math.cube on complex, real part

Average Error: 7.3 → 0.2

Time: 8.2s

Precision: binary64

Cost: 13312

Math

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

↓

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

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 (* -3.0 x.im)) x.im (pow x.re 3.0)))

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 * (-3.0 * x_46_im)), x_46_im, pow(x_46_re, 3.0));
}

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

Target

Original	7.3
Target	0.3
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 7.3

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

2. Simplified 7.3

   \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, x.im \cdot \left(x.im \cdot -3\right), {x.re}^{3}\right)} \]
   Proof
   (fma.f64 x.re (*.f64 x.im (*.f64 x.im -3)) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
   (fma.f64 x.re (*.f64 x.im (*.f64 x.im (Rewrite<= metadata-eval (-.f64 -1 2)))) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
   (fma.f64 x.re (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x.im x.im) (-.f64 -1 2))) (pow.f64 x.re 3)): 11 points increase in error, 9 points decrease in error
   (fma.f64 x.re (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 -1 (*.f64 x.im x.im)) (*.f64 2 (*.f64 x.im x.im)))) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
   (fma.f64 x.re (-.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (*.f64 x.im x.im))) (*.f64 2 (*.f64 x.im x.im))) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
   (fma.f64 x.re (-.f64 (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 x.im) x.im)) (*.f64 2 (*.f64 x.im x.im))) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
   (fma.f64 x.re (-.f64 (*.f64 (neg.f64 x.im) x.im) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 x.im) x.im))) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
   (fma.f64 x.re (-.f64 (*.f64 (neg.f64 x.im) x.im) (*.f64 (Rewrite<= count-2_binary64 (+.f64 x.im x.im)) x.im)) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
   (fma.f64 x.re (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 (neg.f64 x.im) x.im) (neg.f64 (*.f64 (+.f64 x.im x.im) x.im)))) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
   (fma.f64 x.re (+.f64 (*.f64 (neg.f64 x.im) x.im) (Rewrite<= distribute-rgt-neg-out_binary64 (*.f64 (+.f64 x.im x.im) (neg.f64 x.im)))) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
   (fma.f64 x.re (+.f64 (*.f64 (neg.f64 x.im) x.im) (*.f64 (+.f64 x.im x.im) (neg.f64 x.im))) (Rewrite=> unpow3_binary64 (*.f64 (*.f64 x.re x.re) x.re))): 18 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x.re (+.f64 (*.f64 (neg.f64 x.im) x.im) (*.f64 (+.f64 x.im x.im) (neg.f64 x.im)))) (*.f64 (*.f64 x.re x.re) x.re))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 x.re (+.f64 (*.f64 (neg.f64 x.im) x.im) (*.f64 (+.f64 x.im x.im) (neg.f64 x.im)))) (Rewrite=> associate-*l*_binary64 (*.f64 x.re (*.f64 x.re x.re)))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> distribute-lft-out_binary64 (*.f64 x.re (+.f64 (+.f64 (*.f64 (neg.f64 x.im) x.im) (*.f64 (+.f64 x.im x.im) (neg.f64 x.im))) (*.f64 x.re x.re)))): 2 points increase in error, 0 points decrease in error
   (*.f64 x.re (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x.re x.re) (+.f64 (*.f64 (neg.f64 x.im) x.im) (*.f64 (+.f64 x.im x.im) (neg.f64 x.im)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 x.re (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 x.re x.re) (*.f64 (neg.f64 x.im) x.im)) (*.f64 (+.f64 x.im x.im) (neg.f64 x.im))))): 1 points increase in error, 2 points decrease in error
   (*.f64 x.re (+.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (*.f64 (+.f64 x.im x.im) (neg.f64 x.im)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 x.re (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (*.f64 x.re (*.f64 (+.f64 x.im x.im) (neg.f64 x.im))))): 29 points increase in error, 20 points decrease in error
   (+.f64 (*.f64 x.re (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x.re (+.f64 x.im x.im)) (neg.f64 x.im)))): 5 points increase in error, 9 points decrease in error
   (+.f64 (*.f64 x.re (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (*.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 x.im x.re) (*.f64 x.im x.re))) (neg.f64 x.im))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 x.re (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (*.f64 (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 x.re x.im)) (*.f64 x.im x.re)) (neg.f64 x.im))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 x.re (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re)) (neg.f64 (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 7.4

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

4. Applied egg-rr 0.2

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

5. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	7040

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

Alternative 2
Error	0.2
Cost	968

\[\begin{array}{l} \mathbf{if}\;x.im \leq -1 \cdot 10^{+150}:\\ \;\;\;\;\left(1 + -3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\right) + -1\\ \mathbf{elif}\;x.im \leq 10^{+150}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(-3 \cdot x.im\right) + x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 3
Error	12.5
Cost	712

\[\begin{array}{l} t_0 := -3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{if}\;x.im \leq -2.595620580432503 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 2.314281482672583 \cdot 10^{-43}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	5.4
Cost	712

\[\begin{array}{l} t_0 := \left(-3 \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{if}\;x.im \leq -2.595620580432503 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 2.314281482672583 \cdot 10^{-43}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	5.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;x.im \leq -2.595620580432503 \cdot 10^{-37}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 2.314281482672583 \cdot 10^{-43}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-3 \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]

Alternative 6
Error	28.5
Cost	320

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

Alternative 7
Error	47.0
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022291 
(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)))