math.cube on complex, real part

Average Error: 7.1 → 0.7

Time: 53.9s

Precision: binary64

Cost: 968

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x.im \leq -2.3 \cdot 10^{+79}:\\ \;\;\;\;\left(x.im \cdot -3\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq 4.2 \cdot 10^{+36}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot \left(x.im \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot \left(x.im \cdot -3\right)\right) \cdot x.im\\ \end{array} \]

FPCore

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

↓

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im -2.3e+79)
   (* (* x.im -3.0) (* x.re x.im))
   (if (<= x.im 4.2e+36)
     (* x.re (- (* x.re x.re) (* x.im (* x.im 3.0))))
     (* (* x.re (* x.im -3.0)) x.im))))

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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

↓

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -2.3e+79) {
		tmp = (x_46_im * -3.0) * (x_46_re * x_46_im);
	} else if (x_46_im <= 4.2e+36) {
		tmp = x_46_re * ((x_46_re * x_46_re) - (x_46_im * (x_46_im * 3.0)));
	} else {
		tmp = (x_46_re * (x_46_im * -3.0)) * x_46_im;
	}
	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_46re) - (((x_46re * x_46im) + (x_46im * x_46re)) * x_46im)
end function

↓

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= (-2.3d+79)) then
        tmp = (x_46im * (-3.0d0)) * (x_46re * x_46im)
    else if (x_46im <= 4.2d+36) then
        tmp = x_46re * ((x_46re * x_46re) - (x_46im * (x_46im * 3.0d0)))
    else
        tmp = (x_46re * (x_46im * (-3.0d0))) * x_46im
    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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

↓

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -2.3e+79) {
		tmp = (x_46_im * -3.0) * (x_46_re * x_46_im);
	} else if (x_46_im <= 4.2e+36) {
		tmp = x_46_re * ((x_46_re * x_46_re) - (x_46_im * (x_46_im * 3.0)));
	} else {
		tmp = (x_46_re * (x_46_im * -3.0)) * x_46_im;
	}
	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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im)

↓

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= -2.3e+79:
		tmp = (x_46_im * -3.0) * (x_46_re * x_46_im)
	elif x_46_im <= 4.2e+36:
		tmp = x_46_re * ((x_46_re * x_46_re) - (x_46_im * (x_46_im * 3.0)))
	else:
		tmp = (x_46_re * (x_46_im * -3.0)) * x_46_im
	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_re) - Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_im))
end

↓

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= -2.3e+79)
		tmp = Float64(Float64(x_46_im * -3.0) * Float64(x_46_re * x_46_im));
	elseif (x_46_im <= 4.2e+36)
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * Float64(x_46_im * 3.0))));
	else
		tmp = Float64(Float64(x_46_re * Float64(x_46_im * -3.0)) * x_46_im);
	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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
end

↓

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= -2.3e+79)
		tmp = (x_46_im * -3.0) * (x_46_re * x_46_im);
	elseif (x_46_im <= 4.2e+36)
		tmp = x_46_re * ((x_46_re * x_46_re) - (x_46_im * (x_46_im * 3.0)));
	else
		tmp = (x_46_re * (x_46_im * -3.0)) * x_46_im;
	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$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]

↓

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, -2.3e+79], N[(N[(x$46$im * -3.0), $MachinePrecision] * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 4.2e+36], N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]]]

Target

Original	7.1
Target	0.3
Herbie	0.7

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

Derivation

1. Split input into 3 regimes

2. if x.im < -2.3e79

   1. Initial program 29.9

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

   2. Simplified 0.4

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

      [Start] 29.9	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best-simplify-1 [=>] 29.9	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best-simplify-111 [=>] 29.9	\[ x.re \cdot \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best-simplify-50 [=>] 0.4	\[ \color{blue}{\left(x.re - x.im\right) \cdot \left(\left(x.im + x.re\right) \cdot x.re\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best-simplify-3 [=>] 0.4	\[ \left(x.re - x.im\right) \cdot \left(\color{blue}{\left(x.re + x.im\right)} \cdot x.re\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best-simplify-1 [=>] 0.4	\[ \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational_best-simplify-1 [=>] 0.4	\[ \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) - x.im \cdot \left(\color{blue}{x.im \cdot x.re} + x.im \cdot x.re\right) \]
      rational_best-simplify-63 [=>] 0.4	\[ \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   3. Taylor expanded in x.re around 0 0.9

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

   4. Applied egg-rr 0.9

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

   5. Taylor expanded in x.re around 0 0.8

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

   6. Simplified 0.8

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

      [Start] 0.8	\[ \left(-3 \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) \]
      rational_best-simplify-1 [=>] 0.8	\[ \color{blue}{\left(x.im \cdot -3\right)} \cdot \left(x.re \cdot x.im\right) \]

   if -2.3e79 < x.im < 4.20000000000000009e36

   1. Initial program 0.2

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

   2. Simplified 0.2

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

      [Start] 0.2	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best-simplify-1 [=>] 0.2	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational_best-simplify-1 [<=] 0.2	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - x.im \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \]
      rational_best-simplify-63 [=>] 0.2	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - x.im \cdot \color{blue}{\left(x.im \cdot \left(x.re + x.re\right)\right)} \]
      rational_best-simplify-50 [=>] 0.2	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\left(x.re + x.re\right) \cdot \left(x.im \cdot x.im\right)} \]
      rational_best-simplify-32 [=>] 0.2	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re + x.re\right) \cdot \color{blue}{\left|x.im \cdot x.im\right|} \]
      rational_best-simplify-103 [=>] 0.2	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re + x.re\right) \cdot \color{blue}{\left(\left|x.im\right| \cdot \left|x.im\right|\right)} \]
      rational_best-simplify-80 [=>] 0.2	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\left(\left|x.im\right| + \left|x.im\right|\right) \cdot \left(\left|x.im\right| \cdot x.re\right)} \]
      rational_best-simplify-50 [=>] 0.2	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{x.re \cdot \left(\left|x.im\right| \cdot \left(\left|x.im\right| + \left|x.im\right|\right)\right)} \]
      rational_best-simplify-62 [=>] 0.2	\[ \color{blue}{x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - \left|x.im\right| \cdot \left(\left|x.im\right| + \left|x.im\right|\right)\right)} \]
      rational_best-simplify-63 [<=] 0.2	\[ x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{\left(\left|x.im\right| \cdot \left|x.im\right| + \left|x.im\right| \cdot \left|x.im\right|\right)}\right) \]
      rational_best-simplify-103 [<=] 0.2	\[ x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(\color{blue}{\left|x.im \cdot x.im\right|} + \left|x.im\right| \cdot \left|x.im\right|\right)\right) \]
      rational_best-simplify-32 [<=] 0.2	\[ x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(\color{blue}{x.im \cdot x.im} + \left|x.im\right| \cdot \left|x.im\right|\right)\right) \]
      rational_best-simplify-103 [<=] 0.2	\[ x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.im \cdot x.im + \color{blue}{\left|x.im \cdot x.im\right|}\right)\right) \]
      rational_best-simplify-32 [<=] 0.2	\[ x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.im \cdot x.im + \color{blue}{x.im \cdot x.im}\right)\right) \]
      rational_best-simplify-63 [=>] 0.2	\[ x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.im \cdot \left(x.im + x.im\right)}\right) \]
      rational_best-simplify-52 [=>] 0.2	\[ x.re \cdot \color{blue}{\left(x.re \cdot x.re - \left(x.im \cdot \left(x.im + x.im\right) + x.im \cdot x.im\right)\right)} \]
      rational_best-simplify-1 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - \left(\color{blue}{\left(x.im + x.im\right) \cdot x.im} + x.im \cdot x.im\right)\right) \]
      rational_best-simplify-63 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - \color{blue}{x.im \cdot \left(\left(x.im + x.im\right) + x.im\right)}\right) \]

   if 4.20000000000000009e36 < x.im

   1. Initial program 21.7

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

   2. Simplified 0.4

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

      [Start] 21.7	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best-simplify-1 [=>] 21.7	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best-simplify-111 [=>] 21.7	\[ x.re \cdot \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best-simplify-50 [=>] 0.4	\[ \color{blue}{\left(x.re - x.im\right) \cdot \left(\left(x.im + x.re\right) \cdot x.re\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best-simplify-3 [=>] 0.4	\[ \left(x.re - x.im\right) \cdot \left(\color{blue}{\left(x.re + x.im\right)} \cdot x.re\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best-simplify-1 [=>] 0.4	\[ \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational_best-simplify-1 [=>] 0.4	\[ \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) - x.im \cdot \left(\color{blue}{x.im \cdot x.re} + x.im \cdot x.re\right) \]
      rational_best-simplify-63 [=>] 0.4	\[ \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   3. Taylor expanded in x.re around 0 2.8

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

   4. Applied egg-rr 2.7

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

   5. Taylor expanded in x.re around 0 2.6

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

   6. Simplified 2.6

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

      [Start] 2.6	\[ \left(-3 \cdot \left(x.re \cdot x.im\right)\right) \cdot x.im \]
      rational_best-simplify-1 [=>] 2.6	\[ \left(-3 \cdot \color{blue}{\left(x.im \cdot x.re\right)}\right) \cdot x.im \]
      rational_best-simplify-50 [=>] 2.6	\[ \color{blue}{\left(x.re \cdot \left(x.im \cdot -3\right)\right)} \cdot x.im \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -2.3 \cdot 10^{+79}:\\ \;\;\;\;\left(x.im \cdot -3\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq 4.2 \cdot 10^{+36}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot \left(x.im \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot \left(x.im \cdot -3\right)\right) \cdot x.im\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	1088

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

Alternative 2
Error	0.6
Cost	968

\[\begin{array}{l} \mathbf{if}\;x.im \leq -2 \cdot 10^{+91}:\\ \;\;\;\;\left(x.im \cdot -3\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq 4.2 \cdot 10^{+36}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - 3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot \left(x.im \cdot -3\right)\right) \cdot x.im\\ \end{array} \]

Alternative 3
Error	19.7
Cost	576

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

Alternative 4
Error	19.7
Cost	576

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

Alternative 5
Error	19.7
Cost	448

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

Alternative 6
Error	19.7
Cost	448

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

Alternative 7
Error	46.4
Cost	320

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

Reproduce

herbie shell --seed 2023099 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im))))

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