math.square on complex, real part

Percentage Accurate: 93.5% → 100.0%

Time: 3.7s

Alternatives: 2

Speedup: 1.0×

• 44.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ re \cdot re - im \cdot im \end{array} \]

FPCore

(FPCore re_sqr (re im) :precision binary64 (- (* re re) (* im im)))

C

double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}

Fortran

real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    re_sqr = (re * re) - (im * im)
end function

Java

public static double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}

Python

def re_sqr(re, im):
	return (re * re) - (im * im)

Julia

function re_sqr(re, im)
	return Float64(Float64(re * re) - Float64(im * im))
end

MATLAB

function tmp = re_sqr(re, im)
	tmp = (re * re) - (im * im);
end

Wolfram

re$95$sqr[re_, im_] := N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision]

Initial Program: 93.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ re \cdot re - im \cdot im \end{array} \]

FPCore

(FPCore re_sqr (re im) :precision binary64 (- (* re re) (* im im)))

C

double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}

Fortran

real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    re_sqr = (re * re) - (im * im)
end function

Java

public static double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}

Python

def re_sqr(re, im):
	return (re * re) - (im * im)

Julia

function re_sqr(re, im)
	return Float64(Float64(re * re) - Float64(im * im))
end

MATLAB

function tmp = re_sqr(re, im)
	tmp = (re * re) - (im * im);
end

Wolfram

re$95$sqr[re_, im_] := N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore re_sqr (re im) :precision binary64 (* (+ re im) (- re im)))

C

double re_sqr(double re, double im) {
	return (re + im) * (re - im);
}

Fortran

real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    re_sqr = (re + im) * (re - im)
end function

Java

public static double re_sqr(double re, double im) {
	return (re + im) * (re - im);
}

Python

def re_sqr(re, im):
	return (re + im) * (re - im)

Julia

function re_sqr(re, im)
	return Float64(Float64(re + im) * Float64(re - im))
end

MATLAB

function tmp = re_sqr(re, im)
	tmp = (re + im) * (re - im);
end

Wolfram

re$95$sqr[re_, im_] := N[(N[(re + im), $MachinePrecision] * N[(re - im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.1%

   \[re \cdot re - im \cdot im \]
2. Add Preprocessing
3. Step-by-step derivation

   1. cancel-sign-sub-inv 94.1%

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

   2. *-commutative 94.1%

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

   3. +-commutative 94.1%

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

   4. distribute-rgt-neg-out 94.1%

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

   5. add-cube-cbrt 93.6%

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

   6. distribute-rgt-neg-in 93.6%

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

   7. fma-define 93.6%

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

   8. pow2 93.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{im \cdot im}\right)}^{2}}, -\sqrt[3]{im \cdot im}, re \cdot re\right) \]

   9. cbrt-prod 93.4%

      \[\leadsto \mathsf{fma}\left({\color{blue}{\left(\sqrt[3]{im} \cdot \sqrt[3]{im}\right)}}^{2}, -\sqrt[3]{im \cdot im}, re \cdot re\right) \]

   10. pow2 93.4%

       \[\leadsto \mathsf{fma}\left({\color{blue}{\left({\left(\sqrt[3]{im}\right)}^{2}\right)}}^{2}, -\sqrt[3]{im \cdot im}, re \cdot re\right) \]

   11. pow-pow 93.4%

       \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{im}\right)}^{\left(2 \cdot 2\right)}}, -\sqrt[3]{im \cdot im}, re \cdot re\right) \]

   12. metadata-eval 93.4%

       \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{im}\right)}^{\color{blue}{4}}, -\sqrt[3]{im \cdot im}, re \cdot re\right) \]

   13. cbrt-prod 96.4%

       \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{im}\right)}^{4}, -\color{blue}{\sqrt[3]{im} \cdot \sqrt[3]{im}}, re \cdot re\right) \]

   14. pow2 96.4%

       \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{im}\right)}^{4}, -\color{blue}{{\left(\sqrt[3]{im}\right)}^{2}}, re \cdot re\right) \]

   15. pow2 96.4%

       \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{im}\right)}^{4}, -{\left(\sqrt[3]{im}\right)}^{2}, \color{blue}{{re}^{2}}\right) \]

4. Applied egg-rr 96.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{im}\right)}^{4}, -{\left(\sqrt[3]{im}\right)}^{2}, {re}^{2}\right)} \]
5. Step-by-step derivation

   1. fma-undefine 93.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{im}\right)}^{4} \cdot \left(-{\left(\sqrt[3]{im}\right)}^{2}\right) + {re}^{2}} \]

   2. *-commutative 93.2%

      \[\leadsto \color{blue}{\left(-{\left(\sqrt[3]{im}\right)}^{2}\right) \cdot {\left(\sqrt[3]{im}\right)}^{4}} + {re}^{2} \]

   3. +-commutative 93.2%

      \[\leadsto \color{blue}{{re}^{2} + \left(-{\left(\sqrt[3]{im}\right)}^{2}\right) \cdot {\left(\sqrt[3]{im}\right)}^{4}} \]

   4. remove-double-neg 93.2%

      \[\leadsto {re}^{2} + \color{blue}{\left(-\left(-\left(-{\left(\sqrt[3]{im}\right)}^{2}\right) \cdot {\left(\sqrt[3]{im}\right)}^{4}\right)\right)} \]

   5. unsub-neg 93.2%

      \[\leadsto \color{blue}{{re}^{2} - \left(-\left(-{\left(\sqrt[3]{im}\right)}^{2}\right) \cdot {\left(\sqrt[3]{im}\right)}^{4}\right)} \]

   6. *-commutative 93.2%

      \[\leadsto {re}^{2} - \left(-\color{blue}{{\left(\sqrt[3]{im}\right)}^{4} \cdot \left(-{\left(\sqrt[3]{im}\right)}^{2}\right)}\right) \]

   7. distribute-rgt-neg-in 93.2%

      \[\leadsto {re}^{2} - \color{blue}{{\left(\sqrt[3]{im}\right)}^{4} \cdot \left(-\left(-{\left(\sqrt[3]{im}\right)}^{2}\right)\right)} \]

   8. remove-double-neg 93.2%

      \[\leadsto {re}^{2} - {\left(\sqrt[3]{im}\right)}^{4} \cdot \color{blue}{{\left(\sqrt[3]{im}\right)}^{2}} \]

   9. unpow2 93.2%

      \[\leadsto {re}^{2} - {\left(\sqrt[3]{im}\right)}^{4} \cdot \color{blue}{\left(\sqrt[3]{im} \cdot \sqrt[3]{im}\right)} \]

   10. associate-*r* 93.3%

       \[\leadsto {re}^{2} - \color{blue}{\left({\left(\sqrt[3]{im}\right)}^{4} \cdot \sqrt[3]{im}\right) \cdot \sqrt[3]{im}} \]

   11. pow-plus 93.3%

       \[\leadsto {re}^{2} - \color{blue}{{\left(\sqrt[3]{im}\right)}^{\left(4 + 1\right)}} \cdot \sqrt[3]{im} \]

   12. metadata-eval 93.3%

       \[\leadsto {re}^{2} - {\left(\sqrt[3]{im}\right)}^{\color{blue}{5}} \cdot \sqrt[3]{im} \]

6. Simplified 93.3%

   \[\leadsto \color{blue}{{re}^{2} - {\left(\sqrt[3]{im}\right)}^{5} \cdot \sqrt[3]{im}} \]
7. Step-by-step derivation

   1. pow-plus 93.2%

      \[\leadsto {re}^{2} - \color{blue}{{\left(\sqrt[3]{im}\right)}^{\left(5 + 1\right)}} \]

   2. pow1/3 48.3%

      \[\leadsto {re}^{2} - {\color{blue}{\left({im}^{0.3333333333333333}\right)}}^{\left(5 + 1\right)} \]

   3. metadata-eval 48.3%

      \[\leadsto {re}^{2} - {\left({im}^{0.3333333333333333}\right)}^{\color{blue}{6}} \]

   4. metadata-eval 48.3%

      \[\leadsto {re}^{2} - {\left({im}^{0.3333333333333333}\right)}^{\color{blue}{\left(2 + 4\right)}} \]

   5. pow-pow 94.1%

      \[\leadsto {re}^{2} - \color{blue}{{im}^{\left(0.3333333333333333 \cdot \left(2 + 4\right)\right)}} \]

   6. metadata-eval 94.1%

      \[\leadsto {re}^{2} - {im}^{\left(0.3333333333333333 \cdot \color{blue}{6}\right)} \]

   7. metadata-eval 94.1%

      \[\leadsto {re}^{2} - {im}^{\color{blue}{2}} \]

   8. unpow2 94.1%

      \[\leadsto {re}^{2} - \color{blue}{im \cdot im} \]

   9. unpow2 94.1%

      \[\leadsto \color{blue}{re \cdot re} - im \cdot im \]

   10. difference-of-squares 100.0%

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

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 53.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore re_sqr (re im) :precision binary64 (* (+ re im) (+ re im)))

C

double re_sqr(double re, double im) {
	return (re + im) * (re + im);
}

Fortran

real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    re_sqr = (re + im) * (re + im)
end function

Java

public static double re_sqr(double re, double im) {
	return (re + im) * (re + im);
}

Python

def re_sqr(re, im):
	return (re + im) * (re + im)

Julia

function re_sqr(re, im)
	return Float64(Float64(re + im) * Float64(re + im))
end

MATLAB

function tmp = re_sqr(re, im)
	tmp = (re + im) * (re + im);
end

Wolfram

re$95$sqr[re_, im_] := N[(N[(re + im), $MachinePrecision] * N[(re + im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.1%

   \[re \cdot re - im \cdot im \]
2. Add Preprocessing
3. Step-by-step derivation

   1. difference-of-squares 100.0%

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

   2. sub-neg 100.0%

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

   3. add-sqr-sqrt 48.3%

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

   4. sqrt-unprod 75.5%

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

   5. sqr-neg 75.5%

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

   6. sqrt-prod 28.2%

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

   7. add-sqr-sqrt 55.2%

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

4. Applied egg-rr 55.2%

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

5. Final simplification 55.2%

   \[\leadsto \left(re + im\right) \cdot \left(re + im\right) \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))