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

Average Error: 3.4 → 0.8

Time: 14.7s

Precision: binary64

Cost: 1224

Math

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

↓

\[\begin{array}{l} t_1 := x - \frac{y}{z \cdot 3}\\ \mathbf{if}\;t \leq -6.4 \cdot 10^{+78}:\\ \;\;\;\;t_1 + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-73}:\\ \;\;\;\;x + \frac{\frac{y - \frac{t}{y}}{z}}{-3}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{\frac{t}{z \cdot 3}}{y}\\ \end{array} \]

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
 (let* ((t_1 (- x (/ y (* z 3.0)))))
   (if (<= t -6.4e+78)
     (+ t_1 (/ t (* y (* z 3.0))))
     (if (<= t 4.5e-73)
       (+ x (/ (/ (- y (/ t y)) z) -3.0))
       (+ t_1 (/ (/ t (* z 3.0)) y))))))

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 t_1 = x - (y / (z * 3.0));
	double tmp;
	if (t <= -6.4e+78) {
		tmp = t_1 + (t / (y * (z * 3.0)));
	} else if (t <= 4.5e-73) {
		tmp = x + (((y - (t / y)) / z) / -3.0);
	} else {
		tmp = t_1 + ((t / (z * 3.0)) / y);
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = x - (y / (z * 3.0d0))
    if (t <= (-6.4d+78)) then
        tmp = t_1 + (t / (y * (z * 3.0d0)))
    else if (t <= 4.5d-73) then
        tmp = x + (((y - (t / y)) / z) / (-3.0d0))
    else
        tmp = t_1 + ((t / (z * 3.0d0)) / y)
    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 t_1 = x - (y / (z * 3.0));
	double tmp;
	if (t <= -6.4e+78) {
		tmp = t_1 + (t / (y * (z * 3.0)));
	} else if (t <= 4.5e-73) {
		tmp = x + (((y - (t / y)) / z) / -3.0);
	} else {
		tmp = t_1 + ((t / (z * 3.0)) / y);
	}
	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):
	t_1 = x - (y / (z * 3.0))
	tmp = 0
	if t <= -6.4e+78:
		tmp = t_1 + (t / (y * (z * 3.0)))
	elif t <= 4.5e-73:
		tmp = x + (((y - (t / y)) / z) / -3.0)
	else:
		tmp = t_1 + ((t / (z * 3.0)) / y)
	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)
	t_1 = Float64(x - Float64(y / Float64(z * 3.0)))
	tmp = 0.0
	if (t <= -6.4e+78)
		tmp = Float64(t_1 + Float64(t / Float64(y * Float64(z * 3.0))));
	elseif (t <= 4.5e-73)
		tmp = Float64(x + Float64(Float64(Float64(y - Float64(t / y)) / z) / -3.0));
	else
		tmp = Float64(t_1 + Float64(Float64(t / Float64(z * 3.0)) / y));
	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)
	t_1 = x - (y / (z * 3.0));
	tmp = 0.0;
	if (t <= -6.4e+78)
		tmp = t_1 + (t / (y * (z * 3.0)));
	elseif (t <= 4.5e-73)
		tmp = x + (((y - (t / y)) / z) / -3.0);
	else
		tmp = t_1 + ((t / (z * 3.0)) / y);
	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_] := Block[{t$95$1 = N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.4e+78], N[(t$95$1 + N[(t / N[(y * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.5e-73], N[(x + N[(N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(t / N[(z * 3.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	3.4
Target	1.7
Herbie	0.8

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

Derivation

1. Split input into 3 regimes

2. if t < -6.39999999999999989e78

   1. Initial program 0.6

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

   if -6.39999999999999989e78 < t < 4.5e-73

   1. Initial program 5.4

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

   2. Simplified 0.5

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

      [Start] 5.4	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      associate-+l- [=>] 5.4	\[ \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]
      sub-neg [=>] 5.4	\[ \color{blue}{x + \left(-\left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right)} \]
      neg-mul-1 [=>] 5.4	\[ x + \color{blue}{-1 \cdot \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]
      distribute-lft-out-- [<=] 5.4	\[ x + \color{blue}{\left(-1 \cdot \frac{y}{z \cdot 3} - -1 \cdot \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]
      associate-*r/ [=>] 5.4	\[ x + \left(\color{blue}{\frac{-1 \cdot y}{z \cdot 3}} - -1 \cdot \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]
      associate-*l/ [<=] 5.4	\[ x + \left(\color{blue}{\frac{-1}{z \cdot 3} \cdot y} - -1 \cdot \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]
      associate-*r/ [=>] 5.4	\[ x + \left(\frac{-1}{z \cdot 3} \cdot y - \color{blue}{\frac{-1 \cdot t}{\left(z \cdot 3\right) \cdot y}}\right) \]
      times-frac [=>] 0.4	\[ x + \left(\frac{-1}{z \cdot 3} \cdot y - \color{blue}{\frac{-1}{z \cdot 3} \cdot \frac{t}{y}}\right) \]
      distribute-lft-out-- [=>] 0.4	\[ x + \color{blue}{\frac{-1}{z \cdot 3} \cdot \left(y - \frac{t}{y}\right)} \]
      *-commutative [=>] 0.4	\[ x + \frac{-1}{\color{blue}{3 \cdot z}} \cdot \left(y - \frac{t}{y}\right) \]
      associate-/r* [=>] 0.5	\[ x + \color{blue}{\frac{\frac{-1}{3}}{z}} \cdot \left(y - \frac{t}{y}\right) \]
      metadata-eval [=>] 0.5	\[ x + \frac{\color{blue}{-0.3333333333333333}}{z} \cdot \left(y - \frac{t}{y}\right) \]

   3. Applied egg-rr 0.4

      \[\leadsto x + \color{blue}{\frac{\frac{y - \frac{t}{y}}{z}}{-3}} \]

   if 4.5e-73 < t

   1. Initial program 0.8

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

   2. Simplified 1.9

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

      [Start] 0.8	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      associate-/r* [=>] 1.9	\[ \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.4 \cdot 10^{+78}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-73}:\\ \;\;\;\;x + \frac{\frac{y - \frac{t}{y}}{z}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	1481

\[\begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -1 \cdot 10^{+18} \lor \neg \left(z \cdot 3 \leq 2 \cdot 10^{-97}\right):\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y - \frac{t}{y}}{z}}{-3}\\ \end{array} \]

Alternative 2
Error	29.4
Cost	980

\[\begin{array}{l} t_1 := 0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{+57}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-211}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-197}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{elif}\;x \leq 1.04 \cdot 10^{-165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-56}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	29.6
Cost	980

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+57}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-212}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{t}{y}}{z}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-198}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-166}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-55}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	29.0
Cost	980

\[\begin{array}{l} t_1 := \frac{\frac{t}{z}}{y} \cdot 0.3333333333333333\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{+57}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-204}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-199}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-167}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-57}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	29.0
Cost	980

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+51}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-220}:\\ \;\;\;\;\frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{-198}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-164}:\\ \;\;\;\;\frac{\frac{t}{z}}{y} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-57}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	16.8
Cost	976

\[\begin{array}{l} t_1 := \frac{\frac{t}{z}}{y} \cdot 0.3333333333333333\\ t_2 := x + \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -5.7 \cdot 10^{-132}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-300}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{-225}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.08 \cdot 10^{-48}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	16.8
Cost	976

\[\begin{array}{l} t_1 := \frac{\frac{t}{z}}{y}\\ t_2 := x + \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -1.9 \cdot 10^{-129}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-300}:\\ \;\;\;\;\frac{t_1}{3}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-225}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.08 \cdot 10^{-48}:\\ \;\;\;\;t_1 \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	1.6
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-111} \lor \neg \left(y \leq 8 \cdot 10^{-72}\right):\\ \;\;\;\;x + \left(y - \frac{t}{y}\right) \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{z \cdot 3}}{y}\\ \end{array} \]

Alternative 9
Error	1.6
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-97} \lor \neg \left(y \leq 3.65 \cdot 10^{-118}\right):\\ \;\;\;\;x + \frac{y - \frac{t}{y}}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{z \cdot 3}}{y}\\ \end{array} \]

Alternative 10
Error	1.6
Cost	968

\[\begin{array}{l} t_1 := y - \frac{t}{y}\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{-112}:\\ \;\;\;\;x + t_1 \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-74}:\\ \;\;\;\;x + \frac{\frac{t}{z \cdot 3}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{\frac{z}{t_1}}\\ \end{array} \]

Alternative 11
Error	1.6
Cost	968

\[\begin{array}{l} t_1 := y - \frac{t}{y}\\ \mathbf{if}\;y \leq -8 \cdot 10^{-111}:\\ \;\;\;\;x + \frac{t_1 \cdot -0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-73}:\\ \;\;\;\;x + \frac{\frac{t}{z \cdot 3}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{\frac{z}{t_1}}\\ \end{array} \]

Alternative 12
Error	12.2
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+45} \lor \neg \left(x \leq 6.8 \cdot 10^{-79}\right):\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{t}{y}}{z} \cdot -0.3333333333333333\\ \end{array} \]

Alternative 13
Error	8.0
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-34} \lor \neg \left(y \leq 2.05 \cdot 10^{+23}\right):\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \end{array} \]

Alternative 14
Error	5.8
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-35} \lor \neg \left(y \leq 5.4 \cdot 10^{+19}\right):\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{z \cdot 3}}{y}\\ \end{array} \]

Alternative 15
Error	28.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+51}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-57}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16
Error	28.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+51}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{-56}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 17
Error	28.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+51}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-56}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 18
Error	36.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023073 
(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))))