Average Error: 7.1 → 0.2
Time: 9.6s
Precision: binary64
Cost: 7040
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
↓
\[x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - {x.im}^{3} \]
(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 (* x.im 3.0))) (pow x.im 3.0)))
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 * (x_46_im * 3.0))) - pow(x_46_im, 3.0);
}

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 * (x_46im * 3.0d0))) - (x_46im ** 3.0d0)
end function

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 * (x_46_im * 3.0))) - Math.pow(x_46_im, 3.0);
}

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 * (x_46_im * 3.0))) - math.pow(x_46_im, 3.0)

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(x_46_re * Float64(x_46_im * 3.0))) - (x_46_im ^ 3.0))
end

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 * (x_46_im * 3.0))) - (x_46_im ^ 3.0);
end

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[(x$46$re * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im, 3.0], $MachinePrecision]), $MachinePrecision]

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

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.1
Target0.2
Herbie0.2
\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right) \]

Derivation

  1. Initial program 7.1

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

  2. Simplified0.2

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

    [Start]7.1

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

    +-commutative [=>]7.1

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

    *-commutative [=>]7.1

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

    distribute-rgt-out-- [<=]7.1

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

    associate-+r- [=>]7.1

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

    *-commutative [<=]7.1

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

    *-commutative [=>]7.1

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

    *-commutative [<=]7.1

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

    distribute-lft-out [=>]7.1

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

    associate-*r* [=>]7.2

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

    distribute-lft-out [=>]7.2

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

    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. Final simplification0.2

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

Alternatives

Alternative 1
Error0.2
Cost1600
\[\begin{array}{l} t_0 := x.im \cdot \left(x.re + x.im\right)\\ \left(x.re \cdot t_0 - x.im \cdot t_0\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \end{array} \]
Alternative 2
Error0.2
Cost1088
\[\left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) + x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right) \]
Alternative 3
Error0.4
Cost968
\[\begin{array}{l} \mathbf{if}\;x.re \leq -7.5 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 1.92 \cdot 10^{+59}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right) - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \end{array} \]
Alternative 4
Error5.7
Cost841
\[\begin{array}{l} \mathbf{if}\;x.re \leq -1.15 \cdot 10^{-90} \lor \neg \left(x.re \leq 7 \cdot 10^{-71}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]
Alternative 5
Error12.8
Cost713
\[\begin{array}{l} \mathbf{if}\;x.re \leq -1.15 \cdot 10^{-90} \lor \neg \left(x.re \leq 6.1 \cdot 10^{-78}\right):\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \]
Alternative 6
Error5.8
Cost713
\[\begin{array}{l} \mathbf{if}\;x.re \leq -1.15 \cdot 10^{-90} \lor \neg \left(x.re \leq 1.02 \cdot 10^{-78}\right):\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \]
Alternative 7
Error5.8
Cost713
\[\begin{array}{l} \mathbf{if}\;x.re \leq -1.15 \cdot 10^{-90} \lor \neg \left(x.re \leq 6.1 \cdot 10^{-78}\right):\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \]
Alternative 8
Error5.8
Cost713
\[\begin{array}{l} \mathbf{if}\;x.re \leq -1.12 \cdot 10^{-90} \lor \neg \left(x.re \leq 6.1 \cdot 10^{-78}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \]
Alternative 9
Error26.3
Cost649
\[\begin{array}{l} \mathbf{if}\;x.re \leq -1.7 \cdot 10^{+15} \lor \neg \left(x.re \leq 3.2 \cdot 10^{+18}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \]
Alternative 10
Error44.3
Cost320
\[x.im \cdot \left(x.re \cdot x.re\right) \]
Alternative 11
Error43.1
Cost320
\[x.re \cdot \left(x.re \cdot x.im\right) \]

Error

Reproduce

herbie shell --seed 2023011 
(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)))