math.cube on complex, imaginary part

Average Error: 7.6 → 0.2

Time: 18.3s

Precision: binary64

Cost: 1096

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x.re \leq -4.2 \cdot 10^{+117}:\\ \;\;\;\;\left(x.re \cdot 3\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.re \leq 4.8 \cdot 10^{+145}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right) - x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

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 -4.2e+117)
   (* (* x.re 3.0) (* x.re x.im))
   (if (<= x.re 4.8e+145)
     (- (* x.im (* (* x.re x.re) 3.0)) (* x.im (* x.im x.im)))
     (* x.re (* 3.0 (* x.re 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_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 <= -4.2e+117) {
		tmp = (x_46_re * 3.0) * (x_46_re * x_46_im);
	} else if (x_46_re <= 4.8e+145) {
		tmp = (x_46_im * ((x_46_re * x_46_re) * 3.0)) - (x_46_im * (x_46_im * x_46_im));
	} else {
		tmp = x_46_re * (3.0 * (x_46_re * 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_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 <= (-4.2d+117)) then
        tmp = (x_46re * 3.0d0) * (x_46re * x_46im)
    else if (x_46re <= 4.8d+145) then
        tmp = (x_46im * ((x_46re * x_46re) * 3.0d0)) - (x_46im * (x_46im * x_46im))
    else
        tmp = x_46re * (3.0d0 * (x_46re * 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_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 <= -4.2e+117) {
		tmp = (x_46_re * 3.0) * (x_46_re * x_46_im);
	} else if (x_46_re <= 4.8e+145) {
		tmp = (x_46_im * ((x_46_re * x_46_re) * 3.0)) - (x_46_im * (x_46_im * x_46_im));
	} else {
		tmp = x_46_re * (3.0 * (x_46_re * 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_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 <= -4.2e+117:
		tmp = (x_46_re * 3.0) * (x_46_re * x_46_im)
	elif x_46_re <= 4.8e+145:
		tmp = (x_46_im * ((x_46_re * x_46_re) * 3.0)) - (x_46_im * (x_46_im * x_46_im))
	else:
		tmp = x_46_re * (3.0 * (x_46_re * 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_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 <= -4.2e+117)
		tmp = Float64(Float64(x_46_re * 3.0) * Float64(x_46_re * x_46_im));
	elseif (x_46_re <= 4.8e+145)
		tmp = Float64(Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) * 3.0)) - Float64(x_46_im * Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(x_46_re * Float64(3.0 * Float64(x_46_re * 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_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 <= -4.2e+117)
		tmp = (x_46_re * 3.0) * (x_46_re * x_46_im);
	elseif (x_46_re <= 4.8e+145)
		tmp = (x_46_im * ((x_46_re * x_46_re) * 3.0)) - (x_46_im * (x_46_im * x_46_im));
	else
		tmp = x_46_re * (3.0 * (x_46_re * 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$46$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, -4.2e+117], N[(N[(x$46$re * 3.0), $MachinePrecision] * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 4.8e+145], N[(N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(3.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	7.6
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 < -4.2000000000000002e117

   1. Initial program 44.0

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

   2. Simplified 44.2

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

      [Start] 44.0	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational.json-simplify-39 [=>] 44.0	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.re \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational.json-simplify-35 [=>] 44.0	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.re \cdot \color{blue}{\left(x.im \cdot \left(x.re + x.re\right)\right)} \]
      rational.json-simplify-3 [=>] 44.1	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re + x.re\right)\right)} \]
      rational.json-simplify-35 [<=] 44.1	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \color{blue}{\left(x.re \cdot x.re + x.re \cdot x.re\right)} \]
      rational.json-simplify-35 [=>] 44.2	\[ \color{blue}{x.im \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.re \cdot x.re + x.re \cdot x.re\right)\right)} \]
      rational.json-simplify-41 [=>] 44.2	\[ x.im \cdot \color{blue}{\left(\left(x.re \cdot x.re + x.re \cdot x.re\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]
      rational.json-simplify-14 [=>] 44.2	\[ x.im \cdot \color{blue}{\left(\left(\left(x.re \cdot x.re + x.re \cdot x.re\right) + x.re \cdot x.re\right) - x.im \cdot x.im\right)} \]
      rational.json-simplify-41 [=>] 44.2	\[ x.im \cdot \left(\color{blue}{\left(x.re \cdot x.re + \left(x.re \cdot x.re + x.re \cdot x.re\right)\right)} - x.im \cdot x.im\right) \]
      rational.json-simplify-35 [=>] 44.2	\[ x.im \cdot \left(\left(x.re \cdot x.re + \color{blue}{x.re \cdot \left(x.re + x.re\right)}\right) - x.im \cdot x.im\right) \]
      rational.json-simplify-35 [=>] 44.2	\[ x.im \cdot \left(\color{blue}{x.re \cdot \left(x.re + \left(x.re + x.re\right)\right)} - x.im \cdot x.im\right) \]

   3. Taylor expanded in x.im around 0 44.2

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

   4. Applied egg-rr 0.4

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

   5. Simplified 0.4

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

      [Start] 0.4	\[ x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right) + 0 \]
      rational.json-simplify-65 [=>] 0.4	\[ \color{blue}{x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)} \]
      rational.json-simplify-39 [=>] 0.4	\[ x.re \cdot \color{blue}{\left(\left(x.re \cdot 3\right) \cdot x.im\right)} \]
      rational.json-simplify-3 [=>] 0.4	\[ \color{blue}{\left(x.re \cdot 3\right) \cdot \left(x.re \cdot x.im\right)} \]

   if -4.2000000000000002e117 < x.re < 4.79999999999999984e145

   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

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

      [Start] 0.2	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational.json-simplify-39 [=>] 0.2	\[ \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational.json-simplify-39 [=>] 0.2	\[ x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \color{blue}{x.re \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational.json-simplify-39 [=>] 0.2	\[ x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(\color{blue}{x.im \cdot x.re} + x.im \cdot x.re\right) \]
      rational.json-simplify-39 [<=] 0.2	\[ x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + \color{blue}{x.re \cdot x.im}\right) \]
      rational.json-simplify-35 [=>] 0.2	\[ x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   3. Applied egg-rr 0.2

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

   4. Simplified 0.2

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

      [Start] 0.2	\[ \left(\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot 3\right) - x.im \cdot \left(x.im \cdot x.im\right)\right) - 0 \]
      rational.json-simplify-64 [=>] 0.2	\[ \color{blue}{\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot 3\right) - x.im \cdot \left(x.im \cdot x.im\right)} \]
      rational.json-simplify-3 [=>] 0.2	\[ \color{blue}{x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right)} - x.im \cdot \left(x.im \cdot x.im\right) \]

   if 4.79999999999999984e145 < x.re

   1. Initial program 57.3

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

   2. Simplified 57.5

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

      [Start] 57.3	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational.json-simplify-39 [=>] 57.3	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.re \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational.json-simplify-35 [=>] 57.3	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.re \cdot \color{blue}{\left(x.im \cdot \left(x.re + x.re\right)\right)} \]
      rational.json-simplify-3 [=>] 57.4	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re + x.re\right)\right)} \]
      rational.json-simplify-35 [<=] 57.4	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \color{blue}{\left(x.re \cdot x.re + x.re \cdot x.re\right)} \]
      rational.json-simplify-35 [=>] 57.5	\[ \color{blue}{x.im \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.re \cdot x.re + x.re \cdot x.re\right)\right)} \]
      rational.json-simplify-41 [=>] 57.5	\[ x.im \cdot \color{blue}{\left(\left(x.re \cdot x.re + x.re \cdot x.re\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]
      rational.json-simplify-14 [=>] 57.5	\[ x.im \cdot \color{blue}{\left(\left(\left(x.re \cdot x.re + x.re \cdot x.re\right) + x.re \cdot x.re\right) - x.im \cdot x.im\right)} \]
      rational.json-simplify-41 [=>] 57.5	\[ x.im \cdot \left(\color{blue}{\left(x.re \cdot x.re + \left(x.re \cdot x.re + x.re \cdot x.re\right)\right)} - x.im \cdot x.im\right) \]
      rational.json-simplify-35 [=>] 57.5	\[ x.im \cdot \left(\left(x.re \cdot x.re + \color{blue}{x.re \cdot \left(x.re + x.re\right)}\right) - x.im \cdot x.im\right) \]
      rational.json-simplify-35 [=>] 57.5	\[ x.im \cdot \left(\color{blue}{x.re \cdot \left(x.re + \left(x.re + x.re\right)\right)} - x.im \cdot x.im\right) \]

   3. Taylor expanded in x.im around 0 57.5

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

   4. Applied egg-rr 0.4

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

   5. Applied egg-rr 0.4

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

   6. Simplified 0.4

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

      [Start] 0.4	\[ 3 \cdot \left(x.re \cdot \left(x.im \cdot x.re\right)\right) + 0 \]
      rational.json-simplify-65 [=>] 0.4	\[ \color{blue}{3 \cdot \left(x.re \cdot \left(x.im \cdot x.re\right)\right)} \]
      rational.json-simplify-3 [=>] 0.4	\[ \color{blue}{x.re \cdot \left(3 \cdot \left(x.im \cdot x.re\right)\right)} \]
      rational.json-simplify-40 [=>] 0.4	\[ x.re \cdot \color{blue}{\left(\left(x.im \cdot x.re\right) \cdot 3\right)} \]
      rational.json-simplify-39 [=>] 0.4	\[ x.re \cdot \left(\color{blue}{\left(x.re \cdot x.im\right)} \cdot 3\right) \]
      rational.json-simplify-39 [<=] 0.4	\[ x.re \cdot \color{blue}{\left(3 \cdot \left(x.re \cdot x.im\right)\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -4.2 \cdot 10^{+117}:\\ \;\;\;\;\left(x.re \cdot 3\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.re \leq 4.8 \cdot 10^{+145}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right) - x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	1216

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

Alternative 2
Error	0.2
Cost	968

\[\begin{array}{l} \mathbf{if}\;x.re \leq -5 \cdot 10^{+117}:\\ \;\;\;\;\left(x.re \cdot 3\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.re \leq 7 \cdot 10^{+145}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right) - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 3
Error	0.2
Cost	896

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

Alternative 4
Error	19.1
Cost	448

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

Alternative 5
Error	19.1
Cost	448

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

Reproduce

herbie shell --seed 2023066 
(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)))