math.cube on complex, imaginary part

Average Error: 7.3 → 0.2

Time: 7.4s

Precision: binary64

Cost: 7040

Math

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

↓

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

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
 (- (* (* x.re x.im) (* x.re 3.0)) (pow x.im 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) {
	return ((x_46_re * x_46_im) * (x_46_re * 3.0)) - pow(x_46_im, 3.0);
}

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
    code = ((x_46re * x_46im) * (x_46re * 3.0d0)) - (x_46im ** 3.0d0)
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) {
	return ((x_46_re * x_46_im) * (x_46_re * 3.0)) - Math.pow(x_46_im, 3.0);
}

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):
	return ((x_46_re * x_46_im) * (x_46_re * 3.0)) - math.pow(x_46_im, 3.0)

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)
	return Float64(Float64(Float64(x_46_re * x_46_im) * Float64(x_46_re * 3.0)) - (x_46_im ^ 3.0))
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 = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_im) * (x_46_re * 3.0)) - (x_46_im ^ 3.0);
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_] := N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im, 3.0], $MachinePrecision]), $MachinePrecision]

Target

Original	7.3
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. Initial program 7.3

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

2. Simplified 0.2

   \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - {x.im}^{3}} \]
   Proof
   (-.f64 (*.f64 x.re (*.f64 x.re (*.f64 x.im 3))) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 x.re (*.f64 x.re (Rewrite<= *-commutative_binary64 (*.f64 3 x.im)))) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 x.re (*.f64 x.re (*.f64 (Rewrite<= metadata-eval (+.f64 2 1)) x.im))) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 x.re (*.f64 x.re (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 2 x.im) x.im)))) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 x.re (*.f64 x.re (+.f64 (Rewrite<= count-2_binary64 (+.f64 x.im x.im)) x.im))) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x.re x.re) (+.f64 (+.f64 x.im x.im) x.im))) (pow.f64 x.im 3)): 34 points increase in error, 19 points decrease in error
   (-.f64 (*.f64 (*.f64 x.re x.re) (+.f64 (Rewrite=> count-2_binary64 (*.f64 2 x.im)) x.im)) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (*.f64 x.re x.re) (Rewrite=> distribute-lft1-in_binary64 (*.f64 (+.f64 2 1) x.im))) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (*.f64 x.re x.re) (*.f64 (Rewrite=> metadata-eval 3) x.im)) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 (*.f64 x.re x.re) 3) x.im)) (pow.f64 x.im 3)): 24 points increase in error, 20 points decrease in error
   (-.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 3 (*.f64 x.re x.re))) x.im) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= associate-*r*_binary64 (*.f64 3 (*.f64 (*.f64 x.re x.re) x.im))) (pow.f64 x.im 3)): 20 points increase in error, 20 points decrease in error
   (-.f64 (*.f64 3 (Rewrite=> associate-*l*_binary64 (*.f64 x.re (*.f64 x.re x.im)))) (pow.f64 x.im 3)): 16 points increase in error, 33 points decrease in error
   (-.f64 (*.f64 3 (*.f64 x.re (Rewrite<= *-commutative_binary64 (*.f64 x.im x.re)))) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 3 (*.f64 x.re (*.f64 x.im x.re))) (Rewrite=> unpow3_binary64 (*.f64 (*.f64 x.im x.im) x.im))): 20 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 3 (*.f64 x.re (Rewrite=> *-commutative_binary64 (*.f64 x.re x.im)))) (*.f64 (*.f64 x.im x.im) x.im)): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 3 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x.re x.re) x.im))) (*.f64 (*.f64 x.im x.im) x.im)): 33 points increase in error, 16 points decrease in error
   (-.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 3 (*.f64 x.re x.re)) x.im)) (*.f64 (*.f64 x.im x.im) x.im)): 20 points increase in error, 20 points decrease in error
   (-.f64 (*.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 2 1)) (*.f64 x.re x.re)) x.im) (*.f64 (*.f64 x.im x.im) x.im)): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 2 (*.f64 x.re x.re)) (*.f64 x.re x.re))) x.im) (*.f64 (*.f64 x.im x.im) x.im)): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 x.re) x.re)) (*.f64 x.re x.re)) x.im) (*.f64 (*.f64 x.im x.im) x.im)): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (+.f64 (*.f64 (Rewrite<= count-2_binary64 (+.f64 x.re x.re)) x.re) (*.f64 x.re x.re)) x.im) (*.f64 (*.f64 x.im x.im) x.im)): 0 points increase in error, 0 points decrease in error
   (Rewrite=> distribute-rgt-out--_binary64 (*.f64 x.im (-.f64 (+.f64 (*.f64 (+.f64 x.re x.re) x.re) (*.f64 x.re x.re)) (*.f64 x.im x.im)))): 2 points increase in error, 0 points decrease in error
   (*.f64 x.im (Rewrite<= associate-+r-_binary64 (+.f64 (*.f64 (+.f64 x.re x.re) x.re) (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))))): 1 points increase in error, 1 points decrease in error
   (*.f64 x.im (Rewrite=> +-commutative_binary64 (+.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) (*.f64 (+.f64 x.re x.re) x.re)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 x.im (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (*.f64 x.im (*.f64 (+.f64 x.re x.re) x.re)))): 18 points increase in error, 22 points decrease in error
   (+.f64 (*.f64 x.im (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x.im (+.f64 x.re x.re)) x.re))): 7 points increase in error, 12 points decrease in error
   (+.f64 (*.f64 x.im (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (*.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 x.re x.im) (*.f64 x.re x.im))) x.re)): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 x.im (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (*.f64 (+.f64 (*.f64 x.re x.im) (Rewrite<= *-commutative_binary64 (*.f64 x.im x.re))) x.re)): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im)) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)): 0 points increase in error, 0 points decrease in error

3. Taylor expanded in x.re around 0 7.3

   \[\leadsto \color{blue}{3 \cdot \left({x.re}^{2} \cdot x.im\right)} - {x.im}^{3} \]

4. Simplified 0.2

   \[\leadsto \color{blue}{\left(x.re \cdot x.im\right) \cdot \left(3 \cdot x.re\right)} - {x.im}^{3} \]
   Proof
   (*.f64 (*.f64 x.re x.im) (*.f64 3 x.re)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 x.re x.im) 3) x.re)): 34 points increase in error, 23 points decrease in error
   (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 3 (*.f64 x.re x.im))) x.re): 0 points increase in error, 0 points decrease in error
   (Rewrite=> associate-*l*_binary64 (*.f64 3 (*.f64 (*.f64 x.re x.im) x.re))): 26 points increase in error, 34 points decrease in error
   (*.f64 3 (Rewrite<= *-commutative_binary64 (*.f64 x.re (*.f64 x.re x.im)))): 0 points increase in error, 0 points decrease in error
   (*.f64 3 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x.re x.re) x.im))): 37 points increase in error, 25 points decrease in error
   (*.f64 3 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 x.re 2)) x.im)): 0 points increase in error, 0 points decrease in error

5. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	7040

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

Alternative 2
Error	0.2
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 3
Error	0.9
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x.re \leq -7.5 \cdot 10^{+153}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 15500000000:\\ \;\;\;\;x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right)\right) - x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \end{array} \]

Alternative 4
Error	1.0
Cost	968

\[\begin{array}{l} \mathbf{if}\;x.re \leq -1.55 \cdot 10^{+101}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 950000:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right) - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \end{array} \]

Alternative 5
Error	12.9
Cost	712

\[\begin{array}{l} t_0 := 3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{if}\;x.re \leq -0.0035:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq 6.2 \cdot 10^{-63}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	5.8
Cost	712

\[\begin{array}{l} t_0 := 3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{if}\;x.re \leq -0.0045:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq 4.6 \cdot 10^{-64}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	5.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;x.re \leq -0.0035:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 9.2 \cdot 10^{-63}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \end{array} \]

Alternative 8
Error	5.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;x.re \leq -0.0035:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 2.05 \cdot 10^{-62}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \end{array} \]

Alternative 9
Error	28.6
Cost	384

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

Reproduce

herbie shell --seed 2022332 
(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)))