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

Percentage Accurate: 97.7% → 98.8%

Time: 11.3s

Precision: binary64

Cost: 8004

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

• 32.3% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_1 := \left(\frac{z \cdot t}{16} + x \cdot y\right) - \frac{b \cdot a}{4}\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;c + t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, -0.25 \cdot \left(b \cdot a\right)\right)\\ \end{array} \]

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 (* t 0.0625) z (* -0.25 (* b a))))))

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((t * 0.0625), z, (-0.25 * (b * a)));
	}
	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(Float64(t * 0.0625), z, Float64(-0.25 * Float64(b * a)));
	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[(N[(t * 0.0625), $MachinePrecision] * z + N[(-0.25 * N[(b * a), $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 16.7%

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

      [Start] 0.0%	\[ \left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
      associate-+l- [=>] 0.0%	\[ \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(\frac{a \cdot b}{4} - c\right)} \]
      sub-neg [=>] 0.0%	\[ \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) + \left(-\left(\frac{a \cdot b}{4} - c\right)\right)} \]
      neg-mul-1 [=>] 0.0%	\[ \left(x \cdot y + \frac{z \cdot t}{16}\right) + \color{blue}{-1 \cdot \left(\frac{a \cdot b}{4} - c\right)} \]
      metadata-eval [<=] 0.0%	\[ \left(x \cdot y + \frac{z \cdot t}{16}\right) + \color{blue}{\left(-1\right)} \cdot \left(\frac{a \cdot b}{4} - c\right) \]
      metadata-eval [<=] 0.0%	\[ \left(x \cdot y + \frac{z \cdot t}{16}\right) + \left(-\color{blue}{\left(--1\right)}\right) \cdot \left(\frac{a \cdot b}{4} - c\right) \]
      cancel-sign-sub-inv [<=] 0.0%	\[ \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(--1\right) \cdot \left(\frac{a \cdot b}{4} - c\right)} \]
      fma-def [=>] 16.7%	\[ \color{blue}{\mathsf{fma}\left(x, y, \frac{z \cdot t}{16}\right)} - \left(--1\right) \cdot \left(\frac{a \cdot b}{4} - c\right) \]
      associate-/l* [=>] 16.7%	\[ \mathsf{fma}\left(x, y, \color{blue}{\frac{z}{\frac{16}{t}}}\right) - \left(--1\right) \cdot \left(\frac{a \cdot b}{4} - c\right) \]
      metadata-eval [=>] 16.7%	\[ \mathsf{fma}\left(x, y, \frac{z}{\frac{16}{t}}\right) - \color{blue}{1} \cdot \left(\frac{a \cdot b}{4} - c\right) \]
      *-lft-identity [=>] 16.7%	\[ \mathsf{fma}\left(x, y, \frac{z}{\frac{16}{t}}\right) - \color{blue}{\left(\frac{a \cdot b}{4} - c\right)} \]
      associate-/l* [=>] 16.7%	\[ \mathsf{fma}\left(x, y, \frac{z}{\frac{16}{t}}\right) - \left(\color{blue}{\frac{a}{\frac{4}{b}}} - c\right) \]

   3. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\left(z \cdot \left(t \cdot 0.0625\right) + x \cdot y\right)} - \left(\frac{a}{\frac{4}{b}} - c\right) \]
      Step-by-step derivation

      [Start] 16.7%	\[ \mathsf{fma}\left(x, y, \frac{z}{\frac{16}{t}}\right) - \left(\frac{a}{\frac{4}{b}} - c\right) \]
      fma-udef [=>] 0.0%	\[ \color{blue}{\left(x \cdot y + \frac{z}{\frac{16}{t}}\right)} - \left(\frac{a}{\frac{4}{b}} - c\right) \]
      associate-/l* [<=] 0.0%	\[ \left(x \cdot y + \color{blue}{\frac{z \cdot t}{16}}\right) - \left(\frac{a}{\frac{4}{b}} - c\right) \]
      +-commutative [=>] 0.0%	\[ \color{blue}{\left(\frac{z \cdot t}{16} + x \cdot y\right)} - \left(\frac{a}{\frac{4}{b}} - c\right) \]
      associate-/l* [=>] 0.0%	\[ \left(\color{blue}{\frac{z}{\frac{16}{t}}} + x \cdot y\right) - \left(\frac{a}{\frac{4}{b}} - c\right) \]
      div-inv [=>] 0.0%	\[ \left(\color{blue}{z \cdot \frac{1}{\frac{16}{t}}} + x \cdot y\right) - \left(\frac{a}{\frac{4}{b}} - c\right) \]
      clear-num [<=] 0.0%	\[ \left(z \cdot \color{blue}{\frac{t}{16}} + x \cdot y\right) - \left(\frac{a}{\frac{4}{b}} - c\right) \]
      div-inv [=>] 0.0%	\[ \left(z \cdot \color{blue}{\left(t \cdot \frac{1}{16}\right)} + x \cdot y\right) - \left(\frac{a}{\frac{4}{b}} - c\right) \]
      metadata-eval [=>] 0.0%	\[ \left(z \cdot \left(t \cdot \color{blue}{0.0625}\right) + x \cdot y\right) - \left(\frac{a}{\frac{4}{b}} - c\right) \]

   4. Applied egg-rr 33.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot 0.0625, z, x \cdot y - \left(\frac{a}{4} \cdot b - c\right)\right)} \]
      Step-by-step derivation

      [Start] 0.0%	\[ \left(z \cdot \left(t \cdot 0.0625\right) + x \cdot y\right) - \left(\frac{a}{\frac{4}{b}} - c\right) \]
      associate--l+ [=>] 0.0%	\[ \color{blue}{z \cdot \left(t \cdot 0.0625\right) + \left(x \cdot y - \left(\frac{a}{\frac{4}{b}} - c\right)\right)} \]
      *-commutative [=>] 0.0%	\[ \color{blue}{\left(t \cdot 0.0625\right) \cdot z} + \left(x \cdot y - \left(\frac{a}{\frac{4}{b}} - c\right)\right) \]
      fma-def [=>] 33.3%	\[ \color{blue}{\mathsf{fma}\left(t \cdot 0.0625, z, x \cdot y - \left(\frac{a}{\frac{4}{b}} - c\right)\right)} \]
      associate-/r/ [=>] 33.3%	\[ \mathsf{fma}\left(t \cdot 0.0625, z, x \cdot y - \left(\color{blue}{\frac{a}{4} \cdot b} - c\right)\right) \]

   5. Taylor expanded in a around inf 83.3%

      \[\leadsto \mathsf{fma}\left(t \cdot 0.0625, z, \color{blue}{-0.25 \cdot \left(a \cdot b\right)}\right) \]

   6. Simplified 83.3%

      \[\leadsto \mathsf{fma}\left(t \cdot 0.0625, z, \color{blue}{\left(a \cdot b\right) \cdot -0.25}\right) \]
      Step-by-step derivation

      [Start] 83.3%	\[ \mathsf{fma}\left(t \cdot 0.0625, z, -0.25 \cdot \left(a \cdot b\right)\right) \]
      *-commutative [<=] 83.3%	\[ \mathsf{fma}\left(t \cdot 0.0625, z, \color{blue}{\left(a \cdot b\right) \cdot -0.25}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{z \cdot t}{16} + x \cdot y\right) - \frac{b \cdot a}{4} \leq \infty:\\ \;\;\;\;c + \left(\left(\frac{z \cdot t}{16} + x \cdot y\right) - \frac{b \cdot a}{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, -0.25 \cdot \left(b \cdot a\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.8%
Cost	8004

\[\begin{array}{l} t_1 := \left(\frac{z \cdot t}{16} + x \cdot y\right) - \frac{b \cdot a}{4}\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;c + t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, -0.25 \cdot \left(b \cdot a\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	99.0%
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	98.6%
Cost	7360

\[\mathsf{fma}\left(t \cdot 0.0625, z, x \cdot y + \left(c - b \cdot \frac{a}{4}\right)\right) \]

Alternative 5
Accuracy	98.5%
Cost	2116

\[\begin{array}{l} t_1 := \left(\frac{z \cdot t}{16} + x \cdot y\right) - \frac{b \cdot a}{4}\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;c + t_1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(-0.25 \cdot a\right)\\ \end{array} \]

Alternative 6
Accuracy	66.3%
Cost	1616

\[\begin{array}{l} t_1 := 0.0625 \cdot \left(z \cdot t\right)\\ t_2 := t_1 + x \cdot y\\ t_3 := c + b \cdot \left(-0.25 \cdot a\right)\\ \mathbf{if}\;b \cdot a \leq -1 \cdot 10^{+162}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \cdot a \leq -2 \cdot 10^{-178}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \cdot a \leq -4 \cdot 10^{-263}:\\ \;\;\;\;c + t_1\\ \mathbf{elif}\;b \cdot a \leq 2 \cdot 10^{+164}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 7
Accuracy	36.1%
Cost	1508

\[\begin{array}{l} t_1 := 0.0625 \cdot \left(z \cdot t\right)\\ t_2 := b \cdot \left(-0.25 \cdot a\right)\\ \mathbf{if}\;y \leq -1.9 \cdot 10^{-113}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{-264}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{-302}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-251}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-213}:\\ \;\;\;\;c\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-124}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 11500000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+183}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 8
Accuracy	57.0%
Cost	1504

\[\begin{array}{l} t_1 := c + 0.0625 \cdot \left(z \cdot t\right)\\ t_2 := c + x \cdot y\\ t_3 := c + b \cdot \left(-0.25 \cdot a\right)\\ \mathbf{if}\;y \leq -1.9 \cdot 10^{-113}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1.5 \cdot 10^{-262}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-301}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-230}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-124}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 54000000000000:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+183}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Accuracy	88.5%
Cost	1353

\[\begin{array}{l} t_1 := 0.0625 \cdot \left(z \cdot t\right)\\ \mathbf{if}\;b \cdot a \leq -1 \cdot 10^{+162} \lor \neg \left(b \cdot a \leq 2 \cdot 10^{+49}\right):\\ \;\;\;\;\left(c + t_1\right) - \left(b \cdot a\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;c + \left(t_1 + x \cdot y\right)\\ \end{array} \]

Alternative 10
Accuracy	86.1%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;b \cdot a \leq -1 \cdot 10^{+162} \lor \neg \left(b \cdot a \leq 2 \cdot 10^{+164}\right):\\ \;\;\;\;c + b \cdot \left(-0.25 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;c + \left(0.0625 \cdot \left(z \cdot t\right) + x \cdot y\right)\\ \end{array} \]

Alternative 11
Accuracy	85.7%
Cost	1225

\[\begin{array}{l} t_1 := 0.0625 \cdot \left(z \cdot t\right)\\ \mathbf{if}\;b \cdot a \leq -1 \cdot 10^{+162} \lor \neg \left(b \cdot a \leq 2 \cdot 10^{+49}\right):\\ \;\;\;\;t_1 - \left(b \cdot a\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;c + \left(t_1 + x \cdot y\right)\\ \end{array} \]

Alternative 12
Accuracy	86.6%
Cost	1224

\[\begin{array}{l} t_1 := \left(b \cdot a\right) \cdot 0.25\\ t_2 := 0.0625 \cdot \left(z \cdot t\right)\\ \mathbf{if}\;b \cdot a \leq -1 \cdot 10^{+162}:\\ \;\;\;\;\left(c + x \cdot y\right) - t_1\\ \mathbf{elif}\;b \cdot a \leq 2 \cdot 10^{+49}:\\ \;\;\;\;c + \left(t_2 + x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 - t_1\\ \end{array} \]

Alternative 13
Accuracy	35.4%
Cost	1112

\[\begin{array}{l} t_1 := 0.0625 \cdot \left(z \cdot t\right)\\ \mathbf{if}\;y \leq -1.3 \cdot 10^{-113}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-251}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-125}:\\ \;\;\;\;c\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 0.0135:\\ \;\;\;\;c\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+183}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 14
Accuracy	52.0%
Cost	848

\[\begin{array}{l} t_1 := c + x \cdot y\\ t_2 := 0.0625 \cdot \left(z \cdot t\right)\\ \mathbf{if}\;t \leq -9 \cdot 10^{-29}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{-26}:\\ \;\;\;\;b \cdot \left(-0.25 \cdot a\right)\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+190}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 15
Accuracy	56.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-113} \lor \neg \left(y \leq 6.4 \cdot 10^{+183}\right):\\ \;\;\;\;c + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;c + 0.0625 \cdot \left(z \cdot t\right)\\ \end{array} \]

Alternative 16
Accuracy	34.2%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-182}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+57}:\\ \;\;\;\;c\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 17
Accuracy	22.6%
Cost	64

\[c \]

Reproduce

herbie shell --seed 2023165 
(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))