math.cube on complex, real part

Percentage Accurate: 83.0% → 58.6%

Time: 6.5s

Alternatives: 7

Speedup: N/A×

• 43.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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)))

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);
}

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

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);
}

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)

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

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

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]

Initial Program: 83.0% accurate, 1.0× speedup

Math

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

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)))

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);
}

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

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);
}

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)

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

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

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]

Alternative 1: 58.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ {x.re}^{3} \end{array} \]

FPCore

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

C

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

Java

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

Python

def code(x_46_re, x_46_im):
	return math.pow(x_46_re, 3.0)

Julia

function code(x_46_re, x_46_im)
	return x_46_re ^ 3.0
end

MATLAB

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

Wolfram

code[x$46$re_, x$46$im_] := N[Power[x$46$re, 3.0], $MachinePrecision]

Derivation

1. Initial program 80.3%

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

3. Simplified 77.6%

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

5. Taylor expanded in x.re around inf 57.7%

   \[\leadsto \color{blue}{{x.re}^{3}} \]

6. Final simplification 57.7%

   \[\leadsto {x.re}^{3} \]
7. Add Preprocessing

Alternative 2: 51.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re + -27\right)\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (* x.re (* (+ x.re x.im) (+ x.re -27.0))))

C

double code(double x_46_re, double x_46_im) {
	return x_46_re * ((x_46_re + x_46_im) * (x_46_re + -27.0));
}

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_46re + (-27.0d0)))
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_re + -27.0));
}

Python

def code(x_46_re, x_46_im):
	return x_46_re * ((x_46_re + x_46_im) * (x_46_re + -27.0))

Julia

function code(x_46_re, x_46_im)
	return Float64(x_46_re * Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re + -27.0)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = x_46_re * ((x_46_re + x_46_im) * (x_46_re + -27.0));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(x$46$re * N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re + -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.3%

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

   1. difference-of-squares 83.4%

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

4. Applied egg-rr 83.4%

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

6. Simplified 50.7%

   \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

7. Taylor expanded in x.im around 0 36.4%

   \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]
8. Step-by-step derivation

   1. sub-neg 36.4%

      \[\leadsto x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

   2. metadata-eval 36.4%

      \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

   3. unpow2 36.4%

      \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re + -27\right)\right) + \color{blue}{\left(x.re \cdot x.re\right)} \cdot \left(x.re - 27\right) \]

   4. sub-neg 36.4%

      \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re + -27\right)\right) + \left(x.re \cdot x.re\right) \cdot \color{blue}{\left(x.re + \left(-27\right)\right)} \]

   5. metadata-eval 36.4%

      \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re + -27\right)\right) + \left(x.re \cdot x.re\right) \cdot \left(x.re + \color{blue}{-27}\right) \]

   6. associate-*l* 36.4%

      \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re + -27\right)\right) + \color{blue}{x.re \cdot \left(x.re \cdot \left(x.re + -27\right)\right)} \]

   7. distribute-rgt-in 52.1%

      \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + -27\right)\right) \cdot \left(x.im + x.re\right)} \]

   8. +-commutative 52.1%

      \[\leadsto \left(x.re \cdot \left(x.re + -27\right)\right) \cdot \color{blue}{\left(x.re + x.im\right)} \]

   9. associate-*r* 52.4%

      \[\leadsto \color{blue}{x.re \cdot \left(\left(x.re + -27\right) \cdot \left(x.re + x.im\right)\right)} \]

   10. *-commutative 52.4%

       \[\leadsto x.re \cdot \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re + -27\right)\right)} \]

   11. +-commutative 52.4%

       \[\leadsto x.re \cdot \left(\color{blue}{\left(x.im + x.re\right)} \cdot \left(x.re + -27\right)\right) \]

9. Simplified 52.4%

   \[\leadsto \color{blue}{x.re \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \]

10. Final simplification 52.4%

    \[\leadsto x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re + -27\right)\right) \]
11. Add Preprocessing

Alternative 3: 35.7% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (x.re x.im) :precision binary64 (* x.im (* x.re (* x.im -2.0))))

C

double code(double x_46_re, double x_46_im) {
	return x_46_im * (x_46_re * (x_46_im * -2.0));
}

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 * (-2.0d0)))
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 * -2.0));
}

Python

def code(x_46_re, x_46_im):
	return x_46_im * (x_46_re * (x_46_im * -2.0))

Julia

function code(x_46_re, x_46_im)
	return Float64(x_46_im * Float64(x_46_re * Float64(x_46_im * -2.0)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = x_46_im * (x_46_re * (x_46_im * -2.0));
end

Wolfram

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

Derivation

1. Initial program 80.3%

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

   1. difference-of-squares 83.4%

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

4. Applied egg-rr 83.4%

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

6. Simplified 50.7%

   \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

7. Taylor expanded in x.re around 0 33.0%

   \[\leadsto \color{blue}{x.re \cdot \left(-27 \cdot x.im - 2 \cdot {x.im}^{2}\right)} \]
8. Step-by-step derivation

   1. cancel-sign-sub-inv 33.0%

      \[\leadsto x.re \cdot \color{blue}{\left(-27 \cdot x.im + \left(-2\right) \cdot {x.im}^{2}\right)} \]

   2. *-commutative 33.0%

      \[\leadsto x.re \cdot \left(\color{blue}{x.im \cdot -27} + \left(-2\right) \cdot {x.im}^{2}\right) \]

   3. metadata-eval 33.0%

      \[\leadsto x.re \cdot \left(x.im \cdot -27 + \color{blue}{-2} \cdot {x.im}^{2}\right) \]

   4. *-commutative 33.0%

      \[\leadsto x.re \cdot \left(x.im \cdot -27 + \color{blue}{{x.im}^{2} \cdot -2}\right) \]

   5. unpow2 33.0%

      \[\leadsto x.re \cdot \left(x.im \cdot -27 + \color{blue}{\left(x.im \cdot x.im\right)} \cdot -2\right) \]

   6. associate-*l* 33.0%

      \[\leadsto x.re \cdot \left(x.im \cdot -27 + \color{blue}{x.im \cdot \left(x.im \cdot -2\right)}\right) \]

   7. distribute-lft-out 33.0%

      \[\leadsto x.re \cdot \color{blue}{\left(x.im \cdot \left(-27 + x.im \cdot -2\right)\right)} \]

9. Simplified 33.0%

   \[\leadsto \color{blue}{x.re \cdot \left(x.im \cdot \left(-27 + x.im \cdot -2\right)\right)} \]

10. Taylor expanded in x.re around 0 33.7%

    \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(-2 \cdot x.im - 27\right)\right)} \]

11. Taylor expanded in x.im around inf 35.1%

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

13. Simplified 35.1%

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

14. Final simplification 35.1%

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

Alternative 4: 28.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 (* x.im (* x.re (- x.re 27.0))))

C

double code(double x_46_re, double x_46_im) {
	return x_46_im * (x_46_re * (x_46_re - 27.0));
}

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_46re - 27.0d0))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return x_46_im * (x_46_re * (x_46_re - 27.0));
}

Python

def code(x_46_re, x_46_im):
	return x_46_im * (x_46_re * (x_46_re - 27.0))

Julia

function code(x_46_re, x_46_im)
	return Float64(x_46_im * Float64(x_46_re * Float64(x_46_re - 27.0)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = x_46_im * (x_46_re * (x_46_re - 27.0));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(x$46$im * N[(x$46$re * N[(x$46$re - 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.3%

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

   1. difference-of-squares 83.4%

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

4. Applied egg-rr 83.4%

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

6. Simplified 50.7%

   \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

7. Taylor expanded in x.im around inf 33.9%

   \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
8. Step-by-step derivation

   1. sub-neg 33.9%

      \[\leadsto x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   2. metadata-eval 33.9%

      \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   3. associate-*r* 31.7%

      \[\leadsto \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.re + -27\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

9. Simplified 31.7%

   \[\leadsto \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.re + -27\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

10. Taylor expanded in x.im around 0 28.2%

    \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right)} \]

11. Final simplification 28.2%

    \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) \]
12. Add Preprocessing

Alternative 5: 19.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ -27 \cdot \left(x.re \cdot x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 (* -27.0 (* x.re x.im)))

C

double code(double x_46_re, double x_46_im) {
	return -27.0 * (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 = (-27.0d0) * (x_46re * x_46im)
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return -27.0 * (x_46_re * x_46_im);
}

Python

def code(x_46_re, x_46_im):
	return -27.0 * (x_46_re * x_46_im)

Julia

function code(x_46_re, x_46_im)
	return Float64(-27.0 * Float64(x_46_re * x_46_im))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = -27.0 * (x_46_re * x_46_im);
end

Wolfram

code[x$46$re_, x$46$im_] := N[(-27.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.3%

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

   1. difference-of-squares 83.4%

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

4. Applied egg-rr 83.4%

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

6. Simplified 50.7%

   \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

7. Taylor expanded in x.re around 0 33.0%

   \[\leadsto \color{blue}{x.re \cdot \left(-27 \cdot x.im - 2 \cdot {x.im}^{2}\right)} \]
8. Step-by-step derivation

   1. cancel-sign-sub-inv 33.0%

      \[\leadsto x.re \cdot \color{blue}{\left(-27 \cdot x.im + \left(-2\right) \cdot {x.im}^{2}\right)} \]

   2. *-commutative 33.0%

      \[\leadsto x.re \cdot \left(\color{blue}{x.im \cdot -27} + \left(-2\right) \cdot {x.im}^{2}\right) \]

   3. metadata-eval 33.0%

      \[\leadsto x.re \cdot \left(x.im \cdot -27 + \color{blue}{-2} \cdot {x.im}^{2}\right) \]

   4. *-commutative 33.0%

      \[\leadsto x.re \cdot \left(x.im \cdot -27 + \color{blue}{{x.im}^{2} \cdot -2}\right) \]

   5. unpow2 33.0%

      \[\leadsto x.re \cdot \left(x.im \cdot -27 + \color{blue}{\left(x.im \cdot x.im\right)} \cdot -2\right) \]

   6. associate-*l* 33.0%

      \[\leadsto x.re \cdot \left(x.im \cdot -27 + \color{blue}{x.im \cdot \left(x.im \cdot -2\right)}\right) \]

   7. distribute-lft-out 33.0%

      \[\leadsto x.re \cdot \color{blue}{\left(x.im \cdot \left(-27 + x.im \cdot -2\right)\right)} \]

9. Simplified 33.0%

   \[\leadsto \color{blue}{x.re \cdot \left(x.im \cdot \left(-27 + x.im \cdot -2\right)\right)} \]

10. Taylor expanded in x.im around 0 22.1%

    \[\leadsto \color{blue}{-27 \cdot \left(x.im \cdot x.re\right)} \]

11. Final simplification 22.1%

    \[\leadsto -27 \cdot \left(x.re \cdot x.im\right) \]
12. Add Preprocessing

Alternative 6: 19.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ x.re \cdot \left(x.im \cdot -27\right) \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 (* x.re (* x.im -27.0)))

C

double code(double x_46_re, double x_46_im) {
	return x_46_re * (x_46_im * -27.0);
}

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 * (-27.0d0))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return x_46_re * (x_46_im * -27.0);
}

Python

def code(x_46_re, x_46_im):
	return x_46_re * (x_46_im * -27.0)

Julia

function code(x_46_re, x_46_im)
	return Float64(x_46_re * Float64(x_46_im * -27.0))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = x_46_re * (x_46_im * -27.0);
end

Wolfram

code[x$46$re_, x$46$im_] := N[(x$46$re * N[(x$46$im * -27.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.3%

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

   1. difference-of-squares 83.4%

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

4. Applied egg-rr 83.4%

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

6. Simplified 50.7%

   \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

7. Taylor expanded in x.re around 0 33.0%

   \[\leadsto \color{blue}{x.re \cdot \left(-27 \cdot x.im - 2 \cdot {x.im}^{2}\right)} \]
8. Step-by-step derivation

   1. cancel-sign-sub-inv 33.0%

      \[\leadsto x.re \cdot \color{blue}{\left(-27 \cdot x.im + \left(-2\right) \cdot {x.im}^{2}\right)} \]

   2. *-commutative 33.0%

      \[\leadsto x.re \cdot \left(\color{blue}{x.im \cdot -27} + \left(-2\right) \cdot {x.im}^{2}\right) \]

   3. metadata-eval 33.0%

      \[\leadsto x.re \cdot \left(x.im \cdot -27 + \color{blue}{-2} \cdot {x.im}^{2}\right) \]

   4. *-commutative 33.0%

      \[\leadsto x.re \cdot \left(x.im \cdot -27 + \color{blue}{{x.im}^{2} \cdot -2}\right) \]

   5. unpow2 33.0%

      \[\leadsto x.re \cdot \left(x.im \cdot -27 + \color{blue}{\left(x.im \cdot x.im\right)} \cdot -2\right) \]

   6. associate-*l* 33.0%

      \[\leadsto x.re \cdot \left(x.im \cdot -27 + \color{blue}{x.im \cdot \left(x.im \cdot -2\right)}\right) \]

   7. distribute-lft-out 33.0%

      \[\leadsto x.re \cdot \color{blue}{\left(x.im \cdot \left(-27 + x.im \cdot -2\right)\right)} \]

9. Simplified 33.0%

   \[\leadsto \color{blue}{x.re \cdot \left(x.im \cdot \left(-27 + x.im \cdot -2\right)\right)} \]

10. Taylor expanded in x.im around 0 22.4%

    \[\leadsto x.re \cdot \color{blue}{\left(-27 \cdot x.im\right)} \]

12. Simplified 22.4%

    \[\leadsto x.re \cdot \color{blue}{\left(x.im \cdot -27\right)} \]

13. Final simplification 22.4%

    \[\leadsto x.re \cdot \left(x.im \cdot -27\right) \]
14. Add Preprocessing

Alternative 7: 3.1% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ -x.re \end{array} \]

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return -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
end function

Java

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

Python

def code(x_46_re, x_46_im):
	return -x_46_re

Julia

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

MATLAB

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

Wolfram

code[x$46$re_, x$46$im_] := (-x$46$re)

Derivation

1. Initial program 80.3%

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

3. Simplified 77.6%

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

5. Taylor expanded in x.re around 0 50.9%

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

7. Simplified 3.2%

   \[\leadsto \color{blue}{-x.re} \]

8. Final simplification 3.2%

   \[\leadsto -x.re \]
9. Add Preprocessing

Developer target: 87.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * 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_46re) * (x_46re - x_46im)) + ((x_46re * x_46im) * (x_46re - (3.0d0 * x_46im)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024033 
(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)))