math.cube on complex, imaginary part

Average Error: 7.7 → 0.2

Time: 12.9s

Precision: binary64

Cost: 1224

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x.re \leq -2.4 \cdot 10^{+151}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 2.5 \cdot 10^{+116}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot \left(3 \cdot x.im\right)\right) \cdot x.re\\ \end{array} \]

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
 (if (<= x.re -2.4e+151)
   (* 3.0 (* x.re (* x.re x.im)))
   (if (<= x.re 2.5e+116)
     (* x.im (+ (- (* x.re x.re) (* x.im x.im)) (* x.re (+ x.re x.re))))
     (* (* x.re (* 3.0 x.im)) x.re))))

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 tmp;
	if (x_46_re <= -2.4e+151) {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	} else if (x_46_re <= 2.5e+116) {
		tmp = x_46_im * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) + (x_46_re * (x_46_re + x_46_re)));
	} else {
		tmp = (x_46_re * (3.0 * x_46_im)) * x_46_re;
	}
	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) :: tmp
    if (x_46re <= (-2.4d+151)) then
        tmp = 3.0d0 * (x_46re * (x_46re * x_46im))
    else if (x_46re <= 2.5d+116) then
        tmp = x_46im * (((x_46re * x_46re) - (x_46im * x_46im)) + (x_46re * (x_46re + x_46re)))
    else
        tmp = (x_46re * (3.0d0 * x_46im)) * x_46re
    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 tmp;
	if (x_46_re <= -2.4e+151) {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	} else if (x_46_re <= 2.5e+116) {
		tmp = x_46_im * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) + (x_46_re * (x_46_re + x_46_re)));
	} else {
		tmp = (x_46_re * (3.0 * x_46_im)) * x_46_re;
	}
	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):
	tmp = 0
	if x_46_re <= -2.4e+151:
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im))
	elif x_46_re <= 2.5e+116:
		tmp = x_46_im * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) + (x_46_re * (x_46_re + x_46_re)))
	else:
		tmp = (x_46_re * (3.0 * x_46_im)) * x_46_re
	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)
	tmp = 0.0
	if (x_46_re <= -2.4e+151)
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_re * x_46_im)));
	elseif (x_46_re <= 2.5e+116)
		tmp = Float64(x_46_im * Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)) + Float64(x_46_re * Float64(x_46_re + x_46_re))));
	else
		tmp = Float64(Float64(x_46_re * Float64(3.0 * x_46_im)) * x_46_re);
	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)
	tmp = 0.0;
	if (x_46_re <= -2.4e+151)
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	elseif (x_46_re <= 2.5e+116)
		tmp = x_46_im * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) + (x_46_re * (x_46_re + x_46_re)));
	else
		tmp = (x_46_re * (3.0 * x_46_im)) * x_46_re;
	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_] := If[LessEqual[x$46$re, -2.4e+151], N[(3.0 * N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 2.5e+116], N[(x$46$im * N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(x$46$re + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * N[(3.0 * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]]]

Target

Original	7.7
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 3 regimes

2. if x.re < -2.4000000000000001e151

   1. Initial program 61.6

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

   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] 61.6	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational.json-simplify-2 [=>] 61.6	\[ \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational.json-simplify-34 [=>] 61.6	\[ x.im \cdot \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      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 61.7

      \[\leadsto \color{blue}{\left(x.im + x.im\right) \cdot \left(x.re \cdot x.re + \left(x.re + x.im\right) \cdot \left(x.re \cdot 0.5\right)\right)} \]

   5. Simplified 0.4

      \[\leadsto \color{blue}{\left(x.re + \left(x.re + x.im\right) \cdot 0.5\right) \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]
      Proof

      [Start] 61.7	\[ \left(x.im + x.im\right) \cdot \left(x.re \cdot x.re + \left(x.re + x.im\right) \cdot \left(x.re \cdot 0.5\right)\right) \]
      rational.json-simplify-1 [=>] 61.7	\[ \left(x.im + x.im\right) \cdot \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re \cdot 0.5\right) + x.re \cdot x.re\right)} \]
      rational.json-simplify-43 [=>] 61.7	\[ \left(x.im + x.im\right) \cdot \left(\color{blue}{x.re \cdot \left(0.5 \cdot \left(x.re + x.im\right)\right)} + x.re \cdot x.re\right) \]
      rational.json-simplify-2 [<=] 61.7	\[ \left(x.im + x.im\right) \cdot \left(x.re \cdot \color{blue}{\left(\left(x.re + x.im\right) \cdot 0.5\right)} + x.re \cdot x.re\right) \]
      rational.json-simplify-51 [=>] 61.7	\[ \left(x.im + x.im\right) \cdot \color{blue}{\left(x.re \cdot \left(x.re + \left(x.re + x.im\right) \cdot 0.5\right)\right)} \]
      rational.json-simplify-43 [<=] 0.4	\[ \color{blue}{\left(x.re + \left(x.re + x.im\right) \cdot 0.5\right) \cdot \left(\left(x.im + x.im\right) \cdot x.re\right)} \]
      rational.json-simplify-2 [<=] 0.4	\[ \left(x.re + \left(x.re + x.im\right) \cdot 0.5\right) \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   6. Taylor expanded in x.im around 0 0.4

      \[\leadsto \color{blue}{2 \cdot \left(x.re \cdot \left(\left(0.5 \cdot x.re + x.re\right) \cdot x.im\right)\right)} \]

   7. Simplified 0.4

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

      [Start] 0.4	\[ 2 \cdot \left(x.re \cdot \left(\left(0.5 \cdot x.re + x.re\right) \cdot x.im\right)\right) \]
      rational.json-simplify-43 [<=] 0.4	\[ \color{blue}{\left(\left(0.5 \cdot x.re + x.re\right) \cdot x.im\right) \cdot \left(2 \cdot x.re\right)} \]
      metadata-eval [<=] 0.4	\[ \left(\left(0.5 \cdot x.re + x.re\right) \cdot x.im\right) \cdot \left(\color{blue}{\left(1 + 1\right)} \cdot x.re\right) \]
      rational.json-simplify-7 [<=] 0.4	\[ \left(\left(0.5 \cdot x.re + x.re\right) \cdot x.im\right) \cdot \left(\left(1 + 1\right) \cdot \color{blue}{\frac{x.re}{1}}\right) \]
      rational.json-simplify-30 [=>] 0.4	\[ \left(\left(0.5 \cdot x.re + x.re\right) \cdot x.im\right) \cdot \color{blue}{\left(x.re + \frac{x.re}{1}\right)} \]
      rational.json-simplify-7 [=>] 0.4	\[ \left(\left(0.5 \cdot x.re + x.re\right) \cdot x.im\right) \cdot \left(x.re + \color{blue}{x.re}\right) \]
      rational.json-simplify-2 [=>] 0.4	\[ \color{blue}{\left(x.re + x.re\right) \cdot \left(\left(0.5 \cdot x.re + x.re\right) \cdot x.im\right)} \]
      rational.json-simplify-53 [=>] 0.4	\[ \color{blue}{\left(\left(0.5 \cdot x.re + x.re\right) + \left(0.5 \cdot x.re + x.re\right)\right) \cdot \left(x.im \cdot x.re\right)} \]

   8. Applied egg-rr 0.4

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

   9. Simplified 0.4

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

      [Start] 0.4	\[ 3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right) + 0 \]
      rational.json-simplify-4 [=>] 0.4	\[ \color{blue}{3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)} \]

   if -2.4000000000000001e151 < x.re < 2.50000000000000013e116

   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

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

   if 2.50000000000000013e116 < x.re

   1. Initial program 44.5

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

   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] 44.5	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational.json-simplify-2 [=>] 44.5	\[ \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational.json-simplify-34 [=>] 44.5	\[ x.im \cdot \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      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

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

   5. Taylor expanded in x.im around 0 0.4

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

   6. Simplified 0.4

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

      [Start] 0.4	\[ \left(3 \cdot \left(x.re \cdot x.im\right)\right) \cdot x.re \]
      rational.json-simplify-43 [=>] 0.4	\[ \color{blue}{\left(x.re \cdot \left(x.im \cdot 3\right)\right)} \cdot x.re \]
      rational.json-simplify-2 [=>] 0.4	\[ \left(x.re \cdot \color{blue}{\left(3 \cdot x.im\right)}\right) \cdot x.re \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2.4 \cdot 10^{+151}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 2.5 \cdot 10^{+116}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot \left(3 \cdot x.im\right)\right) \cdot x.re\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	1088

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

Alternative 2
Error	18.8
Cost	576

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

Alternative 3
Error	18.8
Cost	576

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

Alternative 4
Error	26.3
Cost	448

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

Alternative 5
Error	18.9
Cost	448

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

Alternative 6
Error	18.9
Cost	448

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

Alternative 7
Error	46.0
Cost	320

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

Alternative 8
Error	46.0
Cost	320

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

Reproduce

herbie shell --seed 2023065 
(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)))