math.cube on complex, imaginary part

Percentage Accurate: 83.6% → 79.1%

Time: 8.0s

Alternatives: 4

Speedup: N/A×

• 44.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \end{array} \]

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)))

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);
}

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_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Python

def code(x_46_re, 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)

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

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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
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]

Initial Program: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \end{array} \]

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)))

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);
}

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_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Python

def code(x_46_re, 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)

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

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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
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]

Alternative 1: 79.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re - x.im\right) \cdot \left(x.im \cdot \left(x.re + x.im\right)\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (* (- x.re x.im) (* x.im (+ x.re x.im))))

C

double code(double x_46_re, double x_46_im) {
	return (x_46_re - x_46_im) * (x_46_im * (x_46_re + x_46_im));
}

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_46im) * (x_46im * (x_46re + x_46im))
end function

Java

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

Python

def code(x_46_re, x_46_im):
	return (x_46_re - x_46_im) * (x_46_im * (x_46_re + x_46_im))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(x_46_re - x_46_im) * Float64(x_46_im * Float64(x_46_re + x_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = (x_46_re - x_46_im) * (x_46_im * (x_46_re + x_46_im));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$im * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.7%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
2. Add Preprocessing
3. Step-by-step derivation

   1. expm1-log1p-u 65.9%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right)\right)} \]

   2. expm1-udef 56.3%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right)} - 1\right)} \]

   3. *-commutative 56.3%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(\color{blue}{x.re \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)}\right)} - 1\right) \]

   4. *-commutative 56.3%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right)\right)} - 1\right) \]

   5. count-2 56.3%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(2 \cdot \left(x.re \cdot x.im\right)\right)}\right)} - 1\right) \]

   6. *-commutative 56.3%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)}\right)} - 1\right) \]

   7. associate-*r* 56.3%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(x.re \cdot \left(x.im \cdot 2\right)\right)}\right)} - 1\right) \]

   8. associate-*r* 56.3%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)}\right)} - 1\right) \]

   9. *-commutative 56.3%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(2 \cdot \left(x.re \cdot x.im\right)\right)}\right)} - 1\right) \]

   10. count-2 56.3%

       \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(x.re \cdot x.im + x.re \cdot x.im\right)}\right)} - 1\right) \]

   11. flip-+ 0.0%

       \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{x.re \cdot x.im - x.re \cdot x.im}}\right)} - 1\right) \]

   12. +-inverses 0.0%

       \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im}\right)} - 1\right) \]

   13. +-inverses 0.0%

       \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \frac{0}{\color{blue}{0}}\right)} - 1\right) \]

4. Applied egg-rr 0.0%

   \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(x.re \cdot \frac{0}{0}\right)} - 1\right)} \]

6. Simplified 72.2%

   \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{0} \]
7. Step-by-step derivation

   1. difference-of-squares 80.1%

      \[\leadsto \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)} \cdot x.im + 0 \]

   2. *-commutative 80.1%

      \[\leadsto \color{blue}{\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)} \cdot x.im + 0 \]

8. Applied egg-rr 80.1%

   \[\leadsto \color{blue}{\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)} \cdot x.im + 0 \]
9. Step-by-step derivation

   1. *-commutative 80.1%

      \[\leadsto \color{blue}{x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)} + 0 \]

   2. distribute-rgt-in 70.3%

      \[\leadsto x.im \cdot \color{blue}{\left(x.re \cdot \left(x.re - x.im\right) + x.im \cdot \left(x.re - x.im\right)\right)} + 0 \]

   3. distribute-lft-in 64.4%

      \[\leadsto \color{blue}{\left(x.im \cdot \left(x.re \cdot \left(x.re - x.im\right)\right) + x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\right)} + 0 \]

10. Applied egg-rr 64.4%

    \[\leadsto \color{blue}{\left(x.im \cdot \left(x.re \cdot \left(x.re - x.im\right)\right) + x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\right)} + 0 \]

12. Simplified 70.3%

    \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - x.im\right) + x.im \cdot \left(x.re - x.im\right)\right)} + 0 \]
13. Step-by-step derivation

    1. distribute-lft-in 64.4%

       \[\leadsto \color{blue}{\left(x.im \cdot \left(x.re \cdot \left(x.re - x.im\right)\right) + x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\right)} + 0 \]

    2. *-commutative 64.4%

       \[\leadsto \left(x.im \cdot \left(x.re \cdot \left(x.re - x.im\right)\right) + x.im \cdot \color{blue}{\left(\left(x.re - x.im\right) \cdot x.im\right)}\right) + 0 \]

    3. associate-*r* 64.4%

       \[\leadsto \left(x.im \cdot \left(x.re \cdot \left(x.re - x.im\right)\right) + \color{blue}{\left(x.im \cdot \left(x.re - x.im\right)\right) \cdot x.im}\right) + 0 \]

    4. *-commutative 64.4%

       \[\leadsto \left(x.im \cdot \left(x.re \cdot \left(x.re - x.im\right)\right) + \color{blue}{\left(\left(x.re - x.im\right) \cdot x.im\right)} \cdot x.im\right) + 0 \]

    5. associate-*l* 64.4%

       \[\leadsto \left(x.im \cdot \left(x.re \cdot \left(x.re - x.im\right)\right) + \color{blue}{\left(x.re - x.im\right) \cdot \left(x.im \cdot x.im\right)}\right) + 0 \]

    6. unpow2 64.4%

       \[\leadsto \left(x.im \cdot \left(x.re \cdot \left(x.re - x.im\right)\right) + \left(x.re - x.im\right) \cdot \color{blue}{{x.im}^{2}}\right) + 0 \]

14. Applied egg-rr 64.4%

    \[\leadsto \color{blue}{\left(x.im \cdot \left(x.re \cdot \left(x.re - x.im\right)\right) + \left(x.re - x.im\right) \cdot {x.im}^{2}\right)} + 0 \]
15. Step-by-step derivation

    1. associate-*r* 65.0%

       \[\leadsto \left(\color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.re - x.im\right)} + \left(x.re - x.im\right) \cdot {x.im}^{2}\right) + 0 \]

    2. *-commutative 65.0%

       \[\leadsto \left(\left(x.im \cdot x.re\right) \cdot \left(x.re - x.im\right) + \color{blue}{{x.im}^{2} \cdot \left(x.re - x.im\right)}\right) + 0 \]

    3. distribute-rgt-in 73.6%

       \[\leadsto \color{blue}{\left(x.re - x.im\right) \cdot \left(x.im \cdot x.re + {x.im}^{2}\right)} + 0 \]

    4. unpow2 73.6%

       \[\leadsto \left(x.re - x.im\right) \cdot \left(x.im \cdot x.re + \color{blue}{x.im \cdot x.im}\right) + 0 \]

    5. distribute-lft-in 80.6%

       \[\leadsto \left(x.re - x.im\right) \cdot \color{blue}{\left(x.im \cdot \left(x.re + x.im\right)\right)} + 0 \]

    6. +-commutative 80.6%

       \[\leadsto \left(x.re - x.im\right) \cdot \left(x.im \cdot \color{blue}{\left(x.im + x.re\right)}\right) + 0 \]

16. Simplified 80.6%

    \[\leadsto \color{blue}{\left(x.re - x.im\right) \cdot \left(x.im \cdot \left(x.im + x.re\right)\right)} + 0 \]

17. Final simplification 80.6%

    \[\leadsto \left(x.re - x.im\right) \cdot \left(x.im \cdot \left(x.re + x.im\right)\right) \]
18. Add Preprocessing

Alternative 2: 78.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (* x.im (* (- x.re x.im) (+ x.re x.im))))

C

double code(double x_46_re, double x_46_im) {
	return x_46_im * ((x_46_re - x_46_im) * (x_46_re + x_46_im));
}

Fortran

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

Java

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

Python

def code(x_46_re, x_46_im):
	return 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(x_46_im * Float64(Float64(x_46_re - x_46_im) * Float64(x_46_re + x_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_re + x_46_im));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(x$46$im * N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.7%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
2. Add Preprocessing
3. Step-by-step derivation

   1. expm1-log1p-u 65.9%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right)\right)} \]

   2. expm1-udef 56.3%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right)} - 1\right)} \]

   3. *-commutative 56.3%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(\color{blue}{x.re \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)}\right)} - 1\right) \]

   4. *-commutative 56.3%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right)\right)} - 1\right) \]

   5. count-2 56.3%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(2 \cdot \left(x.re \cdot x.im\right)\right)}\right)} - 1\right) \]

   6. *-commutative 56.3%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)}\right)} - 1\right) \]

   7. associate-*r* 56.3%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(x.re \cdot \left(x.im \cdot 2\right)\right)}\right)} - 1\right) \]

   8. associate-*r* 56.3%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)}\right)} - 1\right) \]

   9. *-commutative 56.3%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(2 \cdot \left(x.re \cdot x.im\right)\right)}\right)} - 1\right) \]

   10. count-2 56.3%

       \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\left(x.re \cdot x.im + x.re \cdot x.im\right)}\right)} - 1\right) \]

   11. flip-+ 0.0%

       \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{\frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{x.re \cdot x.im - x.re \cdot x.im}}\right)} - 1\right) \]

   12. +-inverses 0.0%

       \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im}\right)} - 1\right) \]

   13. +-inverses 0.0%

       \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(x.re \cdot \frac{0}{\color{blue}{0}}\right)} - 1\right) \]

4. Applied egg-rr 0.0%

   \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(x.re \cdot \frac{0}{0}\right)} - 1\right)} \]

6. Simplified 72.2%

   \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{0} \]
7. Step-by-step derivation

   1. difference-of-squares 80.1%

      \[\leadsto \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)} \cdot x.im + 0 \]

   2. *-commutative 80.1%

      \[\leadsto \color{blue}{\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)} \cdot x.im + 0 \]

8. Applied egg-rr 80.1%

   \[\leadsto \color{blue}{\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)} \cdot x.im + 0 \]

9. Final simplification 80.1%

   \[\leadsto x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \]
10. Add Preprocessing

Alternative 3: 55.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right) \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 (* x.re (* 3.0 (* x.re x.im))))

C

double code(double x_46_re, double x_46_im) {
	return x_46_re * (3.0 * (x_46_re * x_46_im));
}

Fortran

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_46re * x_46im))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return x_46_re * (3.0 * (x_46_re * x_46_im));
}

Python

def code(x_46_re, x_46_im):
	return x_46_re * (3.0 * (x_46_re * x_46_im))

Julia

function code(x_46_re, x_46_im)
	return Float64(x_46_re * Float64(3.0 * Float64(x_46_re * x_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = x_46_re * (3.0 * (x_46_re * x_46_im));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(x$46$re * N[(3.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.7%

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

3. Taylor expanded in x.re around inf 51.6%

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

5. Simplified 51.6%

   \[\leadsto \color{blue}{{x.re}^{2} \cdot \left(x.im \cdot 3\right)} \]
6. Step-by-step derivation

   1. add-sqr-sqrt 24.6%

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

   2. sqrt-unprod 24.6%

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

   3. sqrt-unprod 24.6%

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

   4. unpow2 24.6%

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

   5. swap-sqr 27.1%

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

   6. pow2 27.1%

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

   7. sqrt-unprod 27.1%

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

7. Applied egg-rr 27.1%

   \[\leadsto \color{blue}{{\left(x.re \cdot \sqrt{x.im \cdot 3}\right)}^{2}} \]
8. Step-by-step derivation

   1. unpow2 27.1%

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

   2. *-commutative 27.1%

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

   3. *-commutative 27.1%

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

   4. swap-sqr 24.6%

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

   5. add-sqr-sqrt 51.6%

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

   6. associate-*r* 56.3%

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

9. Applied egg-rr 56.3%

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

10. Taylor expanded in x.im around 0 56.3%

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

11. Final simplification 56.3%

    \[\leadsto x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right) \]
12. Add Preprocessing

Alternative 4: 55.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right) \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 (* x.re (* x.im (* x.re 3.0))))

C

double code(double x_46_re, double x_46_im) {
	return x_46_re * (x_46_im * (x_46_re * 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_46im * (x_46re * 3.0d0))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return x_46_re * (x_46_im * (x_46_re * 3.0));
}

Python

def code(x_46_re, x_46_im):
	return x_46_re * (x_46_im * (x_46_re * 3.0))

Julia

function code(x_46_re, x_46_im)
	return Float64(x_46_re * Float64(x_46_im * Float64(x_46_re * 3.0)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = x_46_re * (x_46_im * (x_46_re * 3.0));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(x$46$re * N[(x$46$im * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.7%

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

3. Taylor expanded in x.re around inf 51.6%

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

5. Simplified 51.6%

   \[\leadsto \color{blue}{{x.re}^{2} \cdot \left(x.im \cdot 3\right)} \]
6. Step-by-step derivation

   1. add-sqr-sqrt 24.6%

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

   2. sqrt-unprod 24.6%

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

   3. sqrt-unprod 24.6%

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

   4. unpow2 24.6%

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

   5. swap-sqr 27.1%

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

   6. pow2 27.1%

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

   7. sqrt-unprod 27.1%

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

7. Applied egg-rr 27.1%

   \[\leadsto \color{blue}{{\left(x.re \cdot \sqrt{x.im \cdot 3}\right)}^{2}} \]
8. Step-by-step derivation

   1. unpow2 27.1%

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

   2. *-commutative 27.1%

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

   3. *-commutative 27.1%

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

   4. swap-sqr 24.6%

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

   5. add-sqr-sqrt 51.6%

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

   6. associate-*r* 56.3%

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

9. Applied egg-rr 56.3%

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

10. Taylor expanded in x.im around 0 56.3%

    \[\leadsto \color{blue}{\left(3 \cdot \left(x.im \cdot x.re\right)\right)} \cdot x.re \]
11. Step-by-step derivation

    1. associate-*r* 56.3%

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

    2. *-commutative 56.3%

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

    3. associate-*l* 56.3%

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

12. Simplified 56.3%

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

13. Final simplification 56.3%

    \[\leadsto x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right) \]
14. Add Preprocessing

Developer target: 92.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* 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_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

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_46im) * (2.0d0 * x_46re)) + ((x_46im * (x_46re - x_46im)) * (x_46re + x_46im))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return ((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));
}

Python

def code(x_46_re, x_46_im):
	return ((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(x_46_re * x_46_im) * Float64(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

MATLAB

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

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(2.0 * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024033 
(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)))