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

Average Error: 4.0 → 0.5

Time: 15.3s

Precision: binary64

Cost: 1225

Math

\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{+40} \lor \neg \left(t \leq 10^{+20}\right):\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y - \frac{t}{y}}{z}}{-3}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))

↓

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -6.5e+40) (not (<= t 1e+20)))
   (+ (- x (/ y (* z 3.0))) (/ t (* y (* z 3.0))))
   (+ x (/ (/ (- y (/ t y)) z) -3.0))))

C

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -6.5e+40) || !(t <= 1e+20)) {
		tmp = (x - (y / (z * 3.0))) + (t / (y * (z * 3.0)));
	} else {
		tmp = x + (((y - (t / y)) / z) / -3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x - (y / (z * 3.0d0))) + (t / ((z * 3.0d0) * y))
end function

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-6.5d+40)) .or. (.not. (t <= 1d+20))) then
        tmp = (x - (y / (z * 3.0d0))) + (t / (y * (z * 3.0d0)))
    else
        tmp = x + (((y - (t / y)) / z) / (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

↓

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -6.5e+40) || !(t <= 1e+20)) {
		tmp = (x - (y / (z * 3.0))) + (t / (y * (z * 3.0)));
	} else {
		tmp = x + (((y - (t / y)) / z) / -3.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y))

↓

def code(x, y, z, t):
	tmp = 0
	if (t <= -6.5e+40) or not (t <= 1e+20):
		tmp = (x - (y / (z * 3.0))) + (t / (y * (z * 3.0)))
	else:
		tmp = x + (((y - (t / y)) / z) / -3.0)
	return tmp

Julia

function code(x, y, z, t)
	return Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(z * 3.0) * y)))
end

↓

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -6.5e+40) || !(t <= 1e+20))
		tmp = Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(y * Float64(z * 3.0))));
	else
		tmp = Float64(x + Float64(Float64(Float64(y - Float64(t / y)) / z) / -3.0));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
end

↓

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -6.5e+40) || ~((t <= 1e+20)))
		tmp = (x - (y / (z * 3.0))) + (t / (y * (z * 3.0)));
	else
		tmp = x + (((y - (t / y)) / z) / -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -6.5e+40], N[Not[LessEqual[t, 1e+20]], $MachinePrecision]], N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(y * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision]]

Target

Original	4.0
Target	1.6
Herbie	0.5

\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y} \]

Derivation

1. Split input into 2 regimes

2. if t < -6.5000000000000001e40 or 1e20 < t

   1. Initial program 0.8

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   if -6.5000000000000001e40 < t < 1e20

   1. Initial program 5.9

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   2. Simplified 0.3

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]
      Proof

      [Start] 5.9	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      associate-+l- [=>] 5.9	\[ \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]
      sub-neg [=>] 5.9	\[ \color{blue}{x + \left(-\left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right)} \]
      neg-mul-1 [=>] 5.9	\[ x + \color{blue}{-1 \cdot \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]
      distribute-lft-out-- [<=] 5.9	\[ x + \color{blue}{\left(-1 \cdot \frac{y}{z \cdot 3} - -1 \cdot \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]
      associate-*r/ [=>] 5.9	\[ x + \left(\color{blue}{\frac{-1 \cdot y}{z \cdot 3}} - -1 \cdot \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]
      associate-*l/ [<=] 5.9	\[ x + \left(\color{blue}{\frac{-1}{z \cdot 3} \cdot y} - -1 \cdot \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]
      associate-*r/ [=>] 5.9	\[ x + \left(\frac{-1}{z \cdot 3} \cdot y - \color{blue}{\frac{-1 \cdot t}{\left(z \cdot 3\right) \cdot y}}\right) \]
      times-frac [=>] 0.3	\[ x + \left(\frac{-1}{z \cdot 3} \cdot y - \color{blue}{\frac{-1}{z \cdot 3} \cdot \frac{t}{y}}\right) \]
      distribute-lft-out-- [=>] 0.3	\[ x + \color{blue}{\frac{-1}{z \cdot 3} \cdot \left(y - \frac{t}{y}\right)} \]
      *-commutative [=>] 0.3	\[ x + \frac{-1}{\color{blue}{3 \cdot z}} \cdot \left(y - \frac{t}{y}\right) \]
      associate-/r* [=>] 0.3	\[ x + \color{blue}{\frac{\frac{-1}{3}}{z}} \cdot \left(y - \frac{t}{y}\right) \]
      metadata-eval [=>] 0.3	\[ x + \frac{\color{blue}{-0.3333333333333333}}{z} \cdot \left(y - \frac{t}{y}\right) \]

   3. Applied egg-rr 0.3

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{+40} \lor \neg \left(t \leq 10^{+20}\right):\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y - \frac{t}{y}}{z}}{-3}\\ \end{array} \]

Alternatives

Alternative 1
Error	18.3
Cost	1040

\[\begin{array}{l} t_1 := x + \frac{\frac{y}{z}}{-3}\\ \mathbf{if}\;y \leq -3 \cdot 10^{-231}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.32 \cdot 10^{-256}:\\ \;\;\;\;\frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-229}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-49}:\\ \;\;\;\;\frac{\frac{-t}{z}}{y \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	18.1
Cost	976

\[\begin{array}{l} t_1 := \frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ t_2 := x + \frac{\frac{y}{z}}{-3}\\ \mathbf{if}\;y \leq -5.2 \cdot 10^{-226}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-256}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-228}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-49}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	18.1
Cost	976

\[\begin{array}{l} t_1 := x + \frac{\frac{y}{z}}{-3}\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{-231}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-256}:\\ \;\;\;\;\frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-232}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-49}:\\ \;\;\;\;\frac{\frac{t}{z} \cdot 0.3333333333333333}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	1.7
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{-42} \lor \neg \left(y \leq 5.8 \cdot 10^{-91}\right):\\ \;\;\;\;x + \left(y - \frac{t}{y}\right) \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{z \cdot 3}}{y}\\ \end{array} \]

Alternative 5
Error	1.7
Cost	968

\[\begin{array}{l} t_1 := y - \frac{t}{y}\\ \mathbf{if}\;y \leq -2.05 \cdot 10^{-42}:\\ \;\;\;\;x + t_1 \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-95}:\\ \;\;\;\;x + \frac{\frac{t}{z \cdot 3}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t_1}{z}}{-3}\\ \end{array} \]

Alternative 6
Error	1.6
Cost	960

\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y} \]

Alternative 7
Error	1.6
Cost	960

\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z} \cdot 0.3333333333333333}{y} \]

Alternative 8
Error	29.2
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-115}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-229}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-295}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{+27}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	29.2
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-115}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-230}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{-293}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{t}{z}}{y}\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{+27}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	29.3
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-115}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-230}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-295}:\\ \;\;\;\;\frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+27}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	11.8
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-115} \lor \neg \left(x \leq 68000000000\right):\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{t}{y}}{z} \cdot -0.3333333333333333\\ \end{array} \]

Alternative 12
Error	8.8
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+36} \lor \neg \left(y \leq 6.5 \cdot 10^{+30}\right):\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \end{array} \]

Alternative 13
Error	6.1
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+36} \lor \neg \left(y \leq 1.1 \cdot 10^{+18}\right):\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{z \cdot 3}}{y}\\ \end{array} \]

Alternative 14
Error	29.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-115}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+27}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Error	29.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-115}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+27}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16
Error	29.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-115}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+27}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 17
Error	38.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023055 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, H"
  :precision binary64

  :herbie-target
  (+ (- x (/ y (* z 3.0))) (/ (/ t (* z 3.0)) y))

  (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))