math.cube on complex, imaginary part

Average Error: 7.4 → 0.2

Time: 23.0s

Precision: binary64

Cost: 7360

Math

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

↓

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

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

Target

Original	7.4
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 7.4

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

2. Simplified 0.2

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

Alternatives

Alternative 1
Error	0.3
Cost	1224

\[\begin{array}{l} \mathbf{if}\;x.re \leq -7.5 \cdot 10^{+65}:\\ \;\;\;\;\left(x.im \cdot \left(3 \cdot x.re\right)\right) \cdot x.re\\ \mathbf{elif}\;x.re \leq 8 \cdot 10^{+128}:\\ \;\;\;\;x.im \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right) + x.re \cdot \left(2 \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3\\ \end{array} \]

Alternative 2
Error	5.7
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x.re \leq -1.8 \cdot 10^{-72}:\\ \;\;\;\;\left(x.im \cdot \left(3 \cdot x.re\right)\right) \cdot x.re\\ \mathbf{elif}\;x.re \leq 2.95 \cdot 10^{-72}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.im \cdot 3\right) \cdot x.re\right) \cdot x.re\\ \end{array} \]

Alternative 3
Error	0.2
Cost	1088

\[\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \left(\left(x.re \cdot x.im\right) \cdot 2\right) \cdot x.re \]

Alternative 4
Error	18.9
Cost	448

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

Alternative 5
Error	18.9
Cost	448

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

Alternative 6
Error	18.9
Cost	448

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

Reproduce

herbie shell --seed 2023010 
(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)))