math.cube on complex, real part

Average Error: 7.6 → 0.4

Time: 13.3s

Precision: binary64

Cost: 1352

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x.im \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\left(x.im \cdot -3\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq 6 \cdot 10^{+64}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-3 \cdot x.im\right)\right)\\ \end{array} \]

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 -1e+154)
   (* (* x.im -3.0) (* x.re x.im))
   (if (<= x.im 6e+64)
     (-
      (* x.re (- (* x.re x.re) (* x.im x.im)))
      (* x.im (* x.re (+ x.im x.im))))
     (* x.im (* x.re (* -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 <= -1e+154) {
		tmp = (x_46_im * -3.0) * (x_46_re * x_46_im);
	} else if (x_46_im <= 6e+64) {
		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 * (-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 <= (-1d+154)) then
        tmp = (x_46im * (-3.0d0)) * (x_46re * x_46im)
    else if (x_46im <= 6d+64) 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 * ((-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 <= -1e+154) {
		tmp = (x_46_im * -3.0) * (x_46_re * x_46_im);
	} else if (x_46_im <= 6e+64) {
		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 * (-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 <= -1e+154:
		tmp = (x_46_im * -3.0) * (x_46_re * x_46_im)
	elif x_46_im <= 6e+64:
		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 * (-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 <= -1e+154)
		tmp = Float64(Float64(x_46_im * -3.0) * Float64(x_46_re * x_46_im));
	elseif (x_46_im <= 6e+64)
		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(-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 <= -1e+154)
		tmp = (x_46_im * -3.0) * (x_46_re * x_46_im);
	elseif (x_46_im <= 6e+64)
		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 * (-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, -1e+154], N[(N[(x$46$im * -3.0), $MachinePrecision] * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 6e+64], 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[(-3.0 * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	7.6
Target	0.3
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if x.im < -1.00000000000000004e154

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational.json-simplify-2 [=>] 64.0	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational.json-simplify-2 [=>] 64.0	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational.json-simplify-47 [=>] 64.0	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]
      rational.json-simplify-43 [=>] 64.0	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.re \cdot \left(\left(x.im + x.im\right) \cdot x.im\right)} \]
      rational.json-simplify-2 [=>] 64.0	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{\left(\left(x.im + x.im\right) \cdot x.im\right) \cdot x.re} \]
      rational.json-simplify-48 [=>] 64.0	\[ \color{blue}{x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)} \]
      rational.json-simplify-45 [=>] 64.0	\[ x.re \cdot \color{blue}{\left(x.re \cdot x.re - \left(x.im \cdot x.im + \left(x.im + x.im\right) \cdot x.im\right)\right)} \]
      rational.json-simplify-47 [=>] 64.0	\[ x.re \cdot \left(x.re \cdot x.re - \color{blue}{x.im \cdot \left(\left(x.im + x.im\right) + x.im\right)}\right) \]

   3. Taylor expanded in x.re around 0 64.0

      \[\leadsto \color{blue}{-1 \cdot \left(x.re \cdot \left(\left(x.im + 2 \cdot x.im\right) \cdot x.im\right)\right)} \]

   4. Simplified 64.0

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

      [Start] 64.0	\[ -1 \cdot \left(x.re \cdot \left(\left(x.im + 2 \cdot x.im\right) \cdot x.im\right)\right) \]
      rational.json-simplify-43 [<=] 64.0	\[ \color{blue}{\left(\left(x.im + 2 \cdot x.im\right) \cdot x.im\right) \cdot \left(-1 \cdot x.re\right)} \]
      rational.json-simplify-2 [=>] 64.0	\[ \color{blue}{\left(x.im \cdot \left(x.im + 2 \cdot x.im\right)\right)} \cdot \left(-1 \cdot x.re\right) \]
      rational.json-simplify-2 [=>] 64.0	\[ \left(x.im \cdot \left(x.im + \color{blue}{x.im \cdot 2}\right)\right) \cdot \left(-1 \cdot x.re\right) \]
      rational.json-simplify-2 [<=] 64.0	\[ \left(x.im \cdot \left(x.im + x.im \cdot 2\right)\right) \cdot \color{blue}{\left(x.re \cdot -1\right)} \]
      rational.json-simplify-9 [=>] 64.0	\[ \left(x.im \cdot \left(x.im + x.im \cdot 2\right)\right) \cdot \color{blue}{\left(-x.re\right)} \]

   5. Applied egg-rr 0.4

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

   6. Simplified 64.0

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

      [Start] 0.4	\[ x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right) + 0 \]
      rational.json-simplify-4 [=>] 0.4	\[ \color{blue}{x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)} \]
      rational.json-simplify-43 [=>] 64.0	\[ \color{blue}{x.re \cdot \left(\left(x.im \cdot -3\right) \cdot x.im\right)} \]
      rational.json-simplify-2 [<=] 64.0	\[ x.re \cdot \color{blue}{\left(x.im \cdot \left(x.im \cdot -3\right)\right)} \]
      rational.json-simplify-43 [<=] 64.0	\[ x.re \cdot \color{blue}{\left(-3 \cdot \left(x.im \cdot x.im\right)\right)} \]
      rational.json-simplify-43 [=>] 64.0	\[ \color{blue}{-3 \cdot \left(\left(x.im \cdot x.im\right) \cdot x.re\right)} \]
      rational.json-simplify-2 [=>] 64.0	\[ -3 \cdot \color{blue}{\left(x.re \cdot \left(x.im \cdot x.im\right)\right)} \]

   7. Applied egg-rr 64.0

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

   8. Simplified 0.4

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

      [Start] 64.0	\[ x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right) + 0 \]
      rational.json-simplify-4 [=>] 64.0	\[ \color{blue}{x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)} \]
      rational.json-simplify-43 [<=] 0.4	\[ \color{blue}{\left(x.im \cdot -3\right) \cdot \left(x.re \cdot x.im\right)} \]

   if -1.00000000000000004e154 < x.im < 6.0000000000000004e64

   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.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-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.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 6.0000000000000004e64 < x.im

   1. Initial program 26.5

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

   2. Simplified 26.6

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

      [Start] 26.5	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational.json-simplify-2 [=>] 26.5	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational.json-simplify-2 [=>] 26.5	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational.json-simplify-47 [=>] 26.5	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]
      rational.json-simplify-43 [=>] 26.6	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.re \cdot \left(\left(x.im + x.im\right) \cdot x.im\right)} \]
      rational.json-simplify-2 [=>] 26.6	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{\left(\left(x.im + x.im\right) \cdot x.im\right) \cdot x.re} \]
      rational.json-simplify-48 [=>] 26.6	\[ \color{blue}{x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)} \]
      rational.json-simplify-45 [=>] 26.6	\[ x.re \cdot \color{blue}{\left(x.re \cdot x.re - \left(x.im \cdot x.im + \left(x.im + x.im\right) \cdot x.im\right)\right)} \]
      rational.json-simplify-47 [=>] 26.6	\[ x.re \cdot \left(x.re \cdot x.re - \color{blue}{x.im \cdot \left(\left(x.im + x.im\right) + x.im\right)}\right) \]

   3. Taylor expanded in x.re around 0 27.4

      \[\leadsto \color{blue}{-1 \cdot \left(x.re \cdot \left(\left(x.im + 2 \cdot x.im\right) \cdot x.im\right)\right)} \]

   4. Simplified 27.4

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

      [Start] 27.4	\[ -1 \cdot \left(x.re \cdot \left(\left(x.im + 2 \cdot x.im\right) \cdot x.im\right)\right) \]
      rational.json-simplify-43 [<=] 27.4	\[ \color{blue}{\left(\left(x.im + 2 \cdot x.im\right) \cdot x.im\right) \cdot \left(-1 \cdot x.re\right)} \]
      rational.json-simplify-2 [=>] 27.4	\[ \color{blue}{\left(x.im \cdot \left(x.im + 2 \cdot x.im\right)\right)} \cdot \left(-1 \cdot x.re\right) \]
      rational.json-simplify-2 [=>] 27.4	\[ \left(x.im \cdot \left(x.im + \color{blue}{x.im \cdot 2}\right)\right) \cdot \left(-1 \cdot x.re\right) \]
      rational.json-simplify-2 [<=] 27.4	\[ \left(x.im \cdot \left(x.im + x.im \cdot 2\right)\right) \cdot \color{blue}{\left(x.re \cdot -1\right)} \]
      rational.json-simplify-9 [=>] 27.4	\[ \left(x.im \cdot \left(x.im + x.im \cdot 2\right)\right) \cdot \color{blue}{\left(-x.re\right)} \]

   5. Applied egg-rr 1.2

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

   6. Simplified 1.2

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

      [Start] 1.2	\[ x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right) + 0 \]
      rational.json-simplify-4 [=>] 1.2	\[ \color{blue}{x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)} \]
      rational.json-simplify-2 [=>] 1.2	\[ x.im \cdot \left(x.re \cdot \color{blue}{\left(-3 \cdot x.im\right)}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\left(x.im \cdot -3\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq 6 \cdot 10^{+64}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-3 \cdot x.im\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	1216

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

Alternative 2
Error	0.2
Cost	1088

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

Alternative 3
Error	0.4
Cost	968

\[\begin{array}{l} \mathbf{if}\;x.im \leq -1.2 \cdot 10^{+88}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 5.4 \cdot 10^{+64}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - 3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-3 \cdot x.im\right)\right)\\ \end{array} \]

Alternative 4
Error	0.2
Cost	968

\[\begin{array}{l} t_0 := -3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{if}\;x.im \leq -1.1 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 1.05 \cdot 10^{+101}:\\ \;\;\;\;\left(x.re \cdot x.re - x.im \cdot \left(x.im \cdot 3\right)\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	18.8
Cost	448

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

Alternative 6
Error	18.8
Cost	448

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

Reproduce

herbie shell --seed 2023077 
(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)))