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

Average Accuracy: 63.1% → 65.8%

Time: 15.6s

Precision: binary64

Cost: 2632

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

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+270}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \end{array} \]

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
 (let* ((t_1 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0))))
   (if (<= t_1 (- INFINITY))
     (* 0.5 (* y (/ x a)))
     (if (<= t_1 2e+270)
       (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0))
       (* 0.5 (* x (/ y a)))))))

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 t_1 = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 0.5 * (y * (x / a));
	} else if (t_1 <= 2e+270) {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	} else {
		tmp = 0.5 * (x * (y / a));
	}
	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 t_1 = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = 0.5 * (y * (x / a));
	} else if (t_1 <= 2e+270) {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	} else {
		tmp = 0.5 * (x * (y / a));
	}
	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):
	t_1 = ((x * y) - ((z * 9.0) * t)) / (a * 2.0)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = 0.5 * (y * (x / a))
	elif t_1 <= 2e+270:
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0)
	else:
		tmp = 0.5 * (x * (y / a))
	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)
	t_1 = Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a * 2.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(0.5 * Float64(y * Float64(x / a)));
	elseif (t_1 <= 2e+270)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * Float64(9.0 * t))) / Float64(a * 2.0));
	else
		tmp = Float64(0.5 * Float64(x * Float64(y / a)));
	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)
	t_1 = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = 0.5 * (y * (x / a));
	elseif (t_1 <= 2e+270)
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	else
		tmp = 0.5 * (x * (y / a));
	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_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(0.5 * N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+270], N[(N[(N[(x * y), $MachinePrecision] - N[(z * N[(9.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	63.1%
Target	65.4%
Herbie	65.8%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) (*.f64 a 2)) < -inf.0

   1. Initial program 0.0%

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

   2. Taylor expanded in x around inf 0.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y \cdot x}{a}} \]

   3. Simplified 14.3%

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

      [Start] 0.9	\[ 0.5 \cdot \frac{y \cdot x}{a} \]
      *-commutative [<=] 0.9	\[ 0.5 \cdot \frac{\color{blue}{x \cdot y}}{a} \]
      associate-/l* [=>] 14.3	\[ 0.5 \cdot \color{blue}{\frac{x}{\frac{a}{y}}} \]
      associate-/r/ [=>] 14.3	\[ 0.5 \cdot \color{blue}{\left(\frac{x}{a} \cdot y\right)} \]

   if -inf.0 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) (*.f64 a 2)) < 2.0000000000000001e270

   1. Initial program 99.2%

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

   2. Simplified 99.2%

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

      [Start] 99.2	\[ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
      associate-*l* [=>] 99.2	\[ \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   if 2.0000000000000001e270 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) (*.f64 a 2))

   1. Initial program 9.9%

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

   2. Simplified 9.9%

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

      [Start] 9.9	\[ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
      sub-neg [=>] 9.9	\[ \frac{\color{blue}{x \cdot y + \left(-\left(z \cdot 9\right) \cdot t\right)}}{a \cdot 2} \]
      +-commutative [=>] 9.9	\[ \frac{\color{blue}{\left(-\left(z \cdot 9\right) \cdot t\right) + x \cdot y}}{a \cdot 2} \]
      neg-sub0 [=>] 9.9	\[ \frac{\color{blue}{\left(0 - \left(z \cdot 9\right) \cdot t\right)} + x \cdot y}{a \cdot 2} \]
      associate-+l- [=>] 9.9	\[ \frac{\color{blue}{0 - \left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)}}{a \cdot 2} \]
      sub0-neg [=>] 9.9	\[ \frac{\color{blue}{-\left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)}}{a \cdot 2} \]
      neg-mul-1 [=>] 9.9	\[ \frac{\color{blue}{-1 \cdot \left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)}}{a \cdot 2} \]
      associate-/l* [=>] 9.9	\[ \color{blue}{\frac{-1}{\frac{a \cdot 2}{\left(z \cdot 9\right) \cdot t - x \cdot y}}} \]
      associate-/r/ [=>] 9.9	\[ \color{blue}{\frac{-1}{a \cdot 2} \cdot \left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)} \]
      *-commutative [=>] 9.9	\[ \color{blue}{\left(\left(z \cdot 9\right) \cdot t - x \cdot y\right) \cdot \frac{-1}{a \cdot 2}} \]
      sub-neg [=>] 9.9	\[ \color{blue}{\left(\left(z \cdot 9\right) \cdot t + \left(-x \cdot y\right)\right)} \cdot \frac{-1}{a \cdot 2} \]
      +-commutative [=>] 9.9	\[ \color{blue}{\left(\left(-x \cdot y\right) + \left(z \cdot 9\right) \cdot t\right)} \cdot \frac{-1}{a \cdot 2} \]
      neg-sub0 [=>] 9.9	\[ \left(\color{blue}{\left(0 - x \cdot y\right)} + \left(z \cdot 9\right) \cdot t\right) \cdot \frac{-1}{a \cdot 2} \]
      associate-+l- [=>] 9.9	\[ \color{blue}{\left(0 - \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)\right)} \cdot \frac{-1}{a \cdot 2} \]
      sub0-neg [=>] 9.9	\[ \color{blue}{\left(-\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)\right)} \cdot \frac{-1}{a \cdot 2} \]
      distribute-lft-neg-out [=>] 9.9	\[ \color{blue}{-\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \frac{-1}{a \cdot 2}} \]
      distribute-rgt-neg-in [=>] 9.9	\[ \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \left(-\frac{-1}{a \cdot 2}\right)} \]

   3. Taylor expanded in x around inf 11.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y \cdot x}{a}} \]

   4. Simplified 28.8%

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

      [Start] 11.0	\[ 0.5 \cdot \frac{y \cdot x}{a} \]
      associate-*l/ [<=] 28.8	\[ 0.5 \cdot \color{blue}{\left(\frac{y}{a} \cdot x\right)} \]
      *-commutative [=>] 28.8	\[ 0.5 \cdot \color{blue}{\left(x \cdot \frac{y}{a}\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \leq -\infty:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \mathbf{elif}\;\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \leq 2 \cdot 10^{+270}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	43.3%
Cost	976

\[\begin{array}{l} t_1 := 0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{if}\;y \leq -2.9 \cdot 10^{-260}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-74}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+31}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \end{array} \]

Alternative 2
Accuracy	43.3%
Cost	976

\[\begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{-260}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-74}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+19}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+31}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \end{array} \]

Alternative 3
Accuracy	45.1%
Cost	976

\[\begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+259}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{+144}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-51}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-153}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \end{array} \]

Alternative 4
Accuracy	45.1%
Cost	976

\[\begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+259}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{+144}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a}\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-50}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-153}:\\ \;\;\;\;\frac{z \cdot \frac{t}{-0.2222222222222222}}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \end{array} \]

Alternative 5
Accuracy	45.1%
Cost	976

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+260}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{+144}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-52}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-153}:\\ \;\;\;\;\frac{z \cdot \left(t \cdot -4.5\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \end{array} \]

Alternative 6
Accuracy	45.1%
Cost	976

\[\begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+259}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{+144}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-53}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-153}:\\ \;\;\;\;\frac{t \cdot \left(z \cdot -4.5\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \end{array} \]

Alternative 7
Accuracy	45.6%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-52} \lor \neg \left(x \leq 4.2 \cdot 10^{-142}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \end{array} \]

Alternative 8
Accuracy	35.3%
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq 1.32 \cdot 10^{+74}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{+250}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \]

Alternative 9
Accuracy	34.6%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-197}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \end{array} \]

Alternative 10
Accuracy	34.6%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-197}:\\ \;\;\;\;-4.5 \cdot \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \end{array} \]

Alternative 11
Accuracy	34.5%
Cost	448

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

Alternative 12
Accuracy	7.4%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023157 
(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)))