math.cube on complex, imaginary part

Average Error: 7.5 → 0.2

Time: 6.6s

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	7.5
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 7.5

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

2. Simplified 7.5

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

3. Applied egg-rr 11.4

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

4. Applied egg-rr 0.2

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

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	0.3
Cost	1352

\[\begin{array}{l} t_0 := x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 3\right)\\ \mathbf{if}\;x.re \leq -2.4640217948772817 \cdot 10^{+73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq 10^{+130}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	4.9
Cost	1096

\[\begin{array}{l} t_0 := x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 3\right)\\ \mathbf{if}\;x.re \leq -2.7434489407218423 \cdot 10^{-38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq 1.798724804872214 \cdot 10^{-46}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 2\right)\right) - x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	18.6
Cost	448

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

Alternative 4
Error	18.6
Cost	448

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

Reproduce

herbie shell --seed 2022225 
(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)))