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

Average Error: 3.7 → 0.4

Time: 16.0s

Precision: binary64

Cost: 1481

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -5 \cdot 10^{-22} \lor \neg \left(z \cdot 3 \leq 10^{-18}\right):\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y - \frac{t}{y}}{z}}{-3}\\ \end{array} \]

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 (<= (* z 3.0) -5e-22) (not (<= (* z 3.0) 1e-18)))
   (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y)))
   (+ 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 (((z * 3.0) <= -5e-22) || !((z * 3.0) <= 1e-18)) {
		tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
	} 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 (((z * 3.0d0) <= (-5d-22)) .or. (.not. ((z * 3.0d0) <= 1d-18))) then
        tmp = (x - (y / (z * 3.0d0))) + (t / ((z * 3.0d0) * y))
    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 (((z * 3.0) <= -5e-22) || !((z * 3.0) <= 1e-18)) {
		tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
	} 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 ((z * 3.0) <= -5e-22) or not ((z * 3.0) <= 1e-18):
		tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y))
	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 ((Float64(z * 3.0) <= -5e-22) || !(Float64(z * 3.0) <= 1e-18))
		tmp = Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(z * 3.0) * y)));
	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 (((z * 3.0) <= -5e-22) || ~(((z * 3.0) <= 1e-18)))
		tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
	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[N[(z * 3.0), $MachinePrecision], -5e-22], N[Not[LessEqual[N[(z * 3.0), $MachinePrecision], 1e-18]], $MachinePrecision]], N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision]]

Target

Original	3.7
Target	1.8
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z 3) < -4.99999999999999954e-22 or 1.0000000000000001e-18 < (*.f64 z 3)

   1. Initial program 0.5

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

   if -4.99999999999999954e-22 < (*.f64 z 3) < 1.0000000000000001e-18

   1. Initial program 11.5

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

   2. Simplified 0.4

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

      [Start] 11.5	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      associate-+l- [=>] 11.5	\[ \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]
      sub-neg [=>] 11.5	\[ \color{blue}{x + \left(-\left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right)} \]
      neg-mul-1 [=>] 11.5	\[ x + \color{blue}{-1 \cdot \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]
      distribute-lft-out-- [<=] 11.5	\[ x + \color{blue}{\left(-1 \cdot \frac{y}{z \cdot 3} - -1 \cdot \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]
      associate-*r/ [=>] 11.5	\[ x + \left(\color{blue}{\frac{-1 \cdot y}{z \cdot 3}} - -1 \cdot \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]
      associate-*l/ [<=] 11.5	\[ x + \left(\color{blue}{\frac{-1}{z \cdot 3} \cdot y} - -1 \cdot \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]
      associate-*r/ [=>] 11.5	\[ x + \left(\frac{-1}{z \cdot 3} \cdot y - \color{blue}{\frac{-1 \cdot t}{\left(z \cdot 3\right) \cdot y}}\right) \]
      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* [=>] 0.4	\[ x + \color{blue}{\frac{\frac{-1}{3}}{z}} \cdot \left(y - \frac{t}{y}\right) \]
      metadata-eval [=>] 0.4	\[ x + \frac{\color{blue}{-0.3333333333333333}}{z} \cdot \left(y - \frac{t}{y}\right) \]

   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 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -5 \cdot 10^{-22} \lor \neg \left(z \cdot 3 \leq 10^{-18}\right):\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y - \frac{t}{y}}{z}}{-3}\\ \end{array} \]

Alternatives

Alternative 1
Error	30.2
Cost	1372

\[\begin{array}{l} t_1 := 0.3333333333333333 \cdot \frac{t}{z \cdot y}\\ t_2 := \frac{y}{z \cdot -3}\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{+114}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-84}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-251}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-293}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-292}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-186}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 19000000000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	30.2
Cost	1372

\[\begin{array}{l} t_1 := 0.3333333333333333 \cdot \frac{t}{z \cdot y}\\ t_2 := \frac{y}{z \cdot -3}\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{+114}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-83}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-253}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-290}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-292}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.05 \cdot 10^{-188}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+18}:\\ \;\;\;\;t \cdot \frac{0.3333333333333333}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	31.8
Cost	1244

\[\begin{array}{l} t_1 := \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -4.7 \cdot 10^{+70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.4 \cdot 10^{-83}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-179}:\\ \;\;\;\;\frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-139}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-47}:\\ \;\;\;\;t \cdot \frac{0.3333333333333333}{z \cdot y}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+32}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+178}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	10.0
Cost	1104

\[\begin{array}{l} t_1 := x + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{if}\;y \leq -4 \cdot 10^{-64}:\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \mathbf{elif}\;y \leq -1.28 \cdot 10^{-278}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-209}:\\ \;\;\;\;\frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \mathbf{elif}\;y \leq 1.56 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{-3}}{z}\\ \end{array} \]

Alternative 5
Error	10.0
Cost	1104

\[\begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-65}:\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-279}:\\ \;\;\;\;x + 0.3333333333333333 \cdot \frac{t}{z \cdot y}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-211}:\\ \;\;\;\;\frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+20}:\\ \;\;\;\;x + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{-3}}{z}\\ \end{array} \]

Alternative 6
Error	17.1
Cost	977

\[\begin{array}{l} t_1 := x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{if}\;y \leq -8.8 \cdot 10^{-85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-179}:\\ \;\;\;\;\frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-121} \lor \neg \left(y \leq 4.4 \cdot 10^{-47}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{0.3333333333333333}{z \cdot y}\\ \end{array} \]

Alternative 7
Error	17.1
Cost	977

\[\begin{array}{l} t_1 := x + \frac{\frac{y}{-3}}{z}\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{-85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-179}:\\ \;\;\;\;\frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-118} \lor \neg \left(y \leq 4.4 \cdot 10^{-47}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{0.3333333333333333}{z \cdot y}\\ \end{array} \]

Alternative 8
Error	17.1
Cost	977

\[\begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-84}:\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-179}:\\ \;\;\;\;\frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-120} \lor \neg \left(y \leq 4.4 \cdot 10^{-47}\right):\\ \;\;\;\;x + \frac{\frac{y}{-3}}{z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{0.3333333333333333}{z \cdot y}\\ \end{array} \]

Alternative 9
Error	1.8
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{-64} \lor \neg \left(y \leq 8.6 \cdot 10^{-106}\right):\\ \;\;\;\;x + \left(y - \frac{t}{y}\right) \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \end{array} \]

Alternative 10
Error	1.8
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-64} \lor \neg \left(y \leq 1.7 \cdot 10^{-104}\right):\\ \;\;\;\;x + \frac{\left(y - \frac{t}{y}\right) \cdot -0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \end{array} \]

Alternative 11
Error	1.7
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{-64} \lor \neg \left(y \leq 8 \cdot 10^{-105}\right):\\ \;\;\;\;x + \frac{y - \frac{t}{y}}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \end{array} \]

Alternative 12
Error	11.6
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-93} \lor \neg \left(x \leq 1.2 \cdot 10^{-35}\right):\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t}{y} - y\right) \cdot \frac{0.3333333333333333}{z}\\ \end{array} \]

Alternative 13
Error	11.5
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-91} \lor \neg \left(x \leq 10^{-42}\right):\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{t}{y} - y}{z}\\ \end{array} \]

Alternative 14
Error	6.2
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-46}:\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+22}:\\ \;\;\;\;x + \frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{-3}}{z}\\ \end{array} \]

Alternative 15
Error	28.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+114}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-38}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16
Error	28.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+114}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-34}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 17
Error	28.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+114}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-42}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 18
Error	37.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022356 
(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))))