math.cube on complex, imaginary part

Average Error: 7.6 → 0.2

Time: 12.4s

Precision: binary64

Cost: 1352

Math

\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

↓

\[\begin{array}{l} t_0 := \left(3 \cdot x.re\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{if}\;x.re \leq -4 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq 4 \cdot 10^{+126}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + 2 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+
  (* (- (* x.re x.re) (* x.im x.im)) x.im)
  (* (+ (* x.re x.im) (* x.im x.re)) x.re)))

↓

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x.re) (* x.re x.im))))
   (if (<= x.re -4e+153)
     t_0
     (if (<= x.re 4e+126)
       (+
        (* (+ x.re x.im) (* (- x.re x.im) x.im))
        (* 2.0 (* (* x.re x.re) x.im)))
       t_0))))

C

double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

↓

double code(double x_46_re, double x_46_im) {
	double t_0 = (3.0 * x_46_re) * (x_46_re * x_46_im);
	double tmp;
	if (x_46_re <= -4e+153) {
		tmp = t_0;
	} else if (x_46_re <= 4e+126) {
		tmp = ((x_46_re + x_46_im) * ((x_46_re - x_46_im) * x_46_im)) + (2.0 * ((x_46_re * x_46_re) * x_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = (((x_46re * x_46re) - (x_46im * x_46im)) * x_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
end function

↓

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (3.0d0 * x_46re) * (x_46re * x_46im)
    if (x_46re <= (-4d+153)) then
        tmp = t_0
    else if (x_46re <= 4d+126) then
        tmp = ((x_46re + x_46im) * ((x_46re - x_46im) * x_46im)) + (2.0d0 * ((x_46re * x_46re) * x_46im))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

↓

public static double code(double x_46_re, double x_46_im) {
	double t_0 = (3.0 * x_46_re) * (x_46_re * x_46_im);
	double tmp;
	if (x_46_re <= -4e+153) {
		tmp = t_0;
	} else if (x_46_re <= 4e+126) {
		tmp = ((x_46_re + x_46_im) * ((x_46_re - x_46_im) * x_46_im)) + (2.0 * ((x_46_re * x_46_re) * x_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re)

↓

def code(x_46_re, x_46_im):
	t_0 = (3.0 * x_46_re) * (x_46_re * x_46_im)
	tmp = 0
	if x_46_re <= -4e+153:
		tmp = t_0
	elif x_46_re <= 4e+126:
		tmp = ((x_46_re + x_46_im) * ((x_46_re - x_46_im) * x_46_im)) + (2.0 * ((x_46_re * x_46_re) * x_46_im))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)) * x_46_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re))
end

↓

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(3.0 * x_46_re) * Float64(x_46_re * x_46_im))
	tmp = 0.0
	if (x_46_re <= -4e+153)
		tmp = t_0;
	elseif (x_46_re <= 4e+126)
		tmp = Float64(Float64(Float64(x_46_re + x_46_im) * Float64(Float64(x_46_re - x_46_im) * x_46_im)) + Float64(2.0 * Float64(Float64(x_46_re * x_46_re) * x_46_im)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
end

↓

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = (3.0 * x_46_re) * (x_46_re * x_46_im);
	tmp = 0.0;
	if (x_46_re <= -4e+153)
		tmp = t_0;
	elseif (x_46_re <= 4e+126)
		tmp = ((x_46_re + x_46_im) * ((x_46_re - x_46_im) * x_46_im)) + (2.0 * ((x_46_re * x_46_re) * x_46_im));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]

↓

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[(3.0 * x$46$re), $MachinePrecision] * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -4e+153], t$95$0, If[LessEqual[x$46$re, 4e+126], N[(N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(N[(x$46$re - x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	7.6
Target	0.2
Herbie	0.2

\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right) \]

Derivation

1. Split input into 2 regimes

2. if x.re < -4e153 or 3.9999999999999997e126 < x.re

   1. Initial program 52.9

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Simplified 0.4

      \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + x.re \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]
      Proof

      [Start] 52.9	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational.json-simplify-2 [=>] 52.9	\[ \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational.json-simplify-34 [=>] 52.9	\[ x.im \cdot \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational.json-simplify-43 [=>] 0.4	\[ \color{blue}{\left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational.json-simplify-1 [=>] 0.4	\[ \color{blue}{\left(x.re + x.im\right)} \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational.json-simplify-2 [=>] 0.4	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \color{blue}{x.re \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational.json-simplify-51 [=>] 0.4	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + x.re \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   3. Taylor expanded in x.re around inf 0.4

      \[\leadsto \left(x.re + x.im\right) \cdot \color{blue}{\left(x.re \cdot x.im\right)} + x.re \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   4. Applied egg-rr 0.4

      \[\leadsto \color{blue}{x.re \cdot \left(x.im \cdot \left(x.im + x.re \cdot 3\right)\right) - 0} \]

   5. Simplified 0.4

      \[\leadsto \color{blue}{\left(x.im + x.re \cdot 3\right) \cdot \left(x.re \cdot x.im\right)} \]
      Proof

      [Start] 0.4	\[ x.re \cdot \left(x.im \cdot \left(x.im + x.re \cdot 3\right)\right) - 0 \]
      rational.json-simplify-5 [=>] 0.4	\[ \color{blue}{x.re \cdot \left(x.im \cdot \left(x.im + x.re \cdot 3\right)\right)} \]
      rational.json-simplify-43 [<=] 0.4	\[ \color{blue}{\left(x.im + x.re \cdot 3\right) \cdot \left(x.re \cdot x.im\right)} \]

   6. Taylor expanded in x.im around 0 0.4

      \[\leadsto \color{blue}{\left(3 \cdot x.re\right)} \cdot \left(x.re \cdot x.im\right) \]

   if -4e153 < x.re < 3.9999999999999997e126

   1. Initial program 0.2

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Simplified 0.2

      \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + x.re \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]
      Proof

      [Start] 0.2	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational.json-simplify-2 [=>] 0.2	\[ \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational.json-simplify-34 [=>] 0.2	\[ x.im \cdot \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational.json-simplify-43 [=>] 0.2	\[ \color{blue}{\left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational.json-simplify-1 [=>] 0.2	\[ \color{blue}{\left(x.re + x.im\right)} \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational.json-simplify-2 [=>] 0.2	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \color{blue}{x.re \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational.json-simplify-51 [=>] 0.2	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + x.re \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   3. Applied egg-rr 0.2

      \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \color{blue}{\frac{x.im + x.im}{\frac{1}{x.re \cdot x.re}}} \]

   4. Simplified 0.2

      \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \color{blue}{\frac{x.re \cdot x.re}{\frac{1}{x.im + x.im}}} \]
      Proof

      [Start] 0.2	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \frac{x.im + x.im}{\frac{1}{x.re \cdot x.re}} \]
      rational.json-simplify-61 [=>] 0.2	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \color{blue}{\frac{x.re \cdot x.re}{\frac{1}{x.im + x.im}}} \]

   5. Applied egg-rr 0.2

      \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \color{blue}{2 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -4 \cdot 10^{+153}:\\ \;\;\;\;\left(3 \cdot x.re\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.re \leq 4 \cdot 10^{+126}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + 2 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot x.re\right) \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	1352

\[\begin{array}{l} t_0 := \left(3 \cdot x.re\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{if}\;x.re \leq -2 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq 4 \cdot 10^{+125}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.2
Cost	1224

\[\begin{array}{l} t_0 := \left(3 \cdot x.re\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{if}\;x.re \leq -4.7 \cdot 10^{+117}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq 1.6 \cdot 10^{+130}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.2
Cost	1088

\[\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + x.re \cdot \frac{x.re}{\frac{0.5}{x.im}} \]

Alternative 4
Error	0.2
Cost	1088

\[\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + x.re \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

Alternative 5
Error	19.1
Cost	576

\[\left(x.im \cdot \left(x.im + x.re \cdot 3\right)\right) \cdot x.re \]

Alternative 6
Error	19.1
Cost	576

\[\left(x.im + x.re \cdot 3\right) \cdot \left(x.re \cdot x.im\right) \]

Alternative 7
Error	19.1
Cost	448

\[3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right) \]

Alternative 8
Error	19.1
Cost	448

\[\left(3 \cdot x.re\right) \cdot \left(x.re \cdot x.im\right) \]

Alternative 9
Error	46.4
Cost	320

\[x.im \cdot \left(x.re \cdot x.im\right) \]

Reproduce

herbie shell --seed 2023066 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im)))

  (+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re)))