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

Average Error: 3.6 → 0.6

Time: 22.4s

Precision: binary64

Cost: 1608

Math

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

↓

\[\begin{array}{l} t_1 := \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{if}\;z \cdot 3 \leq -500000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot 3 \leq 5 \cdot 10^{-95}:\\ \;\;\;\;\left(x - \left(y \cdot \frac{2}{z}\right) \cdot 0.16666666666666666\right) + \frac{\frac{t}{y}}{z \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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))) (/ t (* (* z 3.0) y)))))
   (if (<= (* z 3.0) -500000000.0)
     t_1
     (if (<= (* z 3.0) 5e-95)
       (+ (- x (* (* y (/ 2.0 z)) 0.16666666666666666)) (/ (/ t y) (* z 3.0)))
       t_1))))

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))) + (t / ((z * 3.0) * y));
	double tmp;
	if ((z * 3.0) <= -500000000.0) {
		tmp = t_1;
	} else if ((z * 3.0) <= 5e-95) {
		tmp = (x - ((y * (2.0 / z)) * 0.16666666666666666)) + ((t / y) / (z * 3.0));
	} else {
		tmp = t_1;
	}
	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))) + (t / ((z * 3.0d0) * y))
    if ((z * 3.0d0) <= (-500000000.0d0)) then
        tmp = t_1
    else if ((z * 3.0d0) <= 5d-95) then
        tmp = (x - ((y * (2.0d0 / z)) * 0.16666666666666666d0)) + ((t / y) / (z * 3.0d0))
    else
        tmp = t_1
    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))) + (t / ((z * 3.0) * y));
	double tmp;
	if ((z * 3.0) <= -500000000.0) {
		tmp = t_1;
	} else if ((z * 3.0) <= 5e-95) {
		tmp = (x - ((y * (2.0 / z)) * 0.16666666666666666)) + ((t / y) / (z * 3.0));
	} else {
		tmp = t_1;
	}
	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))) + (t / ((z * 3.0) * y))
	tmp = 0
	if (z * 3.0) <= -500000000.0:
		tmp = t_1
	elif (z * 3.0) <= 5e-95:
		tmp = (x - ((y * (2.0 / z)) * 0.16666666666666666)) + ((t / y) / (z * 3.0))
	else:
		tmp = t_1
	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(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(z * 3.0) * y)))
	tmp = 0.0
	if (Float64(z * 3.0) <= -500000000.0)
		tmp = t_1;
	elseif (Float64(z * 3.0) <= 5e-95)
		tmp = Float64(Float64(x - Float64(Float64(y * Float64(2.0 / z)) * 0.16666666666666666)) + Float64(Float64(t / y) / Float64(z * 3.0)));
	else
		tmp = t_1;
	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))) + (t / ((z * 3.0) * y));
	tmp = 0.0;
	if ((z * 3.0) <= -500000000.0)
		tmp = t_1;
	elseif ((z * 3.0) <= 5e-95)
		tmp = (x - ((y * (2.0 / z)) * 0.16666666666666666)) + ((t / y) / (z * 3.0));
	else
		tmp = t_1;
	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[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * 3.0), $MachinePrecision], -500000000.0], t$95$1, If[LessEqual[N[(z * 3.0), $MachinePrecision], 5e-95], N[(N[(x - N[(N[(y * N[(2.0 / z), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(t / y), $MachinePrecision] / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Target

Original	3.6
Target	1.7
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z 3) < -5e8 or 4.9999999999999998e-95 < (*.f64 z 3)

   1. Initial program 0.6

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

   if -5e8 < (*.f64 z 3) < 4.9999999999999998e-95

   1. Initial program 12.2

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

   2. Simplified 0.3

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

      [Start] 12.2	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      rational.json-simplify-46 [=>] 3.1	\[ \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}} \]
      rational.json-simplify-44 [=>] 0.3	\[ \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}} \]

   3. Applied egg-rr 0.5

      \[\leadsto \left(x - \color{blue}{\left(y \cdot \frac{2}{z}\right) \cdot 0.16666666666666666}\right) + \frac{\frac{t}{y}}{z \cdot 3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -500000000:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{elif}\;z \cdot 3 \leq 5 \cdot 10^{-95}:\\ \;\;\;\;\left(x - \left(y \cdot \frac{2}{z}\right) \cdot 0.16666666666666666\right) + \frac{\frac{t}{y}}{z \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \end{array} \]

Alternatives

Alternative 1
Error	29.3
Cost	1772

\[\begin{array}{l} t_1 := 0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ t_2 := \frac{-0.3333333333333333}{\frac{z}{y}}\\ t_3 := \frac{\frac{y}{z}}{-3}\\ \mathbf{if}\;x \leq -0.00019:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-138}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-208}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-280}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-238}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-193}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{y}{-3}}{z}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-46}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+92}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	29.3
Cost	1772

\[\begin{array}{l} t_1 := 0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ t_2 := \frac{-0.3333333333333333}{\frac{z}{y}}\\ t_3 := \frac{\frac{y}{z}}{-3}\\ \mathbf{if}\;x \leq -0.00016:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.35 \cdot 10^{-139}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-207}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{t}{y}}{z}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-280}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-237}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-195}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{y}{-3}}{z}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-15}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+91}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	29.3
Cost	1772

\[\begin{array}{l} t_1 := 0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ t_2 := \frac{-0.3333333333333333}{\frac{z}{y}}\\ t_3 := \frac{\frac{y}{z}}{-3}\\ \mathbf{if}\;x \leq -0.00031:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{-138}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-208}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{t}{y}}{z}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-279}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-236}:\\ \;\;\;\;\frac{t}{y} \cdot \frac{0.3333333333333333}{z}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-195}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{-142}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{y}{-3}}{z}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+91}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	29.4
Cost	1772

\[\begin{array}{l} t_1 := \frac{-0.3333333333333333}{\frac{z}{y}}\\ t_2 := \frac{\frac{y}{z}}{-3}\\ \mathbf{if}\;x \leq -0.00016:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.08 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-207}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{t}{y}}{z}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-277}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-236}:\\ \;\;\;\;\frac{t}{y} \cdot \frac{0.3333333333333333}{z}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-195}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{z}}{y} \cdot t\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{y}{-3}}{z}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-49}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+91}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	0.4
Cost	1480

\[\begin{array}{l} t_1 := \left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{0.3333333333333333}{z}}{y} \cdot t\\ \mathbf{if}\;z \cdot 3 \leq -5 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot 3 \leq 0.01:\\ \;\;\;\;x + \left(-\frac{-0.3333333333333333 \cdot \left(\frac{t}{y} - y\right)}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	0.7
Cost	1480

\[\begin{array}{l} t_1 := \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{if}\;z \cdot 3 \leq -5 \cdot 10^{-122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot 3 \leq 10^{-24}:\\ \;\;\;\;x + \left(-\frac{-0.3333333333333333 \cdot \left(\frac{t}{y} - y\right)}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	0.5
Cost	1480

\[\begin{array}{l} t_1 := x - \frac{y}{z \cdot 3}\\ t_2 := t_1 + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{if}\;z \cdot 3 \leq -5 \cdot 10^{-48}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \cdot 3 \leq 5 \cdot 10^{-95}:\\ \;\;\;\;t_1 + \frac{\frac{t}{y}}{z \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	19.7
Cost	1240

\[\begin{array}{l} t_1 := x - 0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-207}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{t}{y}}{z}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-279}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-235}:\\ \;\;\;\;\frac{t}{y} \cdot \frac{0.3333333333333333}{z}\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-193}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-142}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{z}}{y} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	1.6
Cost	1032

\[\begin{array}{l} t_1 := x + \left(-\frac{-0.3333333333333333 \cdot \left(\frac{t}{y} - y\right)}{z}\right)\\ \mathbf{if}\;y \leq -1.48 \cdot 10^{-65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{\frac{t}{z \cdot 3}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	12.0
Cost	840

\[\begin{array}{l} t_1 := x - 0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{if}\;x \leq -5.9 \cdot 10^{-71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+23}:\\ \;\;\;\;\frac{0.3333333333333333}{z} \cdot \left(\frac{t}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	11.2
Cost	840

\[\begin{array}{l} t_1 := x - 0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -7 \cdot 10^{+118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+70}:\\ \;\;\;\;x + t \cdot \frac{\frac{0.3333333333333333}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	10.9
Cost	840

\[\begin{array}{l} t_1 := x - 0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{+118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+70}:\\ \;\;\;\;x + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	6.0
Cost	840

\[\begin{array}{l} t_1 := x - 0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -1900000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-45}:\\ \;\;\;\;x + \frac{\frac{t}{z \cdot 3}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	28.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00016:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+91}:\\ \;\;\;\;\frac{-0.3333333333333333}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Error	28.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00034:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+91}:\\ \;\;\;\;\frac{y}{z} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16
Error	28.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00035:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+91}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 17
Error	28.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0003:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+91}:\\ \;\;\;\;\frac{\frac{y}{z}}{-3}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 18
Error	37.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023074 
(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))))