math.cube on complex, imaginary part

Average Error: 7.3 → 0.2

Time: 8.9s

Precision: binary64

Cost: 7040

Math

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

↓

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

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

Target

Original	7.3
Target	0.3
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

   [Start] 7.3	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   +-commutative [=>] 7.3	\[ \color{blue}{\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im} \]
   *-commutative [=>] 7.3	\[ \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} \]
   distribute-rgt-out-- [<=] 7.3	\[ \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot x.im - \left(x.im \cdot x.im\right) \cdot x.im\right)} \]
   associate-+r- [=>] 7.3	\[ \color{blue}{\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re\right) \cdot x.im\right) - \left(x.im \cdot x.im\right) \cdot x.im} \]
   *-commutative [<=] 7.3	\[ \left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re\right) \cdot x.im\right) - \color{blue}{x.im \cdot \left(x.im \cdot x.im\right)} \]
   *-commutative [=>] 7.3	\[ \left(\color{blue}{x.re \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} + \left(x.re \cdot x.re\right) \cdot x.im\right) - x.im \cdot \left(x.im \cdot x.im\right) \]
   *-commutative [<=] 7.3	\[ \left(x.re \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) + \left(x.re \cdot x.re\right) \cdot x.im\right) - x.im \cdot \left(x.im \cdot x.im\right) \]
   distribute-lft-out [=>] 7.3	\[ \left(x.re \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} + \left(x.re \cdot x.re\right) \cdot x.im\right) - x.im \cdot \left(x.im \cdot x.im\right) \]
   associate-*r* [=>] 7.3	\[ \left(\color{blue}{\left(x.re \cdot x.re\right) \cdot \left(x.im + x.im\right)} + \left(x.re \cdot x.re\right) \cdot x.im\right) - x.im \cdot \left(x.im \cdot x.im\right) \]
   distribute-lft-out [=>] 7.3	\[ \color{blue}{\left(x.re \cdot x.re\right) \cdot \left(\left(x.im + x.im\right) + x.im\right)} - x.im \cdot \left(x.im \cdot x.im\right) \]
   associate-*l* [=>] 0.2	\[ \color{blue}{x.re \cdot \left(x.re \cdot \left(\left(x.im + x.im\right) + x.im\right)\right)} - x.im \cdot \left(x.im \cdot x.im\right) \]
   count-2 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot \left(\color{blue}{2 \cdot x.im} + x.im\right)\right) - x.im \cdot \left(x.im \cdot x.im\right) \]
   distribute-lft1-in [=>] 0.2	\[ x.re \cdot \left(x.re \cdot \color{blue}{\left(\left(2 + 1\right) \cdot x.im\right)}\right) - x.im \cdot \left(x.im \cdot x.im\right) \]
   *-commutative [=>] 0.2	\[ x.re \cdot \left(x.re \cdot \color{blue}{\left(x.im \cdot \left(2 + 1\right)\right)}\right) - x.im \cdot \left(x.im \cdot x.im\right) \]
   metadata-eval [=>] 0.2	\[ x.re \cdot \left(x.re \cdot \left(x.im \cdot \color{blue}{3}\right)\right) - x.im \cdot \left(x.im \cdot x.im\right) \]
   cube-unmult [=>] 0.2	\[ x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - \color{blue}{{x.im}^{3}} \]

3. Taylor expanded in x.re around 0 0.2

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

4. Applied egg-rr 24.6

   \[\leadsto x.re \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot \left(x.re \cdot x.im\right)\right)} - 1\right)} - {x.im}^{3} \]

5. Simplified 0.2

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

   [Start] 24.6	\[ x.re \cdot \left(e^{\mathsf{log1p}\left(3 \cdot \left(x.re \cdot x.im\right)\right)} - 1\right) - {x.im}^{3} \]
   expm1-def [=>] 6.6	\[ x.re \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \left(x.re \cdot x.im\right)\right)\right)} - {x.im}^{3} \]
   expm1-log1p [=>] 0.2	\[ x.re \cdot \color{blue}{\left(3 \cdot \left(x.re \cdot x.im\right)\right)} - {x.im}^{3} \]
   associate-*r* [=>] 0.2	\[ x.re \cdot \color{blue}{\left(\left(3 \cdot x.re\right) \cdot x.im\right)} - {x.im}^{3} \]

6. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.8
Cost	9608

\[\begin{array}{l} t_0 := x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(3 \cdot x.im\right)\right)\\ \mathbf{elif}\;t_0 \leq 10^{+282}:\\ \;\;\;\;x.im \cdot \mathsf{fma}\left(3, x.re \cdot x.re, x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 2
Error	0.2
Cost	7040

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

Alternative 3
Error	0.2
Cost	7040

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

Alternative 4
Error	0.2
Cost	1480

\[\begin{array}{l} \mathbf{if}\;x.re \leq -4.6 \cdot 10^{+123}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(3 \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 4 \cdot 10^{+93}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 5
Error	0.2
Cost	968

\[\begin{array}{l} \mathbf{if}\;x.re \leq -3.8 \cdot 10^{+123}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(3 \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 2 \cdot 10^{+138}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right) - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 6
Error	6.6
Cost	713

\[\begin{array}{l} \mathbf{if}\;x.re \leq -12000000000000 \lor \neg \left(x.re \leq 2.65 \cdot 10^{-93}\right):\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \]

Alternative 7
Error	6.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;x.re \leq -12000000000000:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(3 \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 2.65 \cdot 10^{-93}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 8
Error	6.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;x.re \leq -12000000000000:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot 3\right) \cdot x.im\right)\\ \mathbf{elif}\;x.re \leq 2.65 \cdot 10^{-93}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 9
Error	26.1
Cost	649

\[\begin{array}{l} \mathbf{if}\;x.im \leq -9.5 \cdot 10^{-100} \lor \neg \left(x.im \leq 4.2 \cdot 10^{-104}\right):\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]

Alternative 10
Error	42.4
Cost	320

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

Reproduce

herbie shell --seed 2023060 
(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)))