math.log10 on complex, real part

Percentage Accurate: 51.1% → 99.0%

Time: 7.1s

Alternatives: 4

Speedup: 1.1×

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: 51.1% 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: 99.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ \begin{array}{l} t_0 := \sqrt{re\_m + im\_m}\\ t_1 := \log t\_0\\ t_2 := \log \left(\frac{\mathsf{hypot}\left(im\_m, re\_m\right)}{t\_0}\right)\\ \frac{{t\_1}^{3} + {t\_2}^{3}}{\left(\left(t\_1 - t\_2\right) \cdot t\_1 + {t\_2}^{2}\right) \cdot \log 10} \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
(FPCore (re_m im_m)
 :precision binary64
 (let* ((t_0 (sqrt (+ re_m im_m)))
        (t_1 (log t_0))
        (t_2 (log (/ (hypot im_m re_m) t_0))))
   (/
    (+ (pow t_1 3.0) (pow t_2 3.0))
    (* (+ (* (- t_1 t_2) t_1) (pow t_2 2.0)) (log 10.0)))))

C

im_m = fabs(im);
re_m = fabs(re);
double code(double re_m, double im_m) {
	double t_0 = sqrt((re_m + im_m));
	double t_1 = log(t_0);
	double t_2 = log((hypot(im_m, re_m) / t_0));
	return (pow(t_1, 3.0) + pow(t_2, 3.0)) / ((((t_1 - t_2) * t_1) + pow(t_2, 2.0)) * log(10.0));
}

Java

im_m = Math.abs(im);
re_m = Math.abs(re);
public static double code(double re_m, double im_m) {
	double t_0 = Math.sqrt((re_m + im_m));
	double t_1 = Math.log(t_0);
	double t_2 = Math.log((Math.hypot(im_m, re_m) / t_0));
	return (Math.pow(t_1, 3.0) + Math.pow(t_2, 3.0)) / ((((t_1 - t_2) * t_1) + Math.pow(t_2, 2.0)) * Math.log(10.0));
}

Python

im_m = math.fabs(im)
re_m = math.fabs(re)
def code(re_m, im_m):
	t_0 = math.sqrt((re_m + im_m))
	t_1 = math.log(t_0)
	t_2 = math.log((math.hypot(im_m, re_m) / t_0))
	return (math.pow(t_1, 3.0) + math.pow(t_2, 3.0)) / ((((t_1 - t_2) * t_1) + math.pow(t_2, 2.0)) * math.log(10.0))

Julia

im_m = abs(im)
re_m = abs(re)
function code(re_m, im_m)
	t_0 = sqrt(Float64(re_m + im_m))
	t_1 = log(t_0)
	t_2 = log(Float64(hypot(im_m, re_m) / t_0))
	return Float64(Float64((t_1 ^ 3.0) + (t_2 ^ 3.0)) / Float64(Float64(Float64(Float64(t_1 - t_2) * t_1) + (t_2 ^ 2.0)) * log(10.0)))
end

MATLAB

im_m = abs(im);
re_m = abs(re);
function tmp = code(re_m, im_m)
	t_0 = sqrt((re_m + im_m));
	t_1 = log(t_0);
	t_2 = log((hypot(im_m, re_m) / t_0));
	tmp = ((t_1 ^ 3.0) + (t_2 ^ 3.0)) / ((((t_1 - t_2) * t_1) + (t_2 ^ 2.0)) * log(10.0));
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $MachinePrecision]
code[re$95$m_, im$95$m_] := Block[{t$95$0 = N[Sqrt[N[(re$95$m + im$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Log[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Log[N[(N[Sqrt[im$95$m ^ 2 + re$95$m ^ 2], $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Power[t$95$1, 3.0], $MachinePrecision] + N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(t$95$1 - t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] * N[Log[10.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 50.0%

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

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

6. Applied rewrites 48.0%

   \[\leadsto \color{blue}{\frac{{\log \left(\frac{\mathsf{hypot}\left(im, re\right)}{\sqrt{im + re}}\right)}^{3} + {\log \left(\sqrt{im + re}\right)}^{3}}{\log 10 \cdot \left({\log \left(\frac{\mathsf{hypot}\left(im, re\right)}{\sqrt{im + re}}\right)}^{2} + \log \left(\sqrt{im + re}\right) \cdot \left(\log \left(\sqrt{im + re}\right) - \log \left(\frac{\mathsf{hypot}\left(im, re\right)}{\sqrt{im + re}}\right)\right)\right)}} \]

7. Final simplification 48.0%

   \[\leadsto \frac{{\log \left(\sqrt{re + im}\right)}^{3} + {\log \left(\frac{\mathsf{hypot}\left(im, re\right)}{\sqrt{re + im}}\right)}^{3}}{\left(\left(\log \left(\sqrt{re + im}\right) - \log \left(\frac{\mathsf{hypot}\left(im, re\right)}{\sqrt{re + im}}\right)\right) \cdot \log \left(\sqrt{re + im}\right) + {\log \left(\frac{\mathsf{hypot}\left(im, re\right)}{\sqrt{re + im}}\right)}^{2}\right) \cdot \log 10} \]
8. Add Preprocessing

Alternative 2: 99.1% accurate, 0.8× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
(FPCore (re_m im_m) :precision binary64 (/ (log (hypot re_m im_m)) (log 10.0)))

C

im_m = fabs(im);
re_m = fabs(re);
double code(double re_m, double im_m) {
	return log(hypot(re_m, im_m)) / log(10.0);
}

Java

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

Python

im_m = math.fabs(im)
re_m = math.fabs(re)
def code(re_m, im_m):
	return math.log(math.hypot(re_m, im_m)) / math.log(10.0)

Julia

im_m = abs(im)
re_m = abs(re)
function code(re_m, im_m)
	return Float64(log(hypot(re_m, im_m)) / log(10.0))
end

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $MachinePrecision]
code[re$95$m_, im$95$m_] := N[(N[Log[N[Sqrt[re$95$m ^ 2 + im$95$m ^ 2], $MachinePrecision]], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.0%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 3: 50.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ \frac{\mathsf{fma}\left(\frac{re\_m}{im\_m} \cdot \frac{re\_m}{im\_m}, 0.5, \log im\_m\right)}{-\log 0.1} \end{array} \]

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
(FPCore (re_m im_m)
 :precision binary64
 (/ (fma (* (/ re_m im_m) (/ re_m im_m)) 0.5 (log im_m)) (- (log 0.1))))

C

im_m = fabs(im);
re_m = fabs(re);
double code(double re_m, double im_m) {
	return fma(((re_m / im_m) * (re_m / im_m)), 0.5, log(im_m)) / -log(0.1);
}

Julia

im_m = abs(im)
re_m = abs(re)
function code(re_m, im_m)
	return Float64(fma(Float64(Float64(re_m / im_m) * Float64(re_m / im_m)), 0.5, log(im_m)) / Float64(-log(0.1)))
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $MachinePrecision]
code[re$95$m_, im$95$m_] := N[(N[(N[(N[(re$95$m / im$95$m), $MachinePrecision] * N[(re$95$m / im$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + N[Log[im$95$m], $MachinePrecision]), $MachinePrecision] / (-N[Log[0.1], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 50.0%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 29.4

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

5. Applied rewrites 29.4%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 N/A

      \[\leadsto \frac{\color{blue}{-\log im}}{\mathsf{neg}\left(\log 10\right)} \]

   5. lift-log.f64 N/A

      \[\leadsto \frac{-\log im}{\mathsf{neg}\left(\color{blue}{\log 10}\right)} \]

   6. neg-log N/A

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

   7. lower-log.f64 N/A

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

   8. metadata-eval 29.4

      \[\leadsto \frac{-\log im}{\log \color{blue}{0.1}} \]

7. Applied rewrites 29.4%

   \[\leadsto \color{blue}{\frac{-\log im}{\log 0.1}} \]

8. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-log.f64 27.5

       \[\leadsto \frac{-\mathsf{fma}\left(\frac{re}{im} \cdot \frac{re}{im}, 0.5, \color{blue}{\log im}\right)}{\log 0.1} \]

10. Applied rewrites 27.5%

    \[\leadsto \frac{-\color{blue}{\mathsf{fma}\left(\frac{re}{im} \cdot \frac{re}{im}, 0.5, \log im\right)}}{\log 0.1} \]

11. Final simplification 27.5%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{re}{im} \cdot \frac{re}{im}, 0.5, \log im\right)}{-\log 0.1} \]
12. Add Preprocessing

Alternative 4: 54.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ \frac{\log im\_m}{\log 10} \end{array} \]

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
(FPCore (re_m im_m) :precision binary64 (/ (log im_m) (log 10.0)))

C

im_m = fabs(im);
re_m = fabs(re);
double code(double re_m, double im_m) {
	return log(im_m) / log(10.0);
}

Fortran

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

Java

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

Python

im_m = math.fabs(im)
re_m = math.fabs(re)
def code(re_m, im_m):
	return math.log(im_m) / math.log(10.0)

Julia

im_m = abs(im)
re_m = abs(re)
function code(re_m, im_m)
	return Float64(log(im_m) / log(10.0))
end

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $MachinePrecision]
code[re$95$m_, im$95$m_] := N[(N[Log[im$95$m], $MachinePrecision] / N[Log[10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.0%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
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 29.4

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

5. Applied rewrites 29.4%

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

Reproduce

herbie shell --seed 2024331 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))