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

Percentage Accurate: 90.7% → 92.6%

Time: 7.6s

Precision: binary64

Cost: 1220

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

• 27.7% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq -\infty:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= (* (* z 9.0) t) (- INFINITY))
   (* z (/ (* t -4.5) a))
   (/ (- (* x y) (* z (* 9.0 t))) (* 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);
}

↓

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z * 9.0) * t) <= -((double) INFINITY)) {
		tmp = z * ((t * -4.5) / a);
	} else {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	}
	return tmp;
}

Java

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

↓

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z * 9.0) * t) <= -Double.POSITIVE_INFINITY) {
		tmp = z * ((t * -4.5) / a);
	} else {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0)

↓

def code(x, y, z, t, a):
	tmp = 0
	if ((z * 9.0) * t) <= -math.inf:
		tmp = z * ((t * -4.5) / a)
	else:
		tmp = ((x * y) - (z * (9.0 * t))) / (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

↓

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(z * 9.0) * t) <= Float64(-Inf))
		tmp = Float64(z * Float64(Float64(t * -4.5) / a));
	else
		tmp = Float64(Float64(Float64(x * y) - Float64(z * Float64(9.0 * t))) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
end

↓

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((z * 9.0) * t) <= -Inf)
		tmp = z * ((t * -4.5) / a);
	else
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	end
	tmp_2 = 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]

↓

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision], (-Infinity)], N[(z * N[(N[(t * -4.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * y), $MachinePrecision] - N[(z * N[(9.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Target

Original	90.7%
Target	92.7%
Herbie	92.6%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z 9) t) < -inf.0

   1. Initial program 60.8%

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

   2. Simplified 67.1%

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

      [Start] 60.8%	\[ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
      sub-neg [=>] 60.8%	\[ \frac{\color{blue}{x \cdot y + \left(-\left(z \cdot 9\right) \cdot t\right)}}{a \cdot 2} \]
      +-commutative [=>] 60.8%	\[ \frac{\color{blue}{\left(-\left(z \cdot 9\right) \cdot t\right) + x \cdot y}}{a \cdot 2} \]
      neg-sub0 [=>] 60.8%	\[ \frac{\color{blue}{\left(0 - \left(z \cdot 9\right) \cdot t\right)} + x \cdot y}{a \cdot 2} \]
      associate-+l- [=>] 60.8%	\[ \frac{\color{blue}{0 - \left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)}}{a \cdot 2} \]
      sub0-neg [=>] 60.8%	\[ \frac{\color{blue}{-\left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)}}{a \cdot 2} \]
      neg-mul-1 [=>] 60.8%	\[ \frac{\color{blue}{-1 \cdot \left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)}}{a \cdot 2} \]
      associate-/l* [=>] 60.8%	\[ \color{blue}{\frac{-1}{\frac{a \cdot 2}{\left(z \cdot 9\right) \cdot t - x \cdot y}}} \]
      associate-/r/ [=>] 60.8%	\[ \color{blue}{\frac{-1}{a \cdot 2} \cdot \left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)} \]
      *-commutative [=>] 60.8%	\[ \color{blue}{\left(\left(z \cdot 9\right) \cdot t - x \cdot y\right) \cdot \frac{-1}{a \cdot 2}} \]
      sub-neg [=>] 60.8%	\[ \color{blue}{\left(\left(z \cdot 9\right) \cdot t + \left(-x \cdot y\right)\right)} \cdot \frac{-1}{a \cdot 2} \]
      +-commutative [=>] 60.8%	\[ \color{blue}{\left(\left(-x \cdot y\right) + \left(z \cdot 9\right) \cdot t\right)} \cdot \frac{-1}{a \cdot 2} \]
      neg-sub0 [=>] 60.8%	\[ \left(\color{blue}{\left(0 - x \cdot y\right)} + \left(z \cdot 9\right) \cdot t\right) \cdot \frac{-1}{a \cdot 2} \]
      associate-+l- [=>] 60.8%	\[ \color{blue}{\left(0 - \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)\right)} \cdot \frac{-1}{a \cdot 2} \]
      sub0-neg [=>] 60.8%	\[ \color{blue}{\left(-\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)\right)} \cdot \frac{-1}{a \cdot 2} \]
      distribute-lft-neg-out [=>] 60.8%	\[ \color{blue}{-\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \frac{-1}{a \cdot 2}} \]
      distribute-rgt-neg-in [=>] 60.8%	\[ \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \left(-\frac{-1}{a \cdot 2}\right)} \]

   3. Applied egg-rr 67.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{0.5 \cdot \mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}}} \]
      Step-by-step derivation

      [Start] 67.1%	\[ \mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right) \cdot \frac{0.5}{a} \]
      associate-*r/ [=>] 67.1%	\[ \color{blue}{\frac{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right) \cdot 0.5}{a}} \]
      clear-num [=>] 67.1%	\[ \color{blue}{\frac{1}{\frac{a}{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right) \cdot 0.5}}} \]
      *-commutative [=>] 67.1%	\[ \frac{1}{\frac{a}{\color{blue}{0.5 \cdot \mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}}} \]

   4. Taylor expanded in x around 0 67.1%

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

   5. Simplified 99.9%

      \[\leadsto \color{blue}{z \cdot \frac{t \cdot -4.5}{a}} \]
      Step-by-step derivation

      [Start] 67.1%	\[ -4.5 \cdot \frac{t \cdot z}{a} \]
      *-commutative [=>] 67.1%	\[ \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
      *-commutative [=>] 67.1%	\[ \frac{\color{blue}{z \cdot t}}{a} \cdot -4.5 \]
      associate-*r/ [<=] 99.8%	\[ \color{blue}{\left(z \cdot \frac{t}{a}\right)} \cdot -4.5 \]
      associate-*l* [=>] 99.8%	\[ \color{blue}{z \cdot \left(\frac{t}{a} \cdot -4.5\right)} \]
      associate-*l/ [=>] 99.9%	\[ z \cdot \color{blue}{\frac{t \cdot -4.5}{a}} \]

   if -inf.0 < (*.f64 (*.f64 z 9) t)

   1. Initial program 94.6%

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

   2. Simplified 94.6%

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

      [Start] 94.6%	\[ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
      associate-*l* [=>] 94.6%	\[ \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq -\infty:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	92.6%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq -\infty:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \end{array} \]

Alternative 2
Accuracy	65.4%
Cost	977

\[\begin{array}{l} t_1 := x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{if}\;x \leq -1.12 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{+30}:\\ \;\;\;\;\left(z \cdot t\right) \cdot \frac{-4.5}{a}\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-15} \lor \neg \left(x \leq 1.45 \cdot 10^{-70}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \end{array} \]

Alternative 3
Accuracy	65.9%
Cost	977

\[\begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+99}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{+79}:\\ \;\;\;\;\left(z \cdot t\right) \cdot \frac{-4.5}{a}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{+62} \lor \neg \left(x \leq 2.8 \cdot 10^{-69}\right):\\ \;\;\;\;\frac{y \cdot 0.5}{\frac{a}{x}}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \end{array} \]

Alternative 4
Accuracy	52.2%
Cost	448

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

Alternative 5
Accuracy	52.3%
Cost	448

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

Alternative 6
Accuracy	52.2%
Cost	448

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

Reproduce

herbie shell --seed 2023171 
(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)))