math.cube on complex, real part

Average Error: 7.4 → 0.2

Time: 8.7s

Precision: binary64

Cost: 7040

Math

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

↓

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

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
 (+ (pow x.re 3.0) (* x.im (* x.re (* x.im -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 pow(x_46_re, 3.0) + (x_46_im * (x_46_re * (x_46_im * -3.0)));
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = (((x_46re * x_46re) - (x_46im * x_46im)) * x_46re) - (((x_46re * x_46im) + (x_46im * x_46re)) * x_46im)
end function

↓

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = (x_46re ** 3.0d0) + (x_46im * (x_46re * (x_46im * (-3.0d0))))
end function

Java

public static 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);
}

↓

public static double code(double x_46_re, double x_46_im) {
	return Math.pow(x_46_re, 3.0) + (x_46_im * (x_46_re * (x_46_im * -3.0)));
}

Python

def code(x_46_re, 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)

↓

def code(x_46_re, x_46_im):
	return math.pow(x_46_re, 3.0) + (x_46_im * (x_46_re * (x_46_im * -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 Float64((x_46_re ^ 3.0) + Float64(x_46_im * Float64(x_46_re * Float64(x_46_im * -3.0))))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = (((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);
end

↓

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

Target

Original	7.4
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 7.4

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

2. Simplified 7.4

   \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, x.im \cdot \left(x.im \cdot -3\right), {x.re}^{3}\right)} \]
   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)): 10 points increase in error, 10 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))): 26 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)))): 1 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))))): 0 points increase in error, 0 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))))): 13 points increase in error, 19 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)))): 1 points increase in error, 7 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 7.5

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

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. Applied egg-rr 0.2

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

6. Taylor expanded in x.im around 0 0.2

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

7. Simplified 0.2

   \[\leadsto {x.re}^{3} + x.im \cdot \color{blue}{\left(x.re \cdot \left(-3 \cdot x.im\right)\right)} \]
   Proof
   (*.f64 x.re (*.f64 -3 x.im)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x.re -3) x.im)): 33 points increase in error, 33 points decrease in error
   (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 -3 x.re)) x.im): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*r*_binary64 (*.f64 -3 (*.f64 x.re x.im))): 39 points increase in error, 37 points decrease in error

8. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.7
Cost	1352

\[\begin{array}{l} t_0 := \left(x.im \cdot -3\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{if}\;x.im \leq -5.658254694694056 \cdot 10^{+35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 10^{+154}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.7
Cost	968

\[\begin{array}{l} t_0 := \left(x.im \cdot -3\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{if}\;x.im \leq -3.333978339765394 \cdot 10^{+36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 10^{+154}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + -3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	47.0
Cost	448

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

Alternative 4
Error	19.3
Cost	448

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

Reproduce

herbie shell --seed 2022294 
(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)))