math.cube on complex, real part

Average Error: 7.8 → 0.3

Time: 11.1s

Precision: binary64

Cost: 1352

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x.im \leq -2.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)}{1}\\ \mathbf{elif}\;x.im \leq 2.4 \cdot 10^{+141}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \end{array} \]

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 -2.5e+61)
   (/ (* x.im (* x.im (* x.re -3.0))) 1.0)
   (if (<= x.im 2.4e+141)
     (-
      (* x.re (- (* x.re x.re) (* x.im x.im)))
      (* x.im (* x.re (+ x.im x.im))))
     (* x.im (* x.re (* x.im -3.0))))))

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_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 <= -2.5e+61) {
		tmp = (x_46_im * (x_46_im * (x_46_re * -3.0))) / 1.0;
	} else if (x_46_im <= 2.4e+141) {
		tmp = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	} else {
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0));
	}
	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_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 <= (-2.5d+61)) then
        tmp = (x_46im * (x_46im * (x_46re * (-3.0d0)))) / 1.0d0
    else if (x_46im <= 2.4d+141) then
        tmp = (x_46re * ((x_46re * x_46re) - (x_46im * x_46im))) - (x_46im * (x_46re * (x_46im + x_46im)))
    else
        tmp = x_46im * (x_46re * (x_46im * (-3.0d0)))
    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 <= -2.5e+61) {
		tmp = (x_46_im * (x_46_im * (x_46_re * -3.0))) / 1.0;
	} else if (x_46_im <= 2.4e+141) {
		tmp = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	} else {
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0));
	}
	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_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 <= -2.5e+61:
		tmp = (x_46_im * (x_46_im * (x_46_re * -3.0))) / 1.0
	elif x_46_im <= 2.4e+141:
		tmp = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * (x_46_re * (x_46_im + x_46_im)))
	else:
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0))
	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_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 <= -2.5e+61)
		tmp = Float64(Float64(x_46_im * Float64(x_46_im * Float64(x_46_re * -3.0))) / 1.0);
	elseif (x_46_im <= 2.4e+141)
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(x_46_re * Float64(x_46_im + x_46_im))));
	else
		tmp = Float64(x_46_im * Float64(x_46_re * Float64(x_46_im * -3.0)));
	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_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 <= -2.5e+61)
		tmp = (x_46_im * (x_46_im * (x_46_re * -3.0))) / 1.0;
	elseif (x_46_im <= 2.4e+141)
		tmp = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	else
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0));
	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$46$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, -2.5e+61], N[(N[(x$46$im * N[(x$46$im * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[x$46$im, 2.4e+141], N[(N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(x$46$re * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$re * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	7.8
Target	0.2
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if x.im < -2.50000000000000009e61

   1. Initial program 26.1

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

   2. Simplified 26.2

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

      [Start] 26.1	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best.json-simplify-2 [=>] 26.1	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best.json-simplify-2 [=>] 26.1	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational_best.json-simplify-2 [<=] 26.1	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \]
      rational_best.json-simplify-47 [=>] 26.1	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]
      rational_best.json-simplify-44 [=>] 26.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.re \cdot \left(x.im \cdot \left(x.im + x.im\right)\right)} \]
      rational_best.json-simplify-2 [=>] 26.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{\left(x.im \cdot \left(x.im + x.im\right)\right) \cdot x.re} \]
      rational_best.json-simplify-48 [=>] 26.2	\[ \color{blue}{x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im + x.im\right)\right)} \]

   3. Applied egg-rr 26.2

      \[\leadsto \color{blue}{x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot -2 - \left(x.im \cdot x.im - x.re \cdot x.re\right)\right) + 0} \]

   4. Simplified 26.2

      \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -2 - x.im\right)\right)} \]
      Proof

      [Start] 26.2	\[ x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot -2 - \left(x.im \cdot x.im - x.re \cdot x.re\right)\right) + 0 \]
      rational_best.json-simplify-4 [=>] 26.2	\[ \color{blue}{x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot -2 - \left(x.im \cdot x.im - x.re \cdot x.re\right)\right)} \]
      rational_best.json-simplify-46 [=>] 26.2	\[ x.re \cdot \color{blue}{\left(x.re \cdot x.re + \left(\left(x.im \cdot x.im\right) \cdot -2 - x.im \cdot x.im\right)\right)} \]
      rational_best.json-simplify-2 [=>] 26.2	\[ x.re \cdot \left(x.re \cdot x.re + \left(\color{blue}{-2 \cdot \left(x.im \cdot x.im\right)} - x.im \cdot x.im\right)\right) \]
      rational_best.json-simplify-44 [=>] 26.2	\[ x.re \cdot \left(x.re \cdot x.re + \left(\color{blue}{x.im \cdot \left(-2 \cdot x.im\right)} - x.im \cdot x.im\right)\right) \]
      rational_best.json-simplify-48 [=>] 26.2	\[ x.re \cdot \left(x.re \cdot x.re + \color{blue}{x.im \cdot \left(-2 \cdot x.im - x.im\right)}\right) \]
      rational_best.json-simplify-2 [=>] 26.2	\[ x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(\color{blue}{x.im \cdot -2} - x.im\right)\right) \]

   5. Taylor expanded in x.re around 0 1.2

      \[\leadsto \color{blue}{\left(-2 \cdot x.im - x.im\right) \cdot \left(x.re \cdot x.im\right)} \]

   6. Applied egg-rr 1.2

      \[\leadsto \color{blue}{\frac{x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)}{1}} \]

   if -2.50000000000000009e61 < x.im < 2.39999999999999997e141

   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

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

      [Start] 0.2	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best.json-simplify-2 [=>] 0.2	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best.json-simplify-2 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational_best.json-simplify-2 [<=] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \]
      rational_best.json-simplify-47 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   if 2.39999999999999997e141 < x.im

   1. Initial program 55.7

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

   2. Simplified 55.8

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

      [Start] 55.7	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best.json-simplify-2 [=>] 55.7	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best.json-simplify-2 [=>] 55.7	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational_best.json-simplify-2 [<=] 55.7	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \]
      rational_best.json-simplify-47 [=>] 55.7	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]
      rational_best.json-simplify-44 [=>] 55.8	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.re \cdot \left(x.im \cdot \left(x.im + x.im\right)\right)} \]
      rational_best.json-simplify-2 [=>] 55.8	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{\left(x.im \cdot \left(x.im + x.im\right)\right) \cdot x.re} \]
      rational_best.json-simplify-48 [=>] 55.8	\[ \color{blue}{x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im + x.im\right)\right)} \]

   3. Applied egg-rr 55.8

      \[\leadsto \color{blue}{x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot -2 - \left(x.im \cdot x.im - x.re \cdot x.re\right)\right) + 0} \]

   4. Simplified 55.8

      \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -2 - x.im\right)\right)} \]
      Proof

      [Start] 55.8	\[ x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot -2 - \left(x.im \cdot x.im - x.re \cdot x.re\right)\right) + 0 \]
      rational_best.json-simplify-4 [=>] 55.8	\[ \color{blue}{x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot -2 - \left(x.im \cdot x.im - x.re \cdot x.re\right)\right)} \]
      rational_best.json-simplify-46 [=>] 55.8	\[ x.re \cdot \color{blue}{\left(x.re \cdot x.re + \left(\left(x.im \cdot x.im\right) \cdot -2 - x.im \cdot x.im\right)\right)} \]
      rational_best.json-simplify-2 [=>] 55.8	\[ x.re \cdot \left(x.re \cdot x.re + \left(\color{blue}{-2 \cdot \left(x.im \cdot x.im\right)} - x.im \cdot x.im\right)\right) \]
      rational_best.json-simplify-44 [=>] 55.8	\[ x.re \cdot \left(x.re \cdot x.re + \left(\color{blue}{x.im \cdot \left(-2 \cdot x.im\right)} - x.im \cdot x.im\right)\right) \]
      rational_best.json-simplify-48 [=>] 55.8	\[ x.re \cdot \left(x.re \cdot x.re + \color{blue}{x.im \cdot \left(-2 \cdot x.im - x.im\right)}\right) \]
      rational_best.json-simplify-2 [=>] 55.8	\[ x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(\color{blue}{x.im \cdot -2} - x.im\right)\right) \]

   5. Taylor expanded in x.re around 0 0.4

      \[\leadsto \color{blue}{\left(-2 \cdot x.im - x.im\right) \cdot \left(x.re \cdot x.im\right)} \]

   6. Applied egg-rr 0.3

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

   7. Simplified 0.3

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

      [Start] 0.3	\[ x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right) + 0 \]
      rational_best.json-simplify-4 [=>] 0.3	\[ \color{blue}{x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)} \]
      rational_best.json-simplify-44 [=>] 0.3	\[ x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im \cdot -3\right)\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -2.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)}{1}\\ \mathbf{elif}\;x.im \leq 2.4 \cdot 10^{+141}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	1152

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

Alternative 2
Error	0.8
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x.im \leq -6.5 \cdot 10^{+79}:\\ \;\;\;\;\frac{x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)}{1}\\ \mathbf{elif}\;x.im \leq 5.2 \cdot 10^{+26}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -2 - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 3
Error	0.9
Cost	968

\[\begin{array}{l} \mathbf{if}\;x.im \leq -2 \cdot 10^{+61}:\\ \;\;\;\;\frac{x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)}{1}\\ \mathbf{elif}\;x.im \leq 5.2 \cdot 10^{+26}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + -3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 4
Error	19.0
Cost	448

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

Reproduce

herbie shell --seed 2023090 
(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)))