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

Percentage Accurate: 90.9% → 97.2%

Time: 9.0s

Precision: binary64

Cost: 8649

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

• 27.6% of points produce a very large (infinite) output. You may want to add a precondition.

Math

\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

↓

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 5 \cdot 10^{+273}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, \frac{y}{2}, 0.5 \cdot \frac{t}{\frac{a}{z \cdot -9}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))

↓

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* (* z 9.0) t))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 5e+273)))
     (fma (/ x a) (/ y 2.0) (* 0.5 (/ t (/ a (* z -9.0)))))
     (/ (fma x y (* z (* t -9.0))) (* a 2.0)))))

C

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

↓

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 5e+273)) {
		tmp = fma((x / a), (y / 2.0), (0.5 * (t / (a / (z * -9.0)))));
	} else {
		tmp = fma(x, y, (z * (t * -9.0))) / (a * 2.0);
	}
	return tmp;
}

Julia

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

↓

z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 5e+273))
		tmp = fma(Float64(x / a), Float64(y / 2.0), Float64(0.5 * Float64(t / Float64(a / Float64(z * -9.0)))));
	else
		tmp = Float64(fma(x, y, Float64(z * Float64(t * -9.0))) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

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

↓

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 5e+273]], $MachinePrecision]], N[(N[(x / a), $MachinePrecision] * N[(y / 2.0), $MachinePrecision] + N[(0.5 * N[(t / N[(a / N[(z * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y + N[(z * N[(t * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Target

Original	90.9%
Target	93.3%
Herbie	97.2%

\[\begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < -inf.0 or 4.99999999999999961e273 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t))

   1. Initial program 58.9%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Simplified 60.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right) \cdot \frac{0.5}{a}} \]
      Step-by-step derivation

      [Start] 58.9%	\[ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
      sub-neg [=>] 58.9%	\[ \frac{\color{blue}{x \cdot y + \left(-\left(z \cdot 9\right) \cdot t\right)}}{a \cdot 2} \]
      +-commutative [=>] 58.9%	\[ \frac{\color{blue}{\left(-\left(z \cdot 9\right) \cdot t\right) + x \cdot y}}{a \cdot 2} \]
      neg-sub0 [=>] 58.9%	\[ \frac{\color{blue}{\left(0 - \left(z \cdot 9\right) \cdot t\right)} + x \cdot y}{a \cdot 2} \]
      associate-+l- [=>] 58.9%	\[ \frac{\color{blue}{0 - \left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)}}{a \cdot 2} \]
      sub0-neg [=>] 58.9%	\[ \frac{\color{blue}{-\left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)}}{a \cdot 2} \]
      neg-mul-1 [=>] 58.9%	\[ \frac{\color{blue}{-1 \cdot \left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)}}{a \cdot 2} \]
      associate-/l* [=>] 58.9%	\[ \color{blue}{\frac{-1}{\frac{a \cdot 2}{\left(z \cdot 9\right) \cdot t - x \cdot y}}} \]
      associate-/r/ [=>] 58.9%	\[ \color{blue}{\frac{-1}{a \cdot 2} \cdot \left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)} \]
      *-commutative [=>] 58.9%	\[ \color{blue}{\left(\left(z \cdot 9\right) \cdot t - x \cdot y\right) \cdot \frac{-1}{a \cdot 2}} \]
      sub-neg [=>] 58.9%	\[ \color{blue}{\left(\left(z \cdot 9\right) \cdot t + \left(-x \cdot y\right)\right)} \cdot \frac{-1}{a \cdot 2} \]
      +-commutative [=>] 58.9%	\[ \color{blue}{\left(\left(-x \cdot y\right) + \left(z \cdot 9\right) \cdot t\right)} \cdot \frac{-1}{a \cdot 2} \]
      neg-sub0 [=>] 58.9%	\[ \left(\color{blue}{\left(0 - x \cdot y\right)} + \left(z \cdot 9\right) \cdot t\right) \cdot \frac{-1}{a \cdot 2} \]
      associate-+l- [=>] 58.9%	\[ \color{blue}{\left(0 - \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)\right)} \cdot \frac{-1}{a \cdot 2} \]
      sub0-neg [=>] 58.9%	\[ \color{blue}{\left(-\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)\right)} \cdot \frac{-1}{a \cdot 2} \]
      distribute-lft-neg-out [=>] 58.9%	\[ \color{blue}{-\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \frac{-1}{a \cdot 2}} \]
      distribute-rgt-neg-in [=>] 58.9%	\[ \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \left(-\frac{-1}{a \cdot 2}\right)} \]

   3. Applied egg-rr 74.8%

      \[\leadsto \color{blue}{\frac{x}{a} \cdot \frac{y}{2} + 0.5 \cdot \frac{t \cdot \left(z \cdot -9\right)}{a}} \]
      Step-by-step derivation

      [Start] 60.5%	\[ \mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right) \cdot \frac{0.5}{a} \]
      *-commutative [=>] 60.5%	\[ \color{blue}{\frac{0.5}{a} \cdot \mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)} \]
      fma-udef [=>] 58.9%	\[ \frac{0.5}{a} \cdot \color{blue}{\left(x \cdot y + z \cdot \left(t \cdot -9\right)\right)} \]
      *-commutative [=>] 58.9%	\[ \frac{0.5}{a} \cdot \left(x \cdot y + z \cdot \color{blue}{\left(-9 \cdot t\right)}\right) \]
      metadata-eval [<=] 58.9%	\[ \frac{0.5}{a} \cdot \left(x \cdot y + z \cdot \left(\color{blue}{\left(-9\right)} \cdot t\right)\right) \]
      distribute-lft-neg-in [<=] 58.9%	\[ \frac{0.5}{a} \cdot \left(x \cdot y + z \cdot \color{blue}{\left(-9 \cdot t\right)}\right) \]
      distribute-rgt-neg-in [<=] 58.9%	\[ \frac{0.5}{a} \cdot \left(x \cdot y + \color{blue}{\left(-z \cdot \left(9 \cdot t\right)\right)}\right) \]
      distribute-rgt-in [=>] 55.7%	\[ \color{blue}{\left(x \cdot y\right) \cdot \frac{0.5}{a} + \left(-z \cdot \left(9 \cdot t\right)\right) \cdot \frac{0.5}{a}} \]
      clear-num [=>] 55.7%	\[ \left(x \cdot y\right) \cdot \color{blue}{\frac{1}{\frac{a}{0.5}}} + \left(-z \cdot \left(9 \cdot t\right)\right) \cdot \frac{0.5}{a} \]
      div-inv [=>] 55.7%	\[ \left(x \cdot y\right) \cdot \frac{1}{\color{blue}{a \cdot \frac{1}{0.5}}} + \left(-z \cdot \left(9 \cdot t\right)\right) \cdot \frac{0.5}{a} \]
      metadata-eval [=>] 55.7%	\[ \left(x \cdot y\right) \cdot \frac{1}{a \cdot \color{blue}{2}} + \left(-z \cdot \left(9 \cdot t\right)\right) \cdot \frac{0.5}{a} \]
      div-inv [<=] 55.7%	\[ \color{blue}{\frac{x \cdot y}{a \cdot 2}} + \left(-z \cdot \left(9 \cdot t\right)\right) \cdot \frac{0.5}{a} \]

   4. Simplified 96.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{a}, \frac{y}{2}, 0.5 \cdot \frac{t}{\frac{a}{z \cdot -9}}\right)} \]
      Step-by-step derivation

      [Start] 74.8%	\[ \frac{x}{a} \cdot \frac{y}{2} + 0.5 \cdot \frac{t \cdot \left(z \cdot -9\right)}{a} \]
      fma-def [=>] 76.4%	\[ \color{blue}{\mathsf{fma}\left(\frac{x}{a}, \frac{y}{2}, 0.5 \cdot \frac{t \cdot \left(z \cdot -9\right)}{a}\right)} \]
      associate-/l* [=>] 96.7%	\[ \mathsf{fma}\left(\frac{x}{a}, \frac{y}{2}, 0.5 \cdot \color{blue}{\frac{t}{\frac{a}{z \cdot -9}}}\right) \]

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < 4.99999999999999961e273

   1. Initial program 99.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Simplified 99.6%

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

      [Start] 99.6%	\[ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
      div-sub [=>] 98.5%	\[ \color{blue}{\frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]
      +-rgt-identity [<=] 98.5%	\[ \frac{\color{blue}{x \cdot y + 0}}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
      div-sub [<=] 99.6%	\[ \color{blue}{\frac{\left(x \cdot y + 0\right) - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]
      +-rgt-identity [=>] 99.6%	\[ \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
      fma-neg [=>] 99.6%	\[ \frac{\color{blue}{\mathsf{fma}\left(x, y, -\left(z \cdot 9\right) \cdot t\right)}}{a \cdot 2} \]
      associate-*l* [=>] 99.6%	\[ \frac{\mathsf{fma}\left(x, y, -\color{blue}{z \cdot \left(9 \cdot t\right)}\right)}{a \cdot 2} \]
      distribute-rgt-neg-in [=>] 99.6%	\[ \frac{\mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(-9 \cdot t\right)}\right)}{a \cdot 2} \]
      *-commutative [=>] 99.6%	\[ \frac{\mathsf{fma}\left(x, y, z \cdot \left(-\color{blue}{t \cdot 9}\right)\right)}{a \cdot 2} \]
      distribute-rgt-neg-in [=>] 99.6%	\[ \frac{\mathsf{fma}\left(x, y, z \cdot \color{blue}{\left(t \cdot \left(-9\right)\right)}\right)}{a \cdot 2} \]
      metadata-eval [=>] 99.6%	\[ \frac{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot \color{blue}{-9}\right)\right)}{a \cdot 2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -\infty \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \leq 5 \cdot 10^{+273}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, \frac{y}{2}, 0.5 \cdot \frac{t}{\frac{a}{z \cdot -9}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}{a \cdot 2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.2%
Cost	8649

\[\begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 5 \cdot 10^{+273}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, \frac{y}{2}, 0.5 \cdot \frac{t}{\frac{a}{z \cdot -9}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}{a \cdot 2}\\ \end{array} \]

Alternative 2
Accuracy	91.9%
Cost	7364

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq 4 \cdot 10^{+177}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 3
Accuracy	71.0%
Cost	2008

\[\begin{array}{l} t_1 := -4.5 \cdot \frac{t}{\frac{a}{z}}\\ t_2 := \left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-62}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-120}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-71}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{-24}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 4
Accuracy	71.0%
Cost	2008

\[\begin{array}{l} t_1 := -4.5 \cdot \frac{t}{\frac{a}{z}}\\ t_2 := \left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-62}:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-120}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-71}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{-24}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 5
Accuracy	91.8%
Cost	1732

\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq 5 \cdot 10^{+273}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot \frac{y}{2} + 0.5 \cdot \frac{t \cdot \left(z \cdot -9\right)}{a}\\ \end{array} \]

Alternative 6
Accuracy	91.6%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq 4 \cdot 10^{+177}:\\ \;\;\;\;\left(x \cdot y + t \cdot \left(z \cdot -9\right)\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 7
Accuracy	91.7%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq 4 \cdot 10^{+177}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 8
Accuracy	67.9%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+67}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-110}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{a}{x}}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \end{array} \]

Alternative 9
Accuracy	68.3%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+67}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \end{array} \]

Alternative 10
Accuracy	51.5%
Cost	448

\[-4.5 \cdot \left(t \cdot \frac{z}{a}\right) \]

Alternative 11
Accuracy	51.4%
Cost	448

\[-4.5 \cdot \frac{t}{\frac{a}{z}} \]

Reproduce

herbie shell --seed 2023167 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :herbie-target
  (if (< a -2.090464557976709e+86) (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z)))) (if (< a 2.144030707833976e+99) (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))