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

Average Error: 3.6 → 0.5

Time: 16.1s

Precision: binary64

Cost: 1225

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-22} \lor \neg \left(t \leq 1.05 \cdot 10^{-67}\right):\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y - \frac{t}{y}}{z}}{-3}\\ \end{array} \]

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 -2e-22) (not (<= t 1.05e-67)))
   (+ (- 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 <= -2e-22) || !(t <= 1.05e-67)) {
		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 <= (-2d-22)) .or. (.not. (t <= 1.05d-67))) 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 <= -2e-22) || !(t <= 1.05e-67)) {
		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 <= -2e-22) or not (t <= 1.05e-67):
		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 <= -2e-22) || !(t <= 1.05e-67))
		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 <= -2e-22) || ~((t <= 1.05e-67)))
		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, -2e-22], N[Not[LessEqual[t, 1.05e-67]], $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	3.6
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 < -2.0000000000000001e-22 or 1.0500000000000001e-67 < t

   1. Initial program 0.7

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

   if -2.0000000000000001e-22 < t < 1.0500000000000001e-67

   1. Initial program 6.5

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

   2. Simplified 0.2

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

      [Start] 6.5	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      associate-+l- [=>] 6.5	\[ \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]
      sub-neg [=>] 6.5	\[ \color{blue}{x + \left(-\left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right)} \]
      neg-mul-1 [=>] 6.5	\[ x + \color{blue}{-1 \cdot \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]
      distribute-lft-out-- [<=] 6.5	\[ x + \color{blue}{\left(-1 \cdot \frac{y}{z \cdot 3} - -1 \cdot \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]
      associate-*r/ [=>] 6.5	\[ x + \left(\color{blue}{\frac{-1 \cdot y}{z \cdot 3}} - -1 \cdot \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]
      associate-*l/ [<=] 6.5	\[ x + \left(\color{blue}{\frac{-1}{z \cdot 3} \cdot y} - -1 \cdot \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]
      associate-*r/ [=>] 6.5	\[ x + \left(\frac{-1}{z \cdot 3} \cdot y - \color{blue}{\frac{-1 \cdot t}{\left(z \cdot 3\right) \cdot y}}\right) \]
      times-frac [=>] 0.2	\[ x + \left(\frac{-1}{z \cdot 3} \cdot y - \color{blue}{\frac{-1}{z \cdot 3} \cdot \frac{t}{y}}\right) \]
      distribute-lft-out-- [=>] 0.2	\[ x + \color{blue}{\frac{-1}{z \cdot 3} \cdot \left(y - \frac{t}{y}\right)} \]
      *-commutative [=>] 0.2	\[ x + \frac{-1}{\color{blue}{3 \cdot z}} \cdot \left(y - \frac{t}{y}\right) \]
      associate-/r* [=>] 0.2	\[ x + \color{blue}{\frac{\frac{-1}{3}}{z}} \cdot \left(y - \frac{t}{y}\right) \]
      metadata-eval [=>] 0.2	\[ x + \frac{\color{blue}{-0.3333333333333333}}{z} \cdot \left(y - \frac{t}{y}\right) \]

   3. Applied egg-rr 0.2

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-22} \lor \neg \left(t \leq 1.05 \cdot 10^{-67}\right):\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y - \frac{t}{y}}{z}}{-3}\\ \end{array} \]

Alternatives

Alternative 1
Error	31.0
Cost	1108

\[\begin{array}{l} t_1 := 0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{+78}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -4.3 \cdot 10^{-67}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;x \leq -2.45 \cdot 10^{-237}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-290}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-109}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	30.5
Cost	1108

\[\begin{array}{l} t_1 := \frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{+78}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -7.6 \cdot 10^{-66}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;x \leq -1.28 \cdot 10^{-226}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-291}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-109}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	11.4
Cost	1104

\[\begin{array}{l} t_1 := x + \frac{t}{y} \cdot \frac{0.3333333333333333}{z}\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{+50}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 1.08 \cdot 10^{-296}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-221}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+67}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \end{array} \]

Alternative 4
Error	21.1
Cost	978

\[\begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-69} \lor \neg \left(x \leq -2.8 \cdot 10^{-223} \lor \neg \left(x \leq 1.3 \cdot 10^{-290}\right) \land x \leq 4.5 \cdot 10^{-132}\right):\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \end{array} \]

Alternative 5
Error	21.1
Cost	977

\[\begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-69}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-226} \lor \neg \left(x \leq 5.8 \cdot 10^{-291}\right) \land x \leq 4.2 \cdot 10^{-132}:\\ \;\;\;\;\frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \end{array} \]

Alternative 6
Error	21.1
Cost	977

\[\begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-69}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-225}:\\ \;\;\;\;\frac{\frac{t}{z}}{\frac{y}{0.3333333333333333}}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-290} \lor \neg \left(x \leq 2.6 \cdot 10^{-132}\right):\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \end{array} \]

Alternative 7
Error	1.7
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{-129} \lor \neg \left(y \leq 5 \cdot 10^{-58}\right):\\ \;\;\;\;x + \left(y - \frac{t}{y}\right) \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{0.3333333333333333}{\frac{z}{t}}}{y}\\ \end{array} \]

Alternative 8
Error	1.7
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-119} \lor \neg \left(y \leq 5.5 \cdot 10^{-60}\right):\\ \;\;\;\;x + \frac{\frac{y - \frac{t}{y}}{z}}{-3}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{0.3333333333333333}{\frac{z}{t}}}{y}\\ \end{array} \]

Alternative 9
Error	1.7
Cost	968

\[\begin{array}{l} t_1 := y - \frac{t}{y}\\ \mathbf{if}\;y \leq -4.4 \cdot 10^{-130}:\\ \;\;\;\;x + t_1 \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-60}:\\ \;\;\;\;x + \frac{\frac{0.3333333333333333}{\frac{z}{t}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t_1 \cdot -0.3333333333333333}{z}\\ \end{array} \]

Alternative 10
Error	1.6
Cost	960

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

Alternative 11
Error	12.2
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+31}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-132}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{t}{y} - y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \end{array} \]

Alternative 12
Error	9.5
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+50}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+67}:\\ \;\;\;\;x + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \end{array} \]

Alternative 13
Error	7.3
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+50}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+67}:\\ \;\;\;\;x + \frac{\frac{0.3333333333333333}{\frac{z}{t}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \end{array} \]

Alternative 14
Error	29.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+78}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.52 \cdot 10^{-128}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Error	29.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+78}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.52 \cdot 10^{-128}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16
Error	37.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023010 
(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))))