math.cube on complex, imaginary part

Average Error: 7.0 → 0.3

Time: 7.9s

Precision: binary64

Cost: 1480

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x.re \leq -2.1902738719736264 \cdot 10^{+116}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{elif}\;x.re \leq 4.952149493142814 \cdot 10^{+77}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) + x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]

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 -2.1902738719736264e+116)
   (* (* x.re x.im) (* x.re 3.0))
   (if (<= x.re 4.952149493142814e+77)
     (+
      (* x.re (+ (* x.re x.im) (* x.re x.im)))
      (* x.im (- (* x.re x.re) (* x.im x.im))))
     (* x.re (* x.im (* x.re 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_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 <= -2.1902738719736264e+116) {
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
	} else if (x_46_re <= 4.952149493142814e+77) {
		tmp = (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))) + (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im)));
	} else {
		tmp = x_46_re * (x_46_im * (x_46_re * 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_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 <= (-2.1902738719736264d+116)) then
        tmp = (x_46re * x_46im) * (x_46re * 3.0d0)
    else if (x_46re <= 4.952149493142814d+77) then
        tmp = (x_46re * ((x_46re * x_46im) + (x_46re * x_46im))) + (x_46im * ((x_46re * x_46re) - (x_46im * x_46im)))
    else
        tmp = x_46re * (x_46im * (x_46re * 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_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 <= -2.1902738719736264e+116) {
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
	} else if (x_46_re <= 4.952149493142814e+77) {
		tmp = (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))) + (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im)));
	} else {
		tmp = x_46_re * (x_46_im * (x_46_re * 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_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 <= -2.1902738719736264e+116:
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0)
	elif x_46_re <= 4.952149493142814e+77:
		tmp = (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))) + (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im)))
	else:
		tmp = x_46_re * (x_46_im * (x_46_re * 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_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 <= -2.1902738719736264e+116)
		tmp = Float64(Float64(x_46_re * x_46_im) * Float64(x_46_re * 3.0));
	elseif (x_46_re <= 4.952149493142814e+77)
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im))) + Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))));
	else
		tmp = Float64(x_46_re * Float64(x_46_im * Float64(x_46_re * 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_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 <= -2.1902738719736264e+116)
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
	elseif (x_46_re <= 4.952149493142814e+77)
		tmp = (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))) + (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im)));
	else
		tmp = x_46_re * (x_46_im * (x_46_re * 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$46$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, -2.1902738719736264e+116], N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 4.952149493142814e+77], N[(N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$im * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	7.0
Target	0.3
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if x.re < -2.19027387197362642e116

   1. Initial program 38.7

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

   2. Taylor expanded in x.re around inf 38.7

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

   3. Applied egg-rr 0.4

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

   4. Applied egg-rr 0.4

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

   if -2.19027387197362642e116 < x.re < 4.9521494931428136e77

   1. Initial program 0.2

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

   if 4.9521494931428136e77 < x.re

   1. Initial program 28.4

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

   2. Taylor expanded in x.re around 0 28.5

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

   3. Simplified 28.5

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right)\right)} \]
      Proof
      (*.f64 x.im (*.f64 x.re (*.f64 x.re 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 x.im (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x.re x.re) 2))): 2 points increase in error, 0 points decrease in error
      (*.f64 x.im (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 x.re 2)) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 x.im (Rewrite<= *-commutative_binary64 (*.f64 2 (pow.f64 x.re 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 (pow.f64 x.re 2)) x.im)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 (pow.f64 x.re 2) x.im))): 0 points increase in error, 2 points decrease in error

   4. Applied egg-rr 0.3

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

   5. Taylor expanded in x.im around 0 29.3

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

   6. Simplified 1.1

      \[\leadsto \color{blue}{x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)} \]
      Proof
      (*.f64 x.re (*.f64 x.im (*.f64 x.re 3))): 0 points increase in error, 0 points decrease in error
      (*.f64 x.re (*.f64 x.im (Rewrite<= *-commutative_binary64 (*.f64 3 x.re)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 x.re x.im) (*.f64 3 x.re))): 25 points increase in error, 22 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 3 x.re) (*.f64 x.re x.im))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 3 x.re) x.re) x.im)): 51 points increase in error, 22 points decrease in error
      (*.f64 (Rewrite<= associate-*r*_binary64 (*.f64 3 (*.f64 x.re x.re))) x.im): 19 points increase in error, 15 points decrease in error
      (*.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 2 1)) (*.f64 x.re x.re)) x.im): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (+.f64 2 1) (Rewrite<= unpow2_binary64 (pow.f64 x.re 2))) x.im): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 2 (pow.f64 x.re 2)) (pow.f64 x.re 2))) x.im): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2.1902738719736264 \cdot 10^{+116}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{elif}\;x.re \leq 4.952149493142814 \cdot 10^{+77}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) + x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	7040

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

Alternative 2
Error	0.4
Cost	1352

\[\begin{array}{l} \mathbf{if}\;x.re \leq -1.5576967274412496 \cdot 10^{+97}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{elif}\;x.re \leq 4.952149493142814 \cdot 10^{+77}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]

Alternative 3
Error	0.3
Cost	1352

\[\begin{array}{l} \mathbf{if}\;x.re \leq -1 \cdot 10^{+150}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{elif}\;x.re \leq 4.952149493142814 \cdot 10^{+77}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]

Alternative 4
Error	1.0
Cost	1224

\[\begin{array}{l} \mathbf{if}\;x.re \leq -429.23650920572146:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{elif}\;x.re \leq 4.952149493142814 \cdot 10^{+77}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]

Alternative 5
Error	0.3
Cost	1216

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

Alternative 6
Error	0.4
Cost	968

\[\begin{array}{l} \mathbf{if}\;x.re \leq -1.6142798987345884 \cdot 10^{+112}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{elif}\;x.re \leq 1.6960127640473962 \cdot 10^{+62}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right) - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]

Alternative 7
Error	5.5
Cost	712

\[\begin{array}{l} t_0 := x.re \cdot \left(x.re \cdot \left(3 \cdot x.im\right)\right)\\ \mathbf{if}\;x.re \leq -4.754595221657154 \cdot 10^{-52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq 8.815162233884706 \cdot 10^{-77}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	5.5
Cost	712

\[\begin{array}{l} t_0 := \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{if}\;x.re \leq -4.754595221657154 \cdot 10^{-52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq 8.815162233884706 \cdot 10^{-77}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	5.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;x.re \leq -4.754595221657154 \cdot 10^{-52}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{elif}\;x.re \leq 8.815162233884706 \cdot 10^{-77}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]

Alternative 10
Error	28.2
Cost	384

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

Reproduce

herbie shell --seed 2022291 
(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)))