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

Average Error: 3.2 → 0.7

Time: 22.9s

Precision: binary64

Cost: 1220

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

Math

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

↓

\[\begin{array}{l} t_1 := \left(a \cdot 27\right) \cdot b\\ \mathbf{if}\;z \leq 4 \cdot 10^{-237}:\\ \;\;\;\;\left(x \cdot 2 - y \cdot \left(\left(9 \cdot z\right) \cdot t\right)\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + t_1\\ \end{array} \]

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 (<= z 4e-237)
     (+ (- (* x 2.0) (* y (* (* 9.0 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 (z <= 4e-237) {
		tmp = ((x * 2.0) - (y * ((9.0 * 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 (z <= 4d-237) then
        tmp = ((x * 2.0d0) - (y * ((9.0d0 * 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 (z <= 4e-237) {
		tmp = ((x * 2.0) - (y * ((9.0 * 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 z <= 4e-237:
		tmp = ((x * 2.0) - (y * ((9.0 * 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 (z <= 4e-237)
		tmp = Float64(Float64(Float64(x * 2.0) - Float64(y * Float64(Float64(9.0 * z) * t))) + t_1);
	else
		tmp = Float64(Float64(Float64(x * 2.0) - Float64(Float64(Float64(y * 9.0) * 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 (z <= 4e-237)
		tmp = ((x * 2.0) - (y * ((9.0 * 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[z, 4e-237], N[(N[(N[(x * 2.0), $MachinePrecision] - N[(y * N[(N[(9.0 * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(x * 2.0), $MachinePrecision] - N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]

Target

Original	3.2
Target	3.6
Herbie	0.7

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

Derivation

1. Split input into 2 regimes

2. if z < 4e-237

   1. Initial program 4.7

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

   2. Simplified 0.6

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

      [Start] 4.7	\[ \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
      rational.json-simplify-2 [=>] 4.7	\[ \left(x \cdot 2 - \color{blue}{t \cdot \left(\left(y \cdot 9\right) \cdot z\right)}\right) + \left(a \cdot 27\right) \cdot b \]
      rational.json-simplify-2 [=>] 4.7	\[ \left(x \cdot 2 - t \cdot \color{blue}{\left(z \cdot \left(y \cdot 9\right)\right)}\right) + \left(a \cdot 27\right) \cdot b \]
      rational.json-simplify-43 [=>] 4.7	\[ \left(x \cdot 2 - t \cdot \color{blue}{\left(y \cdot \left(9 \cdot z\right)\right)}\right) + \left(a \cdot 27\right) \cdot b \]
      rational.json-simplify-43 [=>] 0.6	\[ \left(x \cdot 2 - \color{blue}{y \cdot \left(\left(9 \cdot z\right) \cdot t\right)}\right) + \left(a \cdot 27\right) \cdot b \]

   if 4e-237 < z

   1. Initial program 0.9

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{-237}:\\ \;\;\;\;\left(x \cdot 2 - y \cdot \left(\left(9 \cdot z\right) \cdot t\right)\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\\ \end{array} \]

Alternatives

Alternative 1
Error	37.9
Cost	2032

\[\begin{array}{l} t_1 := a \cdot \left(27 \cdot b\right)\\ t_2 := -9 \cdot \left(t \cdot \left(y \cdot z\right)\right)\\ t_3 := 27 \cdot \left(a \cdot b\right)\\ \mathbf{if}\;t \leq -3.8 \cdot 10^{-97}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -3.9 \cdot 10^{-225}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;t \leq -3.8 \cdot 10^{-225}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-292}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-244}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-99}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{+46}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+167}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+194}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+211}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	5.4
Cost	1484

\[\begin{array}{l} t_1 := \left(x - z \cdot \left(t \cdot \left(y \cdot 4.5\right)\right)\right) \cdot 2 + \left(a \cdot 27\right) \cdot b\\ t_2 := 2 \cdot x - y \cdot \left(z \cdot \left(9 \cdot t\right)\right)\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{+275}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -4.4 \cdot 10^{+177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{+152}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	0.6
Cost	1348

\[\begin{array}{l} \mathbf{if}\;y \cdot 9 \leq -5 \cdot 10^{+20}:\\ \;\;\;\;\left(x \cdot 2 - y \cdot \left(9 \cdot \left(z \cdot t\right)\right)\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z \cdot \left(t \cdot \left(y \cdot 4.5\right)\right)\right) \cdot 2 + \left(a \cdot 27\right) \cdot b\\ \end{array} \]

Alternative 4
Error	18.8
Cost	972

\[\begin{array}{l} t_1 := 2 \cdot x + b \cdot \left(27 \cdot a\right)\\ \mathbf{if}\;t \leq -1.35 \cdot 10^{-110}:\\ \;\;\;\;y \cdot \left(t \cdot \left(z \cdot -9\right)\right)\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+211}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{+288}:\\ \;\;\;\;-9 \cdot \left(t \cdot \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	13.7
Cost	968

\[\begin{array}{l} t_1 := 2 \cdot x - t \cdot \left(\left(9 \cdot z\right) \cdot y\right)\\ \mathbf{if}\;t \leq -1.32 \cdot 10^{-107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+46}:\\ \;\;\;\;2 \cdot x + b \cdot \left(27 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	13.7
Cost	968

\[\begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-225}:\\ \;\;\;\;2 \cdot x - y \cdot \left(z \cdot \left(9 \cdot t\right)\right)\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+46}:\\ \;\;\;\;2 \cdot x + b \cdot \left(27 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot x - t \cdot \left(\left(9 \cdot z\right) \cdot y\right)\\ \end{array} \]

Alternative 7
Error	13.6
Cost	968

\[\begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-225}:\\ \;\;\;\;4 \cdot \left(x \cdot 0.5 - z \cdot \left(\left(y \cdot t\right) \cdot 2.25\right)\right)\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+46}:\\ \;\;\;\;2 \cdot x + b \cdot \left(27 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot x - t \cdot \left(\left(9 \cdot z\right) \cdot y\right)\\ \end{array} \]

Alternative 8
Error	28.8
Cost	716

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+14}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-252}:\\ \;\;\;\;y \cdot \left(t \cdot \left(z \cdot -9\right)\right)\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-14}:\\ \;\;\;\;b \cdot \left(27 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot x\\ \end{array} \]

Alternative 9
Error	28.8
Cost	716

\[\begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{+14}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-251}:\\ \;\;\;\;\left(y \cdot \left(t \cdot z\right)\right) \cdot -9\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-19}:\\ \;\;\;\;b \cdot \left(27 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot x\\ \end{array} \]

Alternative 10
Error	27.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+14}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;27 \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot x\\ \end{array} \]

Alternative 11
Error	27.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+14}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-24}:\\ \;\;\;\;a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot x\\ \end{array} \]

Alternative 12
Error	27.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+14}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-15}:\\ \;\;\;\;b \cdot \left(27 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot x\\ \end{array} \]

Alternative 13
Error	37.4
Cost	192

\[2 \cdot x \]

Reproduce

herbie shell --seed 2023068 
(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)))