math.cube on complex, real part

Average Error: 7.0 → 0.2

Time: 8.6s

Precision: binary64

Cost: 13312

Math

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

↓

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

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.im (* x.im (* x.re -3.0)) (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_im, (x_46_im * (x_46_re * -3.0)), 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(x_46_im, Float64(x_46_im * Float64(x_46_re * -3.0)), (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[(x$46$im * N[(x$46$im * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision] + N[Power[x$46$re, 3.0], $MachinePrecision]), $MachinePrecision]

Target

Original	7.0
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 7.0

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

2. Taylor expanded in x.re around 0 7.0

   \[\leadsto \color{blue}{x.re \cdot \left(-1 \cdot {x.im}^{2} - 2 \cdot {x.im}^{2}\right) + {x.re}^{3}} \]

3. Simplified 0.2

   \[\leadsto \color{blue}{\mathsf{fma}\left(x.im, x.im \cdot \left(x.re \cdot -3\right), {x.re}^{3}\right)} \]
   Proof
   (fma.f64 x.im (*.f64 x.im (*.f64 x.re -3)) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
   (fma.f64 x.im (*.f64 x.im (*.f64 x.re (Rewrite<= metadata-eval (-.f64 -1 2)))) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
   (fma.f64 x.im (*.f64 x.im (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 -1 x.re) (*.f64 2 x.re)))) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x.im (*.f64 x.im (-.f64 (*.f64 -1 x.re) (*.f64 2 x.re)))) (pow.f64 x.re 3))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x.im x.im) (-.f64 (*.f64 -1 x.re) (*.f64 2 x.re)))) (pow.f64 x.re 3)): 41 points increase in error, 13 points decrease in error
   (+.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 x.im 2)) (-.f64 (*.f64 -1 x.re) (*.f64 2 x.re))) (pow.f64 x.re 3)): 0 points increase in error, 1 points decrease in error
   (+.f64 (*.f64 (pow.f64 x.im 2) (Rewrite=> distribute-rgt-out--_binary64 (*.f64 x.re (-.f64 -1 2)))) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 (pow.f64 x.im 2) x.re) (-.f64 -1 2))) (pow.f64 x.re 3)): 25 points increase in error, 17 points decrease in error
   (+.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 x.re (pow.f64 x.im 2))) (-.f64 -1 2)) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 x.re (*.f64 (pow.f64 x.im 2) (-.f64 -1 2)))) (pow.f64 x.re 3)): 16 points increase in error, 21 points decrease in error
   (+.f64 (*.f64 x.re (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 -1 (pow.f64 x.im 2)) (*.f64 2 (pow.f64 x.im 2))))) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	13312

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

Alternative 2
Error	0.2
Cost	1480

\[\begin{array}{l} \mathbf{if}\;x.im \leq -1 \cdot 10^{+150}:\\ \;\;\;\;\left(1 + -3 \cdot \left(x.im \cdot \left(x.im \cdot x.re\right)\right)\right) + -1\\ \mathbf{elif}\;x.im \leq 6.2104779927208995 \cdot 10^{+88}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]

Alternative 3
Error	0.3
Cost	1352

\[\begin{array}{l} \mathbf{if}\;x.im \leq -1 \cdot 10^{+140}:\\ \;\;\;\;\left(1 + -3 \cdot \left(x.im \cdot \left(x.im \cdot x.re\right)\right)\right) + -1\\ \mathbf{elif}\;x.im \leq 10^{+150}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.im \cdot x.im\right) \cdot \left(x.re \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]

Alternative 4
Error	0.3
Cost	968

\[\begin{array}{l} t_0 := \left(1 + -3 \cdot \left(x.im \cdot \left(x.im \cdot x.re\right)\right)\right) + -1\\ \mathbf{if}\;x.im \leq -1 \cdot 10^{+150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 10^{+155}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + -3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	12.4
Cost	712

\[\begin{array}{l} t_0 := -3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{if}\;x.im \leq -2.347704217393616 \cdot 10^{-85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 1.409996387777235 \cdot 10^{-47}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	5.7
Cost	712

\[\begin{array}{l} t_0 := \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{if}\;x.im \leq -2.347704217393616 \cdot 10^{-85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 1.409996387777235 \cdot 10^{-47}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	5.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;x.im \leq -2.347704217393616 \cdot 10^{-85}:\\ \;\;\;\;\left(x.im \cdot x.re\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{elif}\;x.im \leq 1.409996387777235 \cdot 10^{-47}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]

Alternative 8
Error	28.4
Cost	320

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

Reproduce

herbie shell --seed 2022300 
(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)))