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

Average Error: 3.0 → 0.5

Time: 22.9s

Precision: binary64

Cost: 1220

\[ \begin{array}{c}[y, z, t] = \mathsf{sort}([y, z, t])\\ \end{array} \]

Math

\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

↓

\[\begin{array}{l} \mathbf{if}\;z \leq 1.3 \cdot 10^{-62}:\\ \;\;\;\;\left(x \cdot 2 - y \cdot \left(9 \cdot \left(z \cdot t\right)\right)\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))

↓

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 1.3e-62)
   (+ (- (* x 2.0) (* y (* 9.0 (* z t)))) (* a (* 27.0 b)))
   (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b);
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 1.3e-62) {
		tmp = ((x * 2.0) - (y * (9.0 * (z * t)))) + (a * (27.0 * b));
	} else {
		tmp = ((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((x * 2.0d0) - (((y * 9.0d0) * z) * t)) + ((a * 27.0d0) * b)
end function

↓

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (z <= 1.3d-62) then
        tmp = ((x * 2.0d0) - (y * (9.0d0 * (z * t)))) + (a * (27.0d0 * b))
    else
        tmp = ((x * 2.0d0) - (((y * 9.0d0) * z) * t)) + ((a * 27.0d0) * b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b);
}

↓

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 1.3e-62) {
		tmp = ((x * 2.0) - (y * (9.0 * (z * t)))) + (a * (27.0 * b));
	} else {
		tmp = ((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	return ((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b)

↓

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= 1.3e-62:
		tmp = ((x * 2.0) - (y * (9.0 * (z * t)))) + (a * (27.0 * b))
	else:
		tmp = ((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(x * 2.0) - Float64(Float64(Float64(y * 9.0) * z) * t)) + Float64(Float64(a * 27.0) * b))
end

↓

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 1.3e-62)
		tmp = Float64(Float64(Float64(x * 2.0) - Float64(y * Float64(9.0 * Float64(z * t)))) + Float64(a * Float64(27.0 * b)));
	else
		tmp = Float64(Float64(Float64(x * 2.0) - Float64(Float64(Float64(y * 9.0) * z) * t)) + Float64(Float64(a * 27.0) * b));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b);
end

↓

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= 1.3e-62)
		tmp = ((x * 2.0) - (y * (9.0 * (z * t)))) + (a * (27.0 * b));
	else
		tmp = ((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x * 2.0), $MachinePrecision] - N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 1.3e-62], N[(N[(N[(x * 2.0), $MachinePrecision] - N[(y * N[(9.0 * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(27.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * 2.0), $MachinePrecision] - N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Target

Original	3.0
Target	3.5
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;y < 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if z < 1.3e-62

   1. Initial program 3.6

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

   2. Simplified 0.6

      \[\leadsto \color{blue}{\left(x \cdot 2 - y \cdot \left(9 \cdot \left(z \cdot t\right)\right)\right) + a \cdot \left(27 \cdot b\right)} \]
      Proof

      [Start] 3.6	\[ \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
      rational.json-simplify-2 [=>] 3.6	\[ \left(x \cdot 2 - \color{blue}{t \cdot \left(\left(y \cdot 9\right) \cdot z\right)}\right) + \left(a \cdot 27\right) \cdot b \]
      rational.json-simplify-43 [=>] 0.6	\[ \left(x \cdot 2 - \color{blue}{\left(y \cdot 9\right) \cdot \left(z \cdot t\right)}\right) + \left(a \cdot 27\right) \cdot b \]
      rational.json-simplify-2 [=>] 0.6	\[ \left(x \cdot 2 - \color{blue}{\left(z \cdot t\right) \cdot \left(y \cdot 9\right)}\right) + \left(a \cdot 27\right) \cdot b \]
      rational.json-simplify-43 [=>] 0.5	\[ \left(x \cdot 2 - \color{blue}{y \cdot \left(9 \cdot \left(z \cdot t\right)\right)}\right) + \left(a \cdot 27\right) \cdot b \]
      rational.json-simplify-2 [=>] 0.5	\[ \left(x \cdot 2 - y \cdot \left(9 \cdot \left(z \cdot t\right)\right)\right) + \color{blue}{b \cdot \left(a \cdot 27\right)} \]
      rational.json-simplify-43 [=>] 0.6	\[ \left(x \cdot 2 - y \cdot \left(9 \cdot \left(z \cdot t\right)\right)\right) + \color{blue}{a \cdot \left(27 \cdot b\right)} \]

   if 1.3e-62 < z

   1. Initial program 0.3

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.3 \cdot 10^{-62}:\\ \;\;\;\;\left(x \cdot 2 - y \cdot \left(9 \cdot \left(z \cdot t\right)\right)\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\\ \end{array} \]

Alternatives

Alternative 1
Error	28.6
Cost	1376

\[\begin{array}{l} t_1 := a \cdot \left(27 \cdot b\right)\\ \mathbf{if}\;x \leq -490:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-26}:\\ \;\;\;\;27 \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-88}:\\ \;\;\;\;y \cdot \left(\left(t \cdot z\right) \cdot -9\right)\\ \mathbf{elif}\;x \leq -1.06 \cdot 10^{-234}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-285}:\\ \;\;\;\;y \cdot \left(t \cdot \left(z \cdot -9\right)\right)\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-246}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-129}:\\ \;\;\;\;\left(-9 \cdot t\right) \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot x\\ \end{array} \]

Alternative 2
Error	28.7
Cost	1376

\[\begin{array}{l} t_1 := a \cdot \left(27 \cdot b\right)\\ \mathbf{if}\;x \leq -1720000:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;x \leq -1.95 \cdot 10^{-26}:\\ \;\;\;\;27 \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-89}:\\ \;\;\;\;\left(y \cdot \left(t \cdot z\right)\right) \cdot -9\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-234}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-284}:\\ \;\;\;\;y \cdot \left(t \cdot \left(z \cdot -9\right)\right)\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-247}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-129}:\\ \;\;\;\;\left(-9 \cdot t\right) \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+28}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot x\\ \end{array} \]

Alternative 3
Error	17.3
Cost	1368

\[\begin{array}{l} t_1 := 2 \cdot x + a \cdot \left(27 \cdot b\right)\\ t_2 := 2 \cdot x - 9 \cdot \left(z \cdot \left(y \cdot t\right)\right)\\ \mathbf{if}\;t \leq -1.05 \cdot 10^{-154}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 0.0075:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+27}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+295}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+303}:\\ \;\;\;\;27 \cdot \left(a \cdot b\right) + 2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(z \cdot -9\right)\right)\\ \end{array} \]

Alternative 4
Error	11.8
Cost	1360

\[\begin{array}{l} t_1 := a \cdot \left(27 \cdot b\right)\\ t_2 := t_1 - y \cdot \left(9 \cdot \left(t \cdot z\right)\right)\\ t_3 := 27 \cdot \left(a \cdot b\right) + 2 \cdot x\\ \mathbf{if}\;x \leq -44000000:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 1.82 \cdot 10^{-57}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-35}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 3.75 \cdot 10^{+28}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot x + t_1\\ \end{array} \]

Alternative 5
Error	15.5
Cost	1232

\[\begin{array}{l} t_1 := 2 \cdot x - y \cdot \left(z \cdot \left(t \cdot 9\right)\right)\\ t_2 := 2 \cdot x + a \cdot \left(27 \cdot b\right)\\ \mathbf{if}\;t \leq -3.6 \cdot 10^{-156}:\\ \;\;\;\;2 \cdot x - 9 \cdot \left(z \cdot \left(y \cdot t\right)\right)\\ \mathbf{elif}\;t \leq 0.015:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.42 \cdot 10^{+79}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	1.3
Cost	1220

\[\begin{array}{l} \mathbf{if}\;z \leq 9.8 \cdot 10^{+72}:\\ \;\;\;\;\left(x \cdot 2 - y \cdot \left(9 \cdot \left(z \cdot t\right)\right)\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot x - 9 \cdot \left(z \cdot \left(y \cdot t\right)\right)\\ \end{array} \]

Alternative 7
Error	1.3
Cost	1220

\[\begin{array}{l} \mathbf{if}\;z \leq 3.6 \cdot 10^{+76}:\\ \;\;\;\;\left(x \cdot 2 - y \cdot \left(\left(9 \cdot z\right) \cdot t\right)\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;2 \cdot x - 9 \cdot \left(z \cdot \left(y \cdot t\right)\right)\\ \end{array} \]

Alternative 8
Error	28.3
Cost	1112

\[\begin{array}{l} t_1 := a \cdot \left(27 \cdot b\right)\\ t_2 := y \cdot \left(t \cdot \left(z \cdot -9\right)\right)\\ \mathbf{if}\;x \leq -1680:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-25}:\\ \;\;\;\;27 \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-88}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-235}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-285}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+28}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot x\\ \end{array} \]

Alternative 9
Error	28.4
Cost	1112

\[\begin{array}{l} t_1 := a \cdot \left(27 \cdot b\right)\\ \mathbf{if}\;x \leq -33000000:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-26}:\\ \;\;\;\;27 \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-89}:\\ \;\;\;\;y \cdot \left(\left(t \cdot z\right) \cdot -9\right)\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-235}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-285}:\\ \;\;\;\;y \cdot \left(t \cdot \left(z \cdot -9\right)\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+28}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot x\\ \end{array} \]

Alternative 10
Error	19.4
Cost	972

\[\begin{array}{l} t_1 := 2 \cdot x + a \cdot \left(27 \cdot b\right)\\ \mathbf{if}\;t \leq 4.3 \cdot 10^{+198}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+239}:\\ \;\;\;\;\left(t \cdot z\right) \cdot \left(-9 \cdot y\right)\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{+303}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(z \cdot -9\right)\right)\\ \end{array} \]

Alternative 11
Error	19.4
Cost	972

\[\begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{+198}:\\ \;\;\;\;2 \cdot x + a \cdot \left(27 \cdot b\right)\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+239}:\\ \;\;\;\;\left(t \cdot z\right) \cdot \left(-9 \cdot y\right)\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+302}:\\ \;\;\;\;27 \cdot \left(a \cdot b\right) + 2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(z \cdot -9\right)\right)\\ \end{array} \]

Alternative 12
Error	28.2
Cost	848

\[\begin{array}{l} t_1 := 27 \cdot \left(a \cdot b\right)\\ \mathbf{if}\;x \leq -10000000:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-88}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+28}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot x\\ \end{array} \]

Alternative 13
Error	28.3
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -56000000:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-31}:\\ \;\;\;\;27 \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-88}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+29}:\\ \;\;\;\;a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot x\\ \end{array} \]

Alternative 14
Error	37.0
Cost	192

\[2 \cdot x \]

Reproduce

herbie shell --seed 2023064 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< y 7.590524218811189e-161) (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* a (* 27.0 b))) (+ (- (* x 2.0) (* 9.0 (* y (* t z)))) (* (* a 27.0) b)))

  (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))