Average Error: 7.6 → 0.2
Time: 2.9s
Precision: binary64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
↓
\[\mathsf{fma}\left(x.re \cdot \left(x.im \cdot 3\right), x.re, -{x.im}^{3}\right) \]
(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 3.0)) x.re (- (pow x.im 3.0))))
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 * 3.0)), x_46_re, -pow(x_46_im, 3.0));
}

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 * Float64(x_46_im * 3.0)), x_46_re, Float64(-(x_46_im ^ 3.0)))
end

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 * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision] * x$46$re + (-N[Power[x$46$im, 3.0], $MachinePrecision])), $MachinePrecision]

\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 \cdot \left(x.im \cdot 3\right), x.re, -{x.im}^{3}\right)

Error

Target

Original7.6
Target0.2
Herbie0.2
\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right) \]

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. Simplified7.6

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

  3. Applied egg-rr0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2022210 
(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)))