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

Average Error: 7.5 → 0.7

Time: 12.9s

Precision: binary64

Cost: 2377

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 10^{+284}\right):\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right) + \frac{z}{a} \cdot \frac{-9}{\frac{2}{t}}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a} + 0.5 \cdot \frac{x \cdot y}{a}\\ \end{array} \]

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))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 1e+284)))
     (+ (* x (* y (/ 0.5 a))) (* (/ z a) (/ -9.0 (/ 2.0 t))))
     (+ (* -4.5 (/ (* z t) a)) (* 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);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 1e+284)) {
		tmp = (x * (y * (0.5 / a))) + ((z / a) * (-9.0 / (2.0 / t)));
	} else {
		tmp = (-4.5 * ((z * t) / a)) + (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);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 1e+284)) {
		tmp = (x * (y * (0.5 / a))) + ((z / a) * (-9.0 / (2.0 / t)));
	} else {
		tmp = (-4.5 * ((z * t) / a)) + (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)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 1e+284):
		tmp = (x * (y * (0.5 / a))) + ((z / a) * (-9.0 / (2.0 / t)))
	else:
		tmp = (-4.5 * ((z * t) / a)) + (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(x * y) - Float64(Float64(z * 9.0) * t))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 1e+284))
		tmp = Float64(Float64(x * Float64(y * Float64(0.5 / a))) + Float64(Float64(z / a) * Float64(-9.0 / Float64(2.0 / t))));
	else
		tmp = Float64(Float64(-4.5 * Float64(Float64(z * t) / a)) + Float64(0.5 * Float64(Float64(x * 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);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 1e+284)))
		tmp = (x * (y * (0.5 / a))) + ((z / a) * (-9.0 / (2.0 / t)));
	else
		tmp = (-4.5 * ((z * t) / a)) + (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[(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, 1e+284]], $MachinePrecision]], N[(N[(x * N[(y * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z / a), $MachinePrecision] * N[(-9.0 / N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	7.5
Target	5.8
Herbie	0.7

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < -inf.0 or 1.00000000000000008e284 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t))

   1. Initial program 57.2

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

   2. Simplified 56.8

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

      [Start] 57.2	\[ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
      associate-*l* [=>] 56.8	\[ \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Applied egg-rr 30.6

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

   4. Simplified 0.6

      \[\leadsto \color{blue}{x \cdot \left(y \cdot \frac{0.5}{a}\right) - \frac{z}{a} \cdot \frac{9}{\frac{2}{t}}} \]
      Proof

      [Start] 30.6	\[ \left(x \cdot y\right) \cdot \frac{0.5}{a} + \left(-\frac{z}{a} \cdot \frac{9 \cdot t}{2}\right) \]
      sub-neg [<=] 30.6	\[ \color{blue}{\left(x \cdot y\right) \cdot \frac{0.5}{a} - \frac{z}{a} \cdot \frac{9 \cdot t}{2}} \]
      associate-*l* [=>] 0.6	\[ \color{blue}{x \cdot \left(y \cdot \frac{0.5}{a}\right)} - \frac{z}{a} \cdot \frac{9 \cdot t}{2} \]
      associate-/l* [=>] 0.6	\[ x \cdot \left(y \cdot \frac{0.5}{a}\right) - \frac{z}{a} \cdot \color{blue}{\frac{9}{\frac{2}{t}}} \]

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < 1.00000000000000008e284

   1. Initial program 0.8

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

   2. Simplified 0.9

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

      [Start] 0.8	\[ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
      sub-neg [=>] 0.8	\[ \frac{\color{blue}{x \cdot y + \left(-\left(z \cdot 9\right) \cdot t\right)}}{a \cdot 2} \]
      remove-double-neg [<=] 0.8	\[ \frac{\color{blue}{\left(-\left(-x \cdot y\right)\right)} + \left(-\left(z \cdot 9\right) \cdot t\right)}{a \cdot 2} \]
      distribute-neg-in [<=] 0.8	\[ \frac{\color{blue}{-\left(\left(-x \cdot y\right) + \left(z \cdot 9\right) \cdot t\right)}}{a \cdot 2} \]
      +-commutative [<=] 0.8	\[ \frac{-\color{blue}{\left(\left(z \cdot 9\right) \cdot t + \left(-x \cdot y\right)\right)}}{a \cdot 2} \]
      sub-neg [<=] 0.8	\[ \frac{-\color{blue}{\left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)}}{a \cdot 2} \]
      neg-mul-1 [=>] 0.8	\[ \frac{\color{blue}{-1 \cdot \left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)}}{a \cdot 2} \]
      associate-/l* [=>] 1.0	\[ \color{blue}{\frac{-1}{\frac{a \cdot 2}{\left(z \cdot 9\right) \cdot t - x \cdot y}}} \]
      associate-/r/ [=>] 0.8	\[ \color{blue}{\frac{-1}{a \cdot 2} \cdot \left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)} \]
      sub-neg [=>] 0.8	\[ \frac{-1}{a \cdot 2} \cdot \color{blue}{\left(\left(z \cdot 9\right) \cdot t + \left(-x \cdot y\right)\right)} \]
      +-commutative [=>] 0.8	\[ \frac{-1}{a \cdot 2} \cdot \color{blue}{\left(\left(-x \cdot y\right) + \left(z \cdot 9\right) \cdot t\right)} \]
      neg-sub0 [=>] 0.8	\[ \frac{-1}{a \cdot 2} \cdot \left(\color{blue}{\left(0 - x \cdot y\right)} + \left(z \cdot 9\right) \cdot t\right) \]
      associate-+l- [=>] 0.8	\[ \frac{-1}{a \cdot 2} \cdot \color{blue}{\left(0 - \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)\right)} \]
      sub0-neg [=>] 0.8	\[ \frac{-1}{a \cdot 2} \cdot \color{blue}{\left(-\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)\right)} \]
      distribute-rgt-neg-out [=>] 0.8	\[ \color{blue}{-\frac{-1}{a \cdot 2} \cdot \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)} \]
      distribute-lft-neg-in [=>] 0.8	\[ \color{blue}{\left(-\frac{-1}{a \cdot 2}\right) \cdot \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)} \]

   3. Taylor expanded in x around 0 0.8

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \frac{y \cdot x}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -\infty \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \leq 10^{+284}\right):\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right) + \frac{z}{a} \cdot \frac{-9}{\frac{2}{t}}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a} + 0.5 \cdot \frac{x \cdot y}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	4.2
Cost	2248

\[\begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t \cdot \left(\frac{z}{a} \cdot -4.5\right)\\ \mathbf{elif}\;t_1 \leq 10^{+303}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a} + 0.5 \cdot \frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{a} \cdot \left(t \cdot -4.5\right)\\ \end{array} \]

Alternative 2
Error	4.4
Cost	2121

\[\begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+306} \lor \neg \left(t_1 \leq 2 \cdot 10^{+291}\right):\\ \;\;\;\;t \cdot \left(\frac{z}{a} \cdot -4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \end{array} \]

Alternative 3
Error	4.9
Cost	1872

\[\begin{array}{l} t_1 := \frac{0.5}{a} \cdot \left(x \cdot y - z \cdot \left(9 \cdot t\right)\right)\\ t_2 := 0.5 \cdot \frac{x}{\frac{a}{y}}\\ \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+280}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot y \leq -4 \cdot 10^{-299}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 10^{-239}:\\ \;\;\;\;\frac{z}{a} \cdot \left(t \cdot -4.5\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{+206}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	31.4
Cost	713

\[\begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{-159} \lor \neg \left(a \leq 500000000\right):\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \end{array} \]

Alternative 5
Error	23.8
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-149} \lor \neg \left(y \leq 27000000000000\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \end{array} \]

Alternative 6
Error	23.6
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{-150} \lor \neg \left(y \leq 8400000000000\right):\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \end{array} \]

Alternative 7
Error	23.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-149}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;y \leq 35000000000000:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 8
Error	23.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-149}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;y \leq 225000000000:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5}{a}\\ \end{array} \]

Alternative 9
Error	23.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-149}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;y \leq 2060000000000:\\ \;\;\;\;t \cdot \left(\frac{z}{a} \cdot -4.5\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5}{a}\\ \end{array} \]

Alternative 10
Error	23.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-149}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;y \leq 4000000000000:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5}{a}\\ \end{array} \]

Alternative 11
Error	23.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-149}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;y \leq 5500000000000:\\ \;\;\;\;\frac{z}{a} \cdot \left(t \cdot -4.5\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5}{a}\\ \end{array} \]

Alternative 12
Error	32.8
Cost	448

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

Alternative 13
Error	32.5
Cost	448

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

Reproduce

herbie shell --seed 2023054 
(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)))