math.cube on complex, imaginary part

Average Error: 7.6 → 0.2

Time: 3.9s

Precision: binary64

Math

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

↓

\[\mathsf{fma}\left(x.re + x.im, x.re \cdot x.im - {x.im}^{2}, x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\right) \]

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
 (fma
  (+ x.re x.im)
  (- (* x.re x.im) (pow x.im 2.0))
  (* x.re (+ (* 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_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 fma((x_46_re + x_46_im), ((x_46_re * x_46_im) - pow(x_46_im, 2.0)), (x_46_re * ((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_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 fma(Float64(x_46_re + x_46_im), Float64(Float64(x_46_re * x_46_im) - (x_46_im ^ 2.0)), Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(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$46$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 + x$46$im), $MachinePrecision] * N[(N[(x$46$re * x$46$im), $MachinePrecision] - N[Power[x$46$im, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x$46$re * 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	7.6
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 7.6

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

2. Applied difference-of-squares_binary64 7.6

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

3. Applied associate-*l*_binary64 0.3

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

4. Applied fma-def_binary64 0.2

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

5. Taylor expanded in x.re around 0 0.2

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

6. Final simplification 0.2

   \[\leadsto \mathsf{fma}\left(x.re + x.im, x.re \cdot x.im - {x.im}^{2}, x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\right) \]

Reproduce

herbie shell --seed 2022131 
(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)))