math.log10 on complex, imaginary part

Average Error: 0.8 → 0.3

Time: 3.5s

Precision: binary64

Cost: 13312

Math

\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10} \]

↓

\[\frac{\frac{1}{\log 0.1}}{\frac{-1}{\tan^{-1}_* \frac{im}{re}}} \]

FPCore

(FPCore (re im) :precision binary64 (/ (atan2 im re) (log 10.0)))

↓

(FPCore (re im)
 :precision binary64
 (/ (/ 1.0 (log 0.1)) (/ -1.0 (atan2 im re))))

C

double code(double re, double im) {
	return atan2(im, re) / log(10.0);
}

↓

double code(double re, double im) {
	return (1.0 / log(0.1)) / (-1.0 / atan2(im, re));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = atan2(im, re) / log(10.0d0)
end function

↓

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (1.0d0 / log(0.1d0)) / ((-1.0d0) / atan2(im, re))
end function

Java

public static double code(double re, double im) {
	return Math.atan2(im, re) / Math.log(10.0);
}

↓

public static double code(double re, double im) {
	return (1.0 / Math.log(0.1)) / (-1.0 / Math.atan2(im, re));
}

Python

def code(re, im):
	return math.atan2(im, re) / math.log(10.0)

↓

def code(re, im):
	return (1.0 / math.log(0.1)) / (-1.0 / math.atan2(im, re))

Julia

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

↓

function code(re, im)
	return Float64(Float64(1.0 / log(0.1)) / Float64(-1.0 / atan(im, re)))
end

MATLAB

function tmp = code(re, im)
	tmp = atan2(im, re) / log(10.0);
end

↓

function tmp = code(re, im)
	tmp = (1.0 / log(0.1)) / (-1.0 / atan2(im, re));
end

Wolfram

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

↓

code[re_, im_] := N[(N[(1.0 / N[Log[0.1], $MachinePrecision]), $MachinePrecision] / N[(-1.0 / N[ArcTan[im / re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.8

   \[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10} \]

2. Applied egg-rr 1.0

   \[\leadsto \color{blue}{{\left(\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}\right)}^{-1}} \]

3. Applied egg-rr 0.3

   \[\leadsto {\color{blue}{\left(\log 0.1 \cdot \frac{1}{-\tan^{-1}_* \frac{im}{re}}\right)}}^{-1} \]

4. Simplified 0.3

   \[\leadsto {\color{blue}{\left(\frac{\log 0.1}{-\tan^{-1}_* \frac{im}{re}}\right)}}^{-1} \]
   Proof
   (/.f64 (log.f64 1/10) (neg.f64 (atan2.f64 im re))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (log.f64 1/10) 1)) (neg.f64 (atan2.f64 im re))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*r/_binary64 (*.f64 (log.f64 1/10) (/.f64 1 (neg.f64 (atan2.f64 im re))))): 7 points increase in error, 6 points decrease in error

5. Applied egg-rr 0.3

   \[\leadsto \color{blue}{\frac{\frac{1}{\log 0.1}}{\frac{-1}{\tan^{-1}_* \frac{im}{re}}}} \]

6. Final simplification 0.3

   \[\leadsto \frac{\frac{1}{\log 0.1}}{\frac{-1}{\tan^{-1}_* \frac{im}{re}}} \]

Alternatives

Alternative 1
Error	0.1
Cost	13120

\[\frac{-\tan^{-1}_* \frac{im}{re}}{\log 0.1} \]

Alternative 2
Error	0.8
Cost	13056

\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10} \]

Reproduce

herbie shell --seed 2022337 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10.0)))