math.cube on complex, imaginary part

Average Error: 7.3 → 0.2

Time: 8.9s

Precision: binary64

Cost: 7040

Math

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

↓

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

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 (* 3.0 (* x.re x.im))) (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 * (3.0 * (x_46_re * x_46_im))) - 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 * (3.0d0 * (x_46re * x_46im))) - (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 * (3.0 * (x_46_re * x_46_im))) - 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 * (3.0 * (x_46_re * x_46_im))) - 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(x_46_re * Float64(3.0 * Float64(x_46_re * x_46_im))) - (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 * (3.0 * (x_46_re * x_46_im))) - (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[(x$46$re * N[(3.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $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 7.2

   \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right)\right) - {x.im}^{3}} \]
   Proof
   (-.f64 (*.f64 x.im (*.f64 x.re (*.f64 x.re 3))) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 x.im (*.f64 x.re (Rewrite<= *-commutative_binary64 (*.f64 3 x.re)))) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 x.im (*.f64 x.re (*.f64 (Rewrite<= metadata-eval (+.f64 2 1)) x.re))) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 x.im (*.f64 x.re (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 2 x.re) x.re)))) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 x.im (*.f64 x.re (+.f64 (Rewrite<= count-2_binary64 (+.f64 x.re x.re)) x.re))) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x.im x.re) (+.f64 (+.f64 x.re x.re) x.re))) (pow.f64 x.im 3)): 16 points increase in error, 36 points decrease in error
   (-.f64 (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 (+.f64 x.re x.re) x.re) (*.f64 x.im x.re))) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (+.f64 (Rewrite=> count-2_binary64 (*.f64 2 x.re)) x.re) (*.f64 x.im x.re)) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (Rewrite=> distribute-lft1-in_binary64 (*.f64 (+.f64 2 1) x.re)) (*.f64 x.im x.re)) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (*.f64 (Rewrite=> metadata-eval 3) x.re) (*.f64 x.im x.re)) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= associate-*r*_binary64 (*.f64 3 (*.f64 x.re (*.f64 x.im x.re)))) (pow.f64 x.im 3)): 19 points increase in error, 20 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))): 24 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)): 36 points increase in error, 10 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)): 9 points increase in error, 17 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)))): 0 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, 0 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)))): 17 points increase in error, 9 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))): 6 points increase in error, 10 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.im around 0 7.2

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

4. Simplified 0.2

   \[\leadsto \color{blue}{x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)} - {x.im}^{3} \]
   Proof
   (*.f64 x.re (*.f64 3 (*.f64 x.re x.im))): 0 points increase in error, 0 points decrease in error
   (*.f64 x.re (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 3 x.re) x.im))): 20 points increase in error, 23 points decrease in error
   (*.f64 x.re (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 x.re 3)) x.im)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x.re (*.f64 x.re 3)) x.im)): 46 points increase in error, 26 points decrease in error
   (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x.re x.re) 3)) x.im): 27 points increase in error, 18 points decrease in error
   (*.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 x.re 2)) 3) x.im): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 3 (pow.f64 x.re 2))) x.im): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*r*_binary64 (*.f64 3 (*.f64 (pow.f64 x.re 2) x.im))): 25 points increase in error, 17 points decrease in error

5. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.3
Cost	1352

\[\begin{array}{l} t_0 := \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{if}\;x.re \leq -1 \cdot 10^{+120}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq 6.523949268922945 \cdot 10^{+93}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	9.2
Cost	1096

\[\begin{array}{l} t_0 := \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + \left(x.re + x.re\right)\\ \mathbf{if}\;x.im \leq -1 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 10^{-20}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right) + x.re \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.2
Cost	1088

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

Alternative 4
Error	19.1
Cost	832

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

Alternative 5
Error	19.1
Cost	448

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

Alternative 6
Error	61.3
Cost	192

\[x.re \cdot 2 \]

Reproduce

herbie shell --seed 2022317 
(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)))