math.cube on complex, real part

Average Accuracy: 88.8% → 99.7%

Time: 24.3s

Precision: binary64

Math

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

↓

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

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
 (fma (* x.re (+ x.re x.im)) (- x.re x.im) (* x.im (* (* x.im x.re) -2.0))))

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 fma((x_46_re * (x_46_re + x_46_im)), (x_46_re - x_46_im), (x_46_im * ((x_46_im * x_46_re) * -2.0)));
}

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 fma(Float64(x_46_re * Float64(x_46_re + x_46_im)), Float64(x_46_re - x_46_im), Float64(x_46_im * Float64(Float64(x_46_im * x_46_re) * -2.0)))
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[(x$46$re * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision] + N[(x$46$im * N[(N[(x$46$im * x$46$re), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	88.8%
Target	99.6%
Herbie	99.7%

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

Derivation

1. Initial program 88.8%

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

2. Simplified 99.7%

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

Reproduce

herbie shell --seed 2023151 
(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)))