math.cube on complex, real part

Average Error: 7.2 → 0.2

Time: 2.7s

Precision: binary64

Math

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

↓

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

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

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_im * (x_46_re * x_46_im))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
}

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_46im * (x_46re * x_46im))) - (x_46im * ((x_46re * x_46im) + (x_46re * x_46im)))
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_im * (x_46_re * x_46_im))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
}

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_im * (x_46_re * x_46_im))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))

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(Float64((x_46_re ^ 3.0) - Float64(x_46_im * Float64(x_46_re * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im))))
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_im * (x_46_re * x_46_im))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
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[(N[Power[x$46$re, 3.0], $MachinePrecision] - N[(x$46$im * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.2
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 7.2

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

2. Applied egg-rr 0.6

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

3. Taylor expanded in x.re around inf 7.1

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

4. Simplified 0.2

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

5. Final simplification 0.2

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

Reproduce

herbie shell --seed 2022166 
(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)))