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

Average Error: 2.8 → 0.6

Time: 33.6s

Precision: binary64

Cost: 1476

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

Math

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

↓

\[\begin{array}{l} t_1 := \left(a \cdot 27\right) \cdot b\\ \mathbf{if}\;\left(y \cdot 9\right) \cdot z \leq 5 \cdot 10^{+176}:\\ \;\;\;\;\left(x \cdot 2 - \left(9 \cdot \left(y \cdot z\right)\right) \cdot t\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right) + t_1\\ \end{array} \]

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
 (let* ((t_1 (* (* a 27.0) b)))
   (if (<= (* (* y 9.0) z) 5e+176)
     (+ (- (* x 2.0) (* (* 9.0 (* y z)) t)) t_1)
     (+ (- (* x 2.0) (* (* y 9.0) (* z t))) t_1))))

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 t_1 = (a * 27.0) * b;
	double tmp;
	if (((y * 9.0) * z) <= 5e+176) {
		tmp = ((x * 2.0) - ((9.0 * (y * z)) * t)) + t_1;
	} else {
		tmp = ((x * 2.0) - ((y * 9.0) * (z * t))) + t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (a * 27.0d0) * b
    if (((y * 9.0d0) * z) <= 5d+176) then
        tmp = ((x * 2.0d0) - ((9.0d0 * (y * z)) * t)) + t_1
    else
        tmp = ((x * 2.0d0) - ((y * 9.0d0) * (z * t))) + t_1
    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 t_1 = (a * 27.0) * b;
	double tmp;
	if (((y * 9.0) * z) <= 5e+176) {
		tmp = ((x * 2.0) - ((9.0 * (y * z)) * t)) + t_1;
	} else {
		tmp = ((x * 2.0) - ((y * 9.0) * (z * t))) + t_1;
	}
	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):
	t_1 = (a * 27.0) * b
	tmp = 0
	if ((y * 9.0) * z) <= 5e+176:
		tmp = ((x * 2.0) - ((9.0 * (y * z)) * t)) + t_1
	else:
		tmp = ((x * 2.0) - ((y * 9.0) * (z * t))) + t_1
	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)
	t_1 = Float64(Float64(a * 27.0) * b)
	tmp = 0.0
	if (Float64(Float64(y * 9.0) * z) <= 5e+176)
		tmp = Float64(Float64(Float64(x * 2.0) - Float64(Float64(9.0 * Float64(y * z)) * t)) + t_1);
	else
		tmp = Float64(Float64(Float64(x * 2.0) - Float64(Float64(y * 9.0) * Float64(z * t))) + t_1);
	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)
	t_1 = (a * 27.0) * b;
	tmp = 0.0;
	if (((y * 9.0) * z) <= 5e+176)
		tmp = ((x * 2.0) - ((9.0 * (y * z)) * t)) + t_1;
	else
		tmp = ((x * 2.0) - ((y * 9.0) * (z * t))) + t_1;
	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_] := Block[{t$95$1 = N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision], 5e+176], N[(N[(N[(x * 2.0), $MachinePrecision] - N[(N[(9.0 * N[(y * z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(x * 2.0), $MachinePrecision] - N[(N[(y * 9.0), $MachinePrecision] * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]

Target

Original	2.8
Target	3.2
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y 9) z) < 5e176

   1. Initial program 0.6

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

   2. Taylor expanded in y around 0 0.6

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

   if 5e176 < (*.f64 (*.f64 y 9) z)

   1. Initial program 23.5

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

   2. Simplified 1.2

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

      [Start] 23.5	\[ \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
      rational.json-simplify-2 [=>] 23.5	\[ \left(x \cdot 2 - \color{blue}{t \cdot \left(\left(y \cdot 9\right) \cdot z\right)}\right) + \left(a \cdot 27\right) \cdot b \]
      rational.json-simplify-43 [=>] 1.2	\[ \left(x \cdot 2 - \color{blue}{\left(y \cdot 9\right) \cdot \left(z \cdot t\right)}\right) + \left(a \cdot 27\right) \cdot b \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot 9\right) \cdot z \leq 5 \cdot 10^{+176}:\\ \;\;\;\;\left(x \cdot 2 - \left(9 \cdot \left(y \cdot z\right)\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array} \]

Alternatives

Alternative 1
Error	29.2
Cost	1240

\[\begin{array}{l} t_1 := t \cdot \left(\left(y \cdot z\right) \cdot -9\right)\\ t_2 := \frac{b \cdot \left(a \cdot 54\right)}{2}\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{+18}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-133}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-241}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-295}:\\ \;\;\;\;a \cdot \left(27 \cdot b\right)\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-305}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+47}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot x\\ \end{array} \]

Alternative 2
Error	14.5
Cost	1232

\[\begin{array}{l} t_1 := 2 \cdot x - t \cdot \left(y \cdot \left(9 \cdot z\right)\right)\\ t_2 := 2 \cdot x + \frac{b \cdot \left(a \cdot 54\right)}{2}\\ \mathbf{if}\;b \leq -1.05 \cdot 10^{-119}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{+46}:\\ \;\;\;\;2 \cdot x + a \cdot \left(27 \cdot b\right)\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+67}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	1.3
Cost	1220

\[\begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+54}:\\ \;\;\;\;2 \cdot x - \left(t \cdot z\right) \cdot \left(y \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right) + x\right) + \left(a \cdot 27\right) \cdot b\\ \end{array} \]

Alternative 4
Error	0.7
Cost	1220

\[\begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-143}:\\ \;\;\;\;\left(x \cdot 2 - y \cdot \left(9 \cdot \left(z \cdot t\right)\right)\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - 9 \cdot \left(t \cdot \left(y \cdot z\right)\right)\right) + x\right) + \left(a \cdot 27\right) \cdot b\\ \end{array} \]

Alternative 5
Error	0.6
Cost	1220

\[\begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+37}:\\ \;\;\;\;\left(x \cdot 2 - y \cdot \left(9 \cdot \left(z \cdot t\right)\right)\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - \left(9 \cdot \left(y \cdot z\right)\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\\ \end{array} \]

Alternative 6
Error	29.2
Cost	1112

\[\begin{array}{l} t_1 := -9 \cdot \left(t \cdot \left(y \cdot z\right)\right)\\ t_2 := a \cdot \left(27 \cdot b\right)\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{+18}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-133}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-240}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-294}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-305}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+49}:\\ \;\;\;\;27 \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot x\\ \end{array} \]

Alternative 7
Error	29.3
Cost	1112

\[\begin{array}{l} t_1 := t \cdot \left(y \cdot \left(z \cdot -9\right)\right)\\ t_2 := a \cdot \left(27 \cdot b\right)\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{+18}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-133}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-240}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-295}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.95 \cdot 10^{-305}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+58}:\\ \;\;\;\;27 \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot x\\ \end{array} \]

Alternative 8
Error	29.3
Cost	1112

\[\begin{array}{l} t_1 := t \cdot \left(\left(y \cdot z\right) \cdot -9\right)\\ t_2 := a \cdot \left(27 \cdot b\right)\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{+18}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-133}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-242}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.3 \cdot 10^{-293}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+54}:\\ \;\;\;\;27 \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot x\\ \end{array} \]

Alternative 9
Error	11.8
Cost	1096

\[\begin{array}{l} t_1 := 2 \cdot x + \frac{b \cdot \left(a \cdot 54\right)}{2}\\ \mathbf{if}\;x \leq -9 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-39}:\\ \;\;\;\;y \cdot \left(\left(t \cdot z\right) \cdot -9\right) + 27 \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	16.5
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{-10}:\\ \;\;\;\;y \cdot \left(t \cdot \left(z \cdot -9\right)\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot x + \frac{b \cdot \left(a \cdot 54\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(t \cdot \left(y \cdot -18\right)\right)}{2}\\ \end{array} \]

Alternative 11
Error	12.8
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-54}:\\ \;\;\;\;2 \cdot x - z \cdot \left(y \cdot \left(9 \cdot t\right)\right)\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-38}:\\ \;\;\;\;2 \cdot x + \frac{b \cdot \left(a \cdot 54\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot x - t \cdot \left(y \cdot \left(9 \cdot z\right)\right)\\ \end{array} \]

Alternative 12
Error	12.7
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-84}:\\ \;\;\;\;2 \cdot x - \left(t \cdot z\right) \cdot \left(y \cdot 9\right)\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-38}:\\ \;\;\;\;2 \cdot x + \frac{b \cdot \left(a \cdot 54\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot x - t \cdot \left(y \cdot \left(9 \cdot z\right)\right)\\ \end{array} \]

Alternative 13
Error	16.5
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{-10}:\\ \;\;\;\;y \cdot \left(t \cdot \left(z \cdot -9\right)\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot x + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(z \cdot -9\right)\right)\\ \end{array} \]

Alternative 14
Error	16.5
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{-10}:\\ \;\;\;\;y \cdot \left(t \cdot \left(z \cdot -9\right)\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot x + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(t \cdot \left(y \cdot -18\right)\right)}{2}\\ \end{array} \]

Alternative 15
Error	28.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+18}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+50}:\\ \;\;\;\;27 \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot x\\ \end{array} \]

Alternative 16
Error	28.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+18}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;x \leq 6.7 \cdot 10^{+53}:\\ \;\;\;\;a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot x\\ \end{array} \]

Alternative 17
Error	37.9
Cost	192

\[2 \cdot x \]

Reproduce

herbie shell --seed 2023075 
(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)))