;c\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+35} \lor \neg \left(t \leq 2.7 \cdot 10^{+58}\right) \land t \leq 7.2 \cdot 10^{+171}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Accuracy	97.8%
Cost	1088

\[c + \left(\left(\frac{z \cdot t}{16} + x \cdot y\right) - \frac{b \cdot a}{4}\right) \]

Alternative 17
Accuracy	37.5%
Cost	456

\[\begin{array}{l} \mathbf{if}\;c \leq -3.7 \cdot 10^{+127}:\\ \;\;\;\;c\\ \mathbf{elif}\;c \leq 8.2 \cdot 10^{+65}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;c\\ \end{array} \]

Alternative 18
Accuracy	22.6%
Cost	64

\[c \]

Reproduce

herbie shell --seed 2023171 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, C"
  :precision binary64
  (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))