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

Percentage Accurate: 97.8% → 98.1%

Time: 12.2s

Precision: binary64

Cost: 7360

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

• 34.0% of points produce a very large (infinite) output. You may want to add a precondition.

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

Math

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

↓

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

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
 (+ (fma x y (/ z (/ 16.0 t))) (- c (/ a (/ 4.0 b)))))

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

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)
	return Float64(fma(x, y, Float64(z / Float64(16.0 / t))) + Float64(c - Float64(a / Float64(4.0 / b))))
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_] := N[(N[(x * y + N[(z / N[(16.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c - N[(a / N[(4.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.8%

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

2. Simplified 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \frac{z}{\frac{16}{t}}\right) - \left(\frac{a}{\frac{4}{b}} - c\right)} \]
   Step-by-step derivation

   [Start] 98.8%	\[ \left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   associate-+l- [=>] 98.8%	\[ \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(\frac{a \cdot b}{4} - c\right)} \]
   sub-neg [=>] 98.8%	\[ \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) + \left(-\left(\frac{a \cdot b}{4} - c\right)\right)} \]
   neg-mul-1 [=>] 98.8%	\[ \left(x \cdot y + \frac{z \cdot t}{16}\right) + \color{blue}{-1 \cdot \left(\frac{a \cdot b}{4} - c\right)} \]
   metadata-eval [<=] 98.8%	\[ \left(x \cdot y + \frac{z \cdot t}{16}\right) + \color{blue}{\left(-1\right)} \cdot \left(\frac{a \cdot b}{4} - c\right) \]
   metadata-eval [<=] 98.8%	\[ \left(x \cdot y + \frac{z \cdot t}{16}\right) + \left(-\color{blue}{\left(--1\right)}\right) \cdot \left(\frac{a \cdot b}{4} - c\right) \]
   cancel-sign-sub-inv [<=] 98.8%	\[ \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(--1\right) \cdot \left(\frac{a \cdot b}{4} - c\right)} \]
   fma-def [=>] 99.6%	\[ \color{blue}{\mathsf{fma}\left(x, y, \frac{z \cdot t}{16}\right)} - \left(--1\right) \cdot \left(\frac{a \cdot b}{4} - c\right) \]
   associate-/l* [=>] 99.6%	\[ \mathsf{fma}\left(x, y, \color{blue}{\frac{z}{\frac{16}{t}}}\right) - \left(--1\right) \cdot \left(\frac{a \cdot b}{4} - c\right) \]
   metadata-eval [=>] 99.6%	\[ \mathsf{fma}\left(x, y, \frac{z}{\frac{16}{t}}\right) - \color{blue}{1} \cdot \left(\frac{a \cdot b}{4} - c\right) \]
   *-lft-identity [=>] 99.6%	\[ \mathsf{fma}\left(x, y, \frac{z}{\frac{16}{t}}\right) - \color{blue}{\left(\frac{a \cdot b}{4} - c\right)} \]
   associate-/l* [=>] 99.5%	\[ \mathsf{fma}\left(x, y, \frac{z}{\frac{16}{t}}\right) - \left(\color{blue}{\frac{a}{\frac{4}{b}}} - c\right) \]

3. Final simplification 99.5%

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

Alternatives

Alternative 1
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 2
Accuracy	66.0%
Cost	2396

\[\begin{array}{l} t_1 := \left(z \cdot t\right) \cdot 0.0625\\ t_2 := c + t_1\\ t_3 := t_1 + x \cdot y\\ \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+68}:\\ \;\;\;\;c + \left(a \cdot b\right) \cdot -0.25\\ \mathbf{elif}\;a \cdot b \leq -2 \cdot 10^{-297}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{-250}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot b \leq 10^{-147}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{-50}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot b \leq 10^{+39}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+144}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - \left(a \cdot b\right) \cdot 0.25\\ \end{array} \]

Alternative 3
Accuracy	89.1%
Cost	1353

\[\begin{array}{l} t_1 := \left(z \cdot t\right) \cdot 0.0625\\ \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+83} \lor \neg \left(a \cdot b \leq 10^{+39}\right):\\ \;\;\;\;\left(c + t_1\right) - \left(a \cdot b\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;c + \left(t_1 + x \cdot y\right)\\ \end{array} \]

Alternative 4
Accuracy	39.1%
Cost	1244

\[\begin{array}{l} t_1 := a \cdot \left(b \cdot -0.25\right)\\ t_2 := \left(z \cdot t\right) \cdot 0.0625\\ \mathbf{if}\;t \leq -1.45 \cdot 10^{-115}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.55 \cdot 10^{-165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.3 \cdot 10^{-196}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;t \leq -1.15 \cdot 10^{-240}:\\ \;\;\;\;c\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{-203}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-79}:\\ \;\;\;\;c\\ \mathbf{elif}\;t \leq 120000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Accuracy	86.9%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+111} \lor \neg \left(a \cdot b \leq 2 \cdot 10^{+144}\right):\\ \;\;\;\;x \cdot y - \left(a \cdot b\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;c + \left(\left(z \cdot t\right) \cdot 0.0625 + x \cdot y\right)\\ \end{array} \]

Alternative 6
Accuracy	89.0%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+83} \lor \neg \left(a \cdot b \leq 2 \cdot 10^{+144}\right):\\ \;\;\;\;\left(c + x \cdot y\right) - \left(a \cdot b\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;c + \left(\left(z \cdot t\right) \cdot 0.0625 + x \cdot y\right)\\ \end{array} \]

Alternative 7
Accuracy	86.9%
Cost	1224

\[\begin{array}{l} t_1 := \left(a \cdot b\right) \cdot 0.25\\ t_2 := \left(z \cdot t\right) \cdot 0.0625\\ \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+111}:\\ \;\;\;\;t_2 - t_1\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+144}:\\ \;\;\;\;c + \left(t_2 + x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - t_1\\ \end{array} \]

Alternative 8
Accuracy	62.8%
Cost	1104

\[\begin{array}{l} t_1 := \left(z \cdot t\right) \cdot 0.0625\\ t_2 := t_1 + x \cdot y\\ t_3 := c + \left(a \cdot b\right) \cdot -0.25\\ \mathbf{if}\;c \leq -3.75 \cdot 10^{+123}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \leq -3.8 \cdot 10^{+23}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq -1.16 \cdot 10^{-11}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c \leq 1.32 \cdot 10^{+17}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;c + t_1\\ \end{array} \]

Alternative 9
Accuracy	97.8%
Cost	1088

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

Alternative 10
Accuracy	65.1%
Cost	969

\[\begin{array}{l} \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+79} \lor \neg \left(a \cdot b \leq 2 \cdot 10^{+144}\right):\\ \;\;\;\;c + \left(a \cdot b\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;c + \left(z \cdot t\right) \cdot 0.0625\\ \end{array} \]

Alternative 11
Accuracy	39.3%
Cost	848

\[\begin{array}{l} t_1 := \left(z \cdot t\right) \cdot 0.0625\\ \mathbf{if}\;t \leq -5.5 \cdot 10^{-146}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-203}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-37}:\\ \;\;\;\;c\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{+62}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Accuracy	60.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-146} \lor \neg \left(t \leq 8.5 \cdot 10^{+50}\right):\\ \;\;\;\;c + \left(z \cdot t\right) \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;c + x \cdot y\\ \end{array} \]

Alternative 13
Accuracy	55.4%
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -100 \lor \neg \left(t \leq 9.8 \cdot 10^{+62}\right):\\ \;\;\;\;\left(z \cdot t\right) \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;c + x \cdot y\\ \end{array} \]

Alternative 14
Accuracy	37.4%
Cost	456

\[\begin{array}{l} \mathbf{if}\;c \leq -8.2 \cdot 10^{+117}:\\ \;\;\;\;c\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{+15}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;c\\ \end{array} \]

Alternative 15
Accuracy	22.7%
Cost	64

\[c \]

Reproduce

herbie shell --seed 2023263 
(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))