math.cube on complex, real part

Average Error: 7.9 → 0.7

Time: 13.4s

Precision: binary64

Cost: 1352

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x.im \leq -7.5 \cdot 10^{+153}:\\ \;\;\;\;\left(x.im \cdot \left(x.re \cdot -3\right)\right) \cdot x.im\\ \mathbf{elif}\;x.im \leq 5 \cdot 10^{+29}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - x.re \cdot \left(\left(x.im + x.im\right) \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-3 \cdot \left(x.re \cdot x.im\right)\right) \cdot x.im\\ \end{array} \]

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 -7.5e+153)
   (* (* x.im (* x.re -3.0)) x.im)
   (if (<= x.im 5e+29)
     (-
      (* (+ x.re x.im) (* (- x.re x.im) x.re))
      (* x.re (* (+ x.im x.im) x.im)))
     (* (* -3.0 (* x.re x.im)) 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 <= -7.5e+153) {
		tmp = (x_46_im * (x_46_re * -3.0)) * x_46_im;
	} else if (x_46_im <= 5e+29) {
		tmp = ((x_46_re + x_46_im) * ((x_46_re - x_46_im) * x_46_re)) - (x_46_re * ((x_46_im + x_46_im) * x_46_im));
	} else {
		tmp = (-3.0 * (x_46_re * x_46_im)) * 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 <= (-7.5d+153)) then
        tmp = (x_46im * (x_46re * (-3.0d0))) * x_46im
    else if (x_46im <= 5d+29) then
        tmp = ((x_46re + x_46im) * ((x_46re - x_46im) * x_46re)) - (x_46re * ((x_46im + x_46im) * x_46im))
    else
        tmp = ((-3.0d0) * (x_46re * x_46im)) * 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 <= -7.5e+153) {
		tmp = (x_46_im * (x_46_re * -3.0)) * x_46_im;
	} else if (x_46_im <= 5e+29) {
		tmp = ((x_46_re + x_46_im) * ((x_46_re - x_46_im) * x_46_re)) - (x_46_re * ((x_46_im + x_46_im) * x_46_im));
	} else {
		tmp = (-3.0 * (x_46_re * x_46_im)) * 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 <= -7.5e+153:
		tmp = (x_46_im * (x_46_re * -3.0)) * x_46_im
	elif x_46_im <= 5e+29:
		tmp = ((x_46_re + x_46_im) * ((x_46_re - x_46_im) * x_46_re)) - (x_46_re * ((x_46_im + x_46_im) * x_46_im))
	else:
		tmp = (-3.0 * (x_46_re * x_46_im)) * 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 <= -7.5e+153)
		tmp = Float64(Float64(x_46_im * Float64(x_46_re * -3.0)) * x_46_im);
	elseif (x_46_im <= 5e+29)
		tmp = Float64(Float64(Float64(x_46_re + x_46_im) * Float64(Float64(x_46_re - x_46_im) * x_46_re)) - Float64(x_46_re * Float64(Float64(x_46_im + x_46_im) * x_46_im)));
	else
		tmp = Float64(Float64(-3.0 * Float64(x_46_re * x_46_im)) * 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 <= -7.5e+153)
		tmp = (x_46_im * (x_46_re * -3.0)) * x_46_im;
	elseif (x_46_im <= 5e+29)
		tmp = ((x_46_re + x_46_im) * ((x_46_re - x_46_im) * x_46_re)) - (x_46_re * ((x_46_im + x_46_im) * x_46_im));
	else
		tmp = (-3.0 * (x_46_re * x_46_im)) * 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, -7.5e+153], N[(N[(x$46$im * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision], If[LessEqual[x$46$im, 5e+29], N[(N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(N[(x$46$re - x$46$im), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] - N[(x$46$re * N[(N[(x$46$im + x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-3.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]]]

Target

Original	7.9
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 < -7.50000000000000065e153

   1. Initial program 63.7

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

   2. Simplified 64.0

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

      [Start] 63.7	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational.json-simplify-2 [=>] 63.7	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational.json-simplify-34 [=>] 63.7	\[ x.re \cdot \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational.json-simplify-43 [=>] 0.4	\[ \color{blue}{\left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational.json-simplify-1 [=>] 0.4	\[ \color{blue}{\left(x.re + x.im\right)} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational.json-simplify-2 [=>] 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.json-simplify-51 [=>] 0.4	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]
      rational.json-simplify-43 [=>] 64.0	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \color{blue}{x.re \cdot \left(\left(x.im + x.im\right) \cdot x.im\right)} \]

   3. Taylor expanded in x.re around 0 64.0

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

   4. Simplified 64.0

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

      [Start] 64.0	\[ \left(x.re + x.im\right) \cdot \left(-1 \cdot \left(x.re \cdot x.im\right)\right) - x.re \cdot \left(\left(x.im + x.im\right) \cdot x.im\right) \]
      rational.json-simplify-43 [=>] 64.0	\[ \left(x.re + x.im\right) \cdot \color{blue}{\left(x.re \cdot \left(x.im \cdot -1\right)\right)} - x.re \cdot \left(\left(x.im + x.im\right) \cdot x.im\right) \]
      rational.json-simplify-9 [=>] 64.0	\[ \left(x.re + x.im\right) \cdot \left(x.re \cdot \color{blue}{\left(-x.im\right)}\right) - x.re \cdot \left(\left(x.im + x.im\right) \cdot x.im\right) \]

   5. Applied egg-rr 0.4

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

   6. Taylor expanded in x.re around 0 0.4

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

   7. Simplified 0.4

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

      [Start] 0.4	\[ \left(\left(-2 \cdot x.im - x.im\right) \cdot x.re\right) \cdot x.im \]
      rational.json-simplify-2 [=>] 0.4	\[ \color{blue}{\left(x.re \cdot \left(-2 \cdot x.im - x.im\right)\right)} \cdot x.im \]
      rational.json-simplify-2 [<=] 0.4	\[ \left(x.re \cdot \left(\color{blue}{x.im \cdot -2} - x.im\right)\right) \cdot x.im \]
      rational.json-simplify-6 [<=] 0.4	\[ \left(x.re \cdot \left(x.im \cdot -2 - \color{blue}{1 \cdot x.im}\right)\right) \cdot x.im \]
      rational.json-simplify-52 [=>] 0.4	\[ \left(x.re \cdot \color{blue}{\left(x.im \cdot \left(-2 - 1\right)\right)}\right) \cdot x.im \]
      metadata-eval [=>] 0.4	\[ \left(x.re \cdot \left(x.im \cdot \color{blue}{-3}\right)\right) \cdot x.im \]
      rational.json-simplify-43 [=>] 0.4	\[ \color{blue}{\left(x.im \cdot \left(-3 \cdot x.re\right)\right)} \cdot x.im \]
      rational.json-simplify-2 [<=] 0.4	\[ \left(x.im \cdot \color{blue}{\left(x.re \cdot -3\right)}\right) \cdot x.im \]

   if -7.50000000000000065e153 < x.im < 5.0000000000000001e29

   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

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

   if 5.0000000000000001e29 < x.im

   1. Initial program 21.9

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

   2. Simplified 21.9

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

      [Start] 21.9	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational.json-simplify-2 [=>] 21.9	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational.json-simplify-34 [=>] 21.9	\[ x.re \cdot \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational.json-simplify-43 [=>] 0.3	\[ \color{blue}{\left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational.json-simplify-1 [=>] 0.3	\[ \color{blue}{\left(x.re + x.im\right)} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational.json-simplify-2 [=>] 0.3	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational.json-simplify-51 [=>] 0.3	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]
      rational.json-simplify-43 [=>] 21.9	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \color{blue}{x.re \cdot \left(\left(x.im + x.im\right) \cdot x.im\right)} \]

   3. Taylor expanded in x.re around 0 24.5

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

   4. Simplified 24.5

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

      [Start] 24.5	\[ \left(x.re + x.im\right) \cdot \left(-1 \cdot \left(x.re \cdot x.im\right)\right) - x.re \cdot \left(\left(x.im + x.im\right) \cdot x.im\right) \]
      rational.json-simplify-43 [=>] 24.5	\[ \left(x.re + x.im\right) \cdot \color{blue}{\left(x.re \cdot \left(x.im \cdot -1\right)\right)} - x.re \cdot \left(\left(x.im + x.im\right) \cdot x.im\right) \]
      rational.json-simplify-9 [=>] 24.5	\[ \left(x.re + x.im\right) \cdot \left(x.re \cdot \color{blue}{\left(-x.im\right)}\right) - x.re \cdot \left(\left(x.im + x.im\right) \cdot x.im\right) \]

   5. Applied egg-rr 3.0

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

   6. Taylor expanded in x.im around inf 2.8

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -7.5 \cdot 10^{+153}:\\ \;\;\;\;\left(x.im \cdot \left(x.re \cdot -3\right)\right) \cdot x.im\\ \mathbf{elif}\;x.im \leq 5 \cdot 10^{+29}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - x.re \cdot \left(\left(x.im + x.im\right) \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-3 \cdot \left(x.re \cdot x.im\right)\right) \cdot x.im\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	1352

\[\begin{array}{l} \mathbf{if}\;x.im \leq -1.9 \cdot 10^{+153}:\\ \;\;\;\;\left(x.im \cdot \left(x.re \cdot -3\right)\right) \cdot x.im\\ \mathbf{elif}\;x.im \leq 6.2 \cdot 10^{+30}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-3 \cdot \left(x.re \cdot x.im\right)\right) \cdot x.im\\ \end{array} \]

Alternative 2
Error	0.2
Cost	1344

\[\frac{\left(x.re - x.im\right) \cdot \left(\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot 4\right)}{4} - \frac{x.im}{\frac{\frac{0.5}{x.im}}{x.re}} \]

Alternative 3
Error	0.2
Cost	1088

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

Alternative 4
Error	0.3
Cost	968

\[\begin{array}{l} \mathbf{if}\;x.im \leq -6.6 \cdot 10^{+139}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 2.05 \cdot 10^{+93}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - 3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot -3\right) \cdot \left(x.im \cdot x.re\right)\\ \end{array} \]

Alternative 5
Error	19.0
Cost	448

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

Alternative 6
Error	46.5
Cost	384

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

Reproduce

herbie shell --seed 2023074 
(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)))