math.cube on complex, real part

Average Error: 12.2% → 0.26%

Time: 10.5s

Precision: binary64

Cost: 7040

Math

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

↓

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

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (-
  (* (- (* x.re x.re) (* x.im x.im)) x.re)
  (* (+ (* x.re x.im) (* x.im x.re)) x.im)))

↓

(FPCore (x.re x.im)
 :precision binary64
 (+ (pow x.re 3.0) (* (* x.re x.im) (* 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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

↓

double code(double x_46_re, double x_46_im) {
	return pow(x_46_re, 3.0) + ((x_46_re * x_46_im) * (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_46re) - (((x_46re * x_46im) + (x_46im * x_46re)) * x_46im)
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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

↓

public static double code(double x_46_re, double x_46_im) {
	return Math.pow(x_46_re, 3.0) + ((x_46_re * x_46_im) * (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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im)

↓

def code(x_46_re, x_46_im):
	return math.pow(x_46_re, 3.0) + ((x_46_re * x_46_im) * (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_re) - Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_im))
end

↓

function code(x_46_re, x_46_im)
	return Float64((x_46_re ^ 3.0) + Float64(Float64(x_46_re * x_46_im) * Float64(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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
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$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]

↓

code[x$46$re_, x$46$im_] := N[(N[Power[x$46$re, 3.0], $MachinePrecision] + N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	12.2%
Target	0.38%
Herbie	0.26%

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

Derivation

1. Initial program 12.2

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

2. Simplified 12.14

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

   [Start] 12.2	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   *-commutative [=>] 12.2	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   sub-neg [=>] 12.2	\[ x.re \cdot \color{blue}{\left(x.re \cdot x.re + \left(-x.im \cdot x.im\right)\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   distribute-rgt-in [=>] 12.2	\[ \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot x.re + \left(-x.im \cdot x.im\right) \cdot x.re\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   associate--l+ [=>] 12.2	\[ \color{blue}{\left(x.re \cdot x.re\right) \cdot x.re + \left(\left(-x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)} \]
   *-commutative [=>] 12.2	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re\right)} + \left(\left(-x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right) \]
   cube-unmult [=>] 12.08	\[ \color{blue}{{x.re}^{3}} + \left(\left(-x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right) \]
   *-commutative [=>] 12.08	\[ {x.re}^{3} + \left(\left(-x.im \cdot x.im\right) \cdot x.re - \left(\color{blue}{x.im \cdot x.re} + x.im \cdot x.re\right) \cdot x.im\right) \]
   distribute-rgt-out [=>] 12.08	\[ {x.re}^{3} + \left(\left(-x.im \cdot x.im\right) \cdot x.re - \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \cdot x.im\right) \]
   associate-*l* [=>] 12.14	\[ {x.re}^{3} + \left(\left(-x.im \cdot x.im\right) \cdot x.re - \color{blue}{x.re \cdot \left(\left(x.im + x.im\right) \cdot x.im\right)}\right) \]
   *-commutative [=>] 12.14	\[ {x.re}^{3} + \left(\left(-x.im \cdot x.im\right) \cdot x.re - \color{blue}{\left(\left(x.im + x.im\right) \cdot x.im\right) \cdot x.re}\right) \]
   distribute-rgt-out-- [=>] 12.14	\[ {x.re}^{3} + \color{blue}{x.re \cdot \left(\left(-x.im \cdot x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)} \]
   neg-mul-1 [=>] 12.14	\[ {x.re}^{3} + x.re \cdot \left(\color{blue}{-1 \cdot \left(x.im \cdot x.im\right)} - \left(x.im + x.im\right) \cdot x.im\right) \]
   count-2 [=>] 12.14	\[ {x.re}^{3} + x.re \cdot \left(-1 \cdot \left(x.im \cdot x.im\right) - \color{blue}{\left(2 \cdot x.im\right)} \cdot x.im\right) \]
   associate-*l* [=>] 12.14	\[ {x.re}^{3} + x.re \cdot \left(-1 \cdot \left(x.im \cdot x.im\right) - \color{blue}{2 \cdot \left(x.im \cdot x.im\right)}\right) \]
   distribute-rgt-out-- [=>] 12.14	\[ {x.re}^{3} + x.re \cdot \color{blue}{\left(\left(x.im \cdot x.im\right) \cdot \left(-1 - 2\right)\right)} \]
   associate-*l* [<=] 12.14	\[ {x.re}^{3} + \color{blue}{\left(x.re \cdot \left(x.im \cdot x.im\right)\right) \cdot \left(-1 - 2\right)} \]
   metadata-eval [=>] 12.14	\[ {x.re}^{3} + \left(x.re \cdot \left(x.im \cdot x.im\right)\right) \cdot \color{blue}{-3} \]

3. Taylor expanded in x.re around 0 12.14

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

4. Simplified 0.26

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

   [Start] 12.14	\[ {x.re}^{3} + -3 \cdot \left(x.re \cdot {x.im}^{2}\right) \]
   *-commutative [=>] 12.14	\[ {x.re}^{3} + \color{blue}{\left(x.re \cdot {x.im}^{2}\right) \cdot -3} \]
   associate-*l* [=>] 12.14	\[ {x.re}^{3} + \color{blue}{x.re \cdot \left({x.im}^{2} \cdot -3\right)} \]
   unpow2 [=>] 12.14	\[ {x.re}^{3} + x.re \cdot \left(\color{blue}{\left(x.im \cdot x.im\right)} \cdot -3\right) \]
   associate-*r* [<=] 12.15	\[ {x.re}^{3} + x.re \cdot \color{blue}{\left(x.im \cdot \left(x.im \cdot -3\right)\right)} \]
   associate-*l* [<=] 0.26	\[ {x.re}^{3} + \color{blue}{\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)} \]

5. Final simplification 0.26

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

Alternatives

Alternative 1
Error	0.27%
Cost	7040

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

Alternative 2
Error	0.26%
Cost	7040

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

Alternative 3
Error	0.57%
Cost	1352

\[\begin{array}{l} \mathbf{if}\;x.im \leq -8.5 \cdot 10^{+59}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{elif}\;x.im \leq 1.6 \cdot 10^{+141}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - 2 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \end{array} \]

Alternative 4
Error	0.53%
Cost	1352

\[\begin{array}{l} \mathbf{if}\;x.im \leq -2.5 \cdot 10^{+61}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{elif}\;x.im \leq 2.4 \cdot 10^{+141}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \end{array} \]

Alternative 5
Error	8.04%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;x.im \leq -1.3 \cdot 10^{-66} \lor \neg \left(x.im \leq 1.6 \cdot 10^{-41}\right):\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \end{array} \]

Alternative 6
Error	0.37%
Cost	1088

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

Alternative 7
Error	0.38%
Cost	1088

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

Alternative 8
Error	8.15%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x.im \leq -1.15 \cdot 10^{-66} \lor \neg \left(x.im \leq 2.25 \cdot 10^{-35}\right):\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 9
Error	44.31%
Cost	320

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

Alternative 10
Error	72.81%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im))))

  (- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im)))