math.log10 on complex, real part

Percentage Accurate: 52.0% → 99.1%

Time: 8.5s

Alternatives: 6

Speedup: 0.8×

Specification

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))

C

double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(sqrt(((re * re) + (im * im)))) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.log(Math.sqrt(((re * re) + (im * im)))) / Math.log(10.0);
}

Python

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Initial Program: 52.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))

C

double code(double re, double im) {
	return log(sqrt(((re * re) + (im * im)))) / log(10.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(sqrt(((re * re) + (im * im)))) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.log(Math.sqrt(((re * re) + (im * im)))) / Math.log(10.0);
}

Python

def code(re, im):
	return math.log(math.sqrt(((re * re) + (im * im)))) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(sqrt(Float64(Float64(re * re) + Float64(im * im)))) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = log(sqrt(((re * re) + (im * im)))) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (log (hypot re im)) (log 10.0)))

C

double code(double re, double im) {
	return log(hypot(re, im)) / log(10.0);
}

Java

public static double code(double re, double im) {
	return Math.log(Math.hypot(re, im)) / Math.log(10.0);
}

Python

def code(re, im):
	return math.log(math.hypot(re, im)) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(hypot(re, im)) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = log(hypot(re, im)) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[Log[N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\log \color{blue}{\left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log 10} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re + im \cdot im}}\right)}{\log 10} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re} + im \cdot im}\right)}{\log 10} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}}\right)}{\log 10} \]

   5. lower-hypot.f64 99.1

      \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

4. Applied rewrites 99.1%

   \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]
5. Add Preprocessing

Alternative 2: 25.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \log im\right) \cdot 0.5}{\log 10} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (* (fma (/ re im) (/ re im) (* 2.0 (log im))) 0.5) (log 10.0)))

C

double code(double re, double im) {
	return (fma((re / im), (re / im), (2.0 * log(im))) * 0.5) / log(10.0);
}

Julia

function code(re, im)
	return Float64(Float64(fma(Float64(re / im), Float64(re / im), Float64(2.0 * log(im))) * 0.5) / log(10.0))
end

Wolfram

code[re_, im_] := N[(N[(N[(N[(re / im), $MachinePrecision] * N[(re / im), $MachinePrecision] + N[(2.0 * N[Log[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing

4. Applied rewrites 54.5%

   \[\leadsto \frac{\color{blue}{\log \left(\mathsf{fma}\left(im, im, re \cdot re\right)\right) \cdot 0.5}}{\log 10} \]

5. Taylor expanded in im around inf

   \[\leadsto \frac{\color{blue}{\left(-2 \cdot \log \left(\frac{1}{im}\right) + \frac{{re}^{2}}{{im}^{2}}\right)} \cdot \frac{1}{2}}{\log 10} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{{re}^{2}}{{im}^{2}} + -2 \cdot \log \left(\frac{1}{im}\right)\right)} \cdot \frac{1}{2}}{\log 10} \]

   2. unpow2 N/A

      \[\leadsto \frac{\left(\frac{\color{blue}{re \cdot re}}{{im}^{2}} + -2 \cdot \log \left(\frac{1}{im}\right)\right) \cdot \frac{1}{2}}{\log 10} \]

   3. unpow2 N/A

      \[\leadsto \frac{\left(\frac{re \cdot re}{\color{blue}{im \cdot im}} + -2 \cdot \log \left(\frac{1}{im}\right)\right) \cdot \frac{1}{2}}{\log 10} \]

   4. times-frac N/A

      \[\leadsto \frac{\left(\color{blue}{\frac{re}{im} \cdot \frac{re}{im}} + -2 \cdot \log \left(\frac{1}{im}\right)\right) \cdot \frac{1}{2}}{\log 10} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, -2 \cdot \log \left(\frac{1}{im}\right)\right)} \cdot \frac{1}{2}}{\log 10} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{re}{im}}, \frac{re}{im}, -2 \cdot \log \left(\frac{1}{im}\right)\right) \cdot \frac{1}{2}}{\log 10} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{re}{im}, \color{blue}{\frac{re}{im}}, -2 \cdot \log \left(\frac{1}{im}\right)\right) \cdot \frac{1}{2}}{\log 10} \]

   8. log-rec N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, -2 \cdot \color{blue}{\left(\mathsf{neg}\left(\log im\right)\right)}\right) \cdot \frac{1}{2}}{\log 10} \]

   9. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, -2 \cdot \color{blue}{\left(-1 \cdot \log im\right)}\right) \cdot \frac{1}{2}}{\log 10} \]

   10. associate-*r* N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{\left(-2 \cdot -1\right) \cdot \log im}\right) \cdot \frac{1}{2}}{\log 10} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{2} \cdot \log im\right) \cdot \frac{1}{2}}{\log 10} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, \color{blue}{2 \cdot \log im}\right) \cdot \frac{1}{2}}{\log 10} \]

   13. lower-log.f64 27.7

       \[\leadsto \frac{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \color{blue}{\log im}\right) \cdot 0.5}{\log 10} \]

7. Applied rewrites 27.7%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{re}{im}, \frac{re}{im}, 2 \cdot \log im\right)} \cdot 0.5}{\log 10} \]
8. Add Preprocessing

Alternative 3: 27.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\log im}{\log 10} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (log im) (log 10.0)))

C

double code(double re, double im) {
	return log(im) / log(10.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = log(im) / log(10.0d0)
end function

Java

public static double code(double re, double im) {
	return Math.log(im) / Math.log(10.0);
}

Python

def code(re, im):
	return math.log(im) / math.log(10.0)

Julia

function code(re, im)
	return Float64(log(im) / log(10.0))
end

MATLAB

function tmp = code(re, im)
	tmp = log(im) / log(10.0);
end

Wolfram

code[re_, im_] := N[(N[Log[im], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing

3. Taylor expanded in re around 0

   \[\leadsto \frac{\color{blue}{\log im}}{\log 10} \]
4. Step-by-step derivation

   1. lower-log.f64 29.5

      \[\leadsto \frac{\color{blue}{\log im}}{\log 10} \]

5. Applied rewrites 29.5%

   \[\leadsto \frac{\color{blue}{\log im}}{\log 10} \]
6. Add Preprocessing

Alternative 4: 3.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\frac{re}{im} \cdot re\right) \cdot -0.5}{\log 0.1 \cdot im} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (/ (* (* (/ re im) re) -0.5) (* (log 0.1) im)))

C

double code(double re, double im) {
	return (((re / im) * re) * -0.5) / (log(0.1) * im);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (((re / im) * re) * (-0.5d0)) / (log(0.1d0) * im)
end function

Java

public static double code(double re, double im) {
	return (((re / im) * re) * -0.5) / (Math.log(0.1) * im);
}

Python

def code(re, im):
	return (((re / im) * re) * -0.5) / (math.log(0.1) * im)

Julia

function code(re, im)
	return Float64(Float64(Float64(Float64(re / im) * re) * -0.5) / Float64(log(0.1) * im))
end

MATLAB

function tmp = code(re, im)
	tmp = (((re / im) * re) * -0.5) / (log(0.1) * im);
end

Wolfram

code[re_, im_] := N[(N[(N[(N[(re / im), $MachinePrecision] * re), $MachinePrecision] * -0.5), $MachinePrecision] / N[(N[Log[0.1], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\log \color{blue}{\left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log 10} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re + im \cdot im}}\right)}{\log 10} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re} + im \cdot im}\right)}{\log 10} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}}\right)}{\log 10} \]

   5. lower-hypot.f64 99.1

      \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

4. Applied rewrites 99.1%

   \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

5. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2} \cdot \log 10} + \frac{\log im}{\log 10}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot {re}^{2}}{{im}^{2} \cdot \log 10}} + \frac{\log im}{\log 10} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{2} \cdot {re}^{2}}{\color{blue}{\log 10 \cdot {im}^{2}}} + \frac{\log im}{\log 10} \]

   3. times-frac N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\log 10} \cdot \frac{{re}^{2}}{{im}^{2}}} + \frac{\log im}{\log 10} \]

   4. remove-double-neg N/A

      \[\leadsto \frac{\frac{1}{2}}{\log 10} \cdot \frac{{re}^{2}}{{im}^{2}} + \frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log im\right)\right)\right)}}{\log 10} \]

   5. log-rec N/A

      \[\leadsto \frac{\frac{1}{2}}{\log 10} \cdot \frac{{re}^{2}}{{im}^{2}} + \frac{\mathsf{neg}\left(\color{blue}{\log \left(\frac{1}{im}\right)}\right)}{\log 10} \]

   6. mul-1-neg N/A

      \[\leadsto \frac{\frac{1}{2}}{\log 10} \cdot \frac{{re}^{2}}{{im}^{2}} + \frac{\color{blue}{-1 \cdot \log \left(\frac{1}{im}\right)}}{\log 10} \]

   7. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2}}{\log 10} \cdot \frac{{re}^{2}}{{im}^{2}} + \color{blue}{-1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \frac{{re}^{2}}{{im}^{2}}, -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right)} \]

   9. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{2}}{\log 10}}, \frac{{re}^{2}}{{im}^{2}}, -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right) \]

   10. lower-log.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\color{blue}{\log 10}}, \frac{{re}^{2}}{{im}^{2}}, -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \frac{\color{blue}{re \cdot re}}{{im}^{2}}, -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \frac{re \cdot re}{\color{blue}{im \cdot im}}, -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right) \]

   13. times-frac N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \color{blue}{\frac{re}{im} \cdot \frac{re}{im}}, -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \color{blue}{\frac{re}{im} \cdot \frac{re}{im}}, -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right) \]

   15. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \color{blue}{\frac{re}{im}} \cdot \frac{re}{im}, -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \frac{re}{im} \cdot \color{blue}{\frac{re}{im}}, -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right) \]

   17. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \frac{re}{im} \cdot \frac{re}{im}, \color{blue}{\frac{-1 \cdot \log \left(\frac{1}{im}\right)}{\log 10}}\right) \]

   18. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \frac{re}{im} \cdot \frac{re}{im}, \frac{\color{blue}{\mathsf{neg}\left(\log \left(\frac{1}{im}\right)\right)}}{\log 10}\right) \]

   19. log-rec N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \frac{re}{im} \cdot \frac{re}{im}, \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log im\right)\right)}\right)}{\log 10}\right) \]

   20. remove-double-neg N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \frac{re}{im} \cdot \frac{re}{im}, \frac{\color{blue}{\log im}}{\log 10}\right) \]

7. Applied rewrites 27.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{\log 10}, \frac{re}{im} \cdot \frac{re}{im}, \frac{\log im}{\log 10}\right)} \]

8. Taylor expanded in re around inf

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{{re}^{2}}{{im}^{2} \cdot \log 10}} \]
9. Step-by-step derivation

10. Applied rewrites 3.0%

    \[\leadsto \frac{0.5 \cdot \left(re \cdot re\right)}{\color{blue}{\left(\log 10 \cdot im\right) \cdot im}} \]
11. Step-by-step derivation

12. Applied rewrites 3.5%

    \[\leadsto \frac{\left(\frac{re}{im} \cdot re\right) \cdot -0.5}{\log 0.1 \cdot \color{blue}{im}} \]
13. Add Preprocessing

Alternative 5: 3.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot re}{im} \cdot \frac{re}{\log 10 \cdot im} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (/ (* 0.5 re) im) (/ re (* (log 10.0) im))))

C

double code(double re, double im) {
	return ((0.5 * re) / im) * (re / (log(10.0) * im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = ((0.5d0 * re) / im) * (re / (log(10.0d0) * im))
end function

Java

public static double code(double re, double im) {
	return ((0.5 * re) / im) * (re / (Math.log(10.0) * im));
}

Python

def code(re, im):
	return ((0.5 * re) / im) * (re / (math.log(10.0) * im))

Julia

function code(re, im)
	return Float64(Float64(Float64(0.5 * re) / im) * Float64(re / Float64(log(10.0) * im)))
end

MATLAB

function tmp = code(re, im)
	tmp = ((0.5 * re) / im) * (re / (log(10.0) * im));
end

Wolfram

code[re_, im_] := N[(N[(N[(0.5 * re), $MachinePrecision] / im), $MachinePrecision] * N[(re / N[(N[Log[10.0], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\log \color{blue}{\left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log 10} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re + im \cdot im}}\right)}{\log 10} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re} + im \cdot im}\right)}{\log 10} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}}\right)}{\log 10} \]

   5. lower-hypot.f64 99.1

      \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

4. Applied rewrites 99.1%

   \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

5. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2} \cdot \log 10} + \frac{\log im}{\log 10}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot {re}^{2}}{{im}^{2} \cdot \log 10}} + \frac{\log im}{\log 10} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{2} \cdot {re}^{2}}{\color{blue}{\log 10 \cdot {im}^{2}}} + \frac{\log im}{\log 10} \]

   3. times-frac N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\log 10} \cdot \frac{{re}^{2}}{{im}^{2}}} + \frac{\log im}{\log 10} \]

   4. remove-double-neg N/A

      \[\leadsto \frac{\frac{1}{2}}{\log 10} \cdot \frac{{re}^{2}}{{im}^{2}} + \frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log im\right)\right)\right)}}{\log 10} \]

   5. log-rec N/A

      \[\leadsto \frac{\frac{1}{2}}{\log 10} \cdot \frac{{re}^{2}}{{im}^{2}} + \frac{\mathsf{neg}\left(\color{blue}{\log \left(\frac{1}{im}\right)}\right)}{\log 10} \]

   6. mul-1-neg N/A

      \[\leadsto \frac{\frac{1}{2}}{\log 10} \cdot \frac{{re}^{2}}{{im}^{2}} + \frac{\color{blue}{-1 \cdot \log \left(\frac{1}{im}\right)}}{\log 10} \]

   7. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2}}{\log 10} \cdot \frac{{re}^{2}}{{im}^{2}} + \color{blue}{-1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \frac{{re}^{2}}{{im}^{2}}, -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right)} \]

   9. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{2}}{\log 10}}, \frac{{re}^{2}}{{im}^{2}}, -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right) \]

   10. lower-log.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\color{blue}{\log 10}}, \frac{{re}^{2}}{{im}^{2}}, -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \frac{\color{blue}{re \cdot re}}{{im}^{2}}, -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \frac{re \cdot re}{\color{blue}{im \cdot im}}, -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right) \]

   13. times-frac N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \color{blue}{\frac{re}{im} \cdot \frac{re}{im}}, -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \color{blue}{\frac{re}{im} \cdot \frac{re}{im}}, -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right) \]

   15. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \color{blue}{\frac{re}{im}} \cdot \frac{re}{im}, -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \frac{re}{im} \cdot \color{blue}{\frac{re}{im}}, -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right) \]

   17. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \frac{re}{im} \cdot \frac{re}{im}, \color{blue}{\frac{-1 \cdot \log \left(\frac{1}{im}\right)}{\log 10}}\right) \]

   18. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \frac{re}{im} \cdot \frac{re}{im}, \frac{\color{blue}{\mathsf{neg}\left(\log \left(\frac{1}{im}\right)\right)}}{\log 10}\right) \]

   19. log-rec N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \frac{re}{im} \cdot \frac{re}{im}, \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log im\right)\right)}\right)}{\log 10}\right) \]

   20. remove-double-neg N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \frac{re}{im} \cdot \frac{re}{im}, \frac{\color{blue}{\log im}}{\log 10}\right) \]

7. Applied rewrites 27.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{\log 10}, \frac{re}{im} \cdot \frac{re}{im}, \frac{\log im}{\log 10}\right)} \]

8. Taylor expanded in re around inf

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{{re}^{2}}{{im}^{2} \cdot \log 10}} \]
9. Step-by-step derivation

10. Applied rewrites 3.0%

    \[\leadsto \frac{0.5 \cdot \left(re \cdot re\right)}{\color{blue}{\left(\log 10 \cdot im\right) \cdot im}} \]
11. Step-by-step derivation

12. Applied rewrites 3.5%

    \[\leadsto \frac{0.5 \cdot re}{im} \cdot \frac{re}{\color{blue}{\log 10 \cdot im}} \]
13. Add Preprocessing

Alternative 6: 3.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot re\right) \cdot \frac{re}{\left(\log 10 \cdot im\right) \cdot im} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 re) (/ re (* (* (log 10.0) im) im))))

C

double code(double re, double im) {
	return (0.5 * re) * (re / ((log(10.0) * im) * im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * re) * (re / ((log(10.0d0) * im) * im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * re) * (re / ((Math.log(10.0) * im) * im));
}

Python

def code(re, im):
	return (0.5 * re) * (re / ((math.log(10.0) * im) * im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * re) * Float64(re / Float64(Float64(log(10.0) * im) * im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * re) * (re / ((log(10.0) * im) * im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * re), $MachinePrecision] * N[(re / N[(N[(N[Log[10.0], $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\log \color{blue}{\left(\sqrt{re \cdot re + im \cdot im}\right)}}{\log 10} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re + im \cdot im}}\right)}{\log 10} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{\color{blue}{re \cdot re} + im \cdot im}\right)}{\log 10} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + \color{blue}{im \cdot im}}\right)}{\log 10} \]

   5. lower-hypot.f64 99.1

      \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

4. Applied rewrites 99.1%

   \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

5. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{re}^{2}}{{im}^{2} \cdot \log 10} + \frac{\log im}{\log 10}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot {re}^{2}}{{im}^{2} \cdot \log 10}} + \frac{\log im}{\log 10} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{2} \cdot {re}^{2}}{\color{blue}{\log 10 \cdot {im}^{2}}} + \frac{\log im}{\log 10} \]

   3. times-frac N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\log 10} \cdot \frac{{re}^{2}}{{im}^{2}}} + \frac{\log im}{\log 10} \]

   4. remove-double-neg N/A

      \[\leadsto \frac{\frac{1}{2}}{\log 10} \cdot \frac{{re}^{2}}{{im}^{2}} + \frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log im\right)\right)\right)}}{\log 10} \]

   5. log-rec N/A

      \[\leadsto \frac{\frac{1}{2}}{\log 10} \cdot \frac{{re}^{2}}{{im}^{2}} + \frac{\mathsf{neg}\left(\color{blue}{\log \left(\frac{1}{im}\right)}\right)}{\log 10} \]

   6. mul-1-neg N/A

      \[\leadsto \frac{\frac{1}{2}}{\log 10} \cdot \frac{{re}^{2}}{{im}^{2}} + \frac{\color{blue}{-1 \cdot \log \left(\frac{1}{im}\right)}}{\log 10} \]

   7. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2}}{\log 10} \cdot \frac{{re}^{2}}{{im}^{2}} + \color{blue}{-1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \frac{{re}^{2}}{{im}^{2}}, -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right)} \]

   9. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{2}}{\log 10}}, \frac{{re}^{2}}{{im}^{2}}, -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right) \]

   10. lower-log.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\color{blue}{\log 10}}, \frac{{re}^{2}}{{im}^{2}}, -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \frac{\color{blue}{re \cdot re}}{{im}^{2}}, -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \frac{re \cdot re}{\color{blue}{im \cdot im}}, -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right) \]

   13. times-frac N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \color{blue}{\frac{re}{im} \cdot \frac{re}{im}}, -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \color{blue}{\frac{re}{im} \cdot \frac{re}{im}}, -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right) \]

   15. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \color{blue}{\frac{re}{im}} \cdot \frac{re}{im}, -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \frac{re}{im} \cdot \color{blue}{\frac{re}{im}}, -1 \cdot \frac{\log \left(\frac{1}{im}\right)}{\log 10}\right) \]

   17. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \frac{re}{im} \cdot \frac{re}{im}, \color{blue}{\frac{-1 \cdot \log \left(\frac{1}{im}\right)}{\log 10}}\right) \]

   18. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \frac{re}{im} \cdot \frac{re}{im}, \frac{\color{blue}{\mathsf{neg}\left(\log \left(\frac{1}{im}\right)\right)}}{\log 10}\right) \]

   19. log-rec N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \frac{re}{im} \cdot \frac{re}{im}, \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log im\right)\right)}\right)}{\log 10}\right) \]

   20. remove-double-neg N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{\log 10}, \frac{re}{im} \cdot \frac{re}{im}, \frac{\color{blue}{\log im}}{\log 10}\right) \]

7. Applied rewrites 27.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{\log 10}, \frac{re}{im} \cdot \frac{re}{im}, \frac{\log im}{\log 10}\right)} \]

8. Taylor expanded in re around inf

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{{re}^{2}}{{im}^{2} \cdot \log 10}} \]
9. Step-by-step derivation

10. Applied rewrites 3.0%

    \[\leadsto \frac{0.5 \cdot \left(re \cdot re\right)}{\color{blue}{\left(\log 10 \cdot im\right) \cdot im}} \]
11. Step-by-step derivation

12. Applied rewrites 3.2%

    \[\leadsto \left(0.5 \cdot re\right) \cdot \frac{re}{\color{blue}{\left(\log 10 \cdot im\right) \cdot im}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024329 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))