math.log10 on complex, imaginary part

Average Error: 0.8 → 0.1

Time: 20.9s

Precision: binary64

Cost: 26368

Math

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

↓

\[\begin{array}{l} t_0 := -\log 10\\ \frac{\frac{\tan^{-1}_* \frac{im}{re}}{t_0}}{t_0 \cdot \frac{1}{\log 10}} \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (log 10.0))))
   (/ (/ (atan2 im re) t_0) (* t_0 (/ 1.0 (log 10.0))))))

C

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

↓

double code(double re, double im) {
	double t_0 = -log(10.0);
	return (atan2(im, re) / t_0) / (t_0 * (1.0 / log(10.0)));
}

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
    real(8) :: t_0
    t_0 = -log(10.0d0)
    code = (atan2(im, re) / t_0) / (t_0 * (1.0d0 / log(10.0d0)))
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) {
	double t_0 = -Math.log(10.0);
	return (Math.atan2(im, re) / t_0) / (t_0 * (1.0 / Math.log(10.0)));
}

Python

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

↓

def code(re, im):
	t_0 = -math.log(10.0)
	return (math.atan2(im, re) / t_0) / (t_0 * (1.0 / math.log(10.0)))

Julia

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

↓

function code(re, im)
	t_0 = Float64(-log(10.0))
	return Float64(Float64(atan(im, re) / t_0) / Float64(t_0 * Float64(1.0 / log(10.0))))
end

MATLAB

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

↓

function tmp = code(re, im)
	t_0 = -log(10.0);
	tmp = (atan2(im, re) / t_0) / (t_0 * (1.0 / log(10.0)));
end

Wolfram

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

↓

code[re_, im_] := Block[{t$95$0 = (-N[Log[10.0], $MachinePrecision])}, N[(N[(N[ArcTan[im / re], $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 * N[(1.0 / N[Log[10.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 0.8

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

2. Applied egg-rr 9.8

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

3. Simplified 0.9

   \[\leadsto \color{blue}{\log 10 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\frac{\log 10}{\frac{1}{\log 10}}}} \]
   Proof

   [Start] 9.8	\[ \log 10 \cdot \frac{\frac{1}{\tan^{-1}_* \frac{im}{re}}}{\log 10 \cdot \frac{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}{\tan^{-1}_* \frac{im}{re}}} \]
   rational.json-simplify-44 [=>] 9.6	\[ \log 10 \cdot \color{blue}{\frac{\frac{1}{\log 10 \cdot \frac{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}{\tan^{-1}_* \frac{im}{re}}}}{\tan^{-1}_* \frac{im}{re}}} \]
   rational.json-simplify-46 [=>] 10.0	\[ \log 10 \cdot \frac{\color{blue}{\frac{\frac{1}{\log 10}}{\frac{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}{\tan^{-1}_* \frac{im}{re}}}}}{\tan^{-1}_* \frac{im}{re}} \]
   rational.json-simplify-47 [=>] 10.0	\[ \log 10 \cdot \frac{\frac{\frac{1}{\log 10}}{\color{blue}{\frac{\log 10}{\tan^{-1}_* \frac{im}{re} \cdot \tan^{-1}_* \frac{im}{re}}}}}{\tan^{-1}_* \frac{im}{re}} \]
   rational.json-simplify-61 [=>] 9.7	\[ \log 10 \cdot \frac{\color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot \tan^{-1}_* \frac{im}{re}}{\frac{\log 10}{\frac{1}{\log 10}}}}}{\tan^{-1}_* \frac{im}{re}} \]
   rational.json-simplify-44 [=>] 9.3	\[ \log 10 \cdot \color{blue}{\frac{\frac{\tan^{-1}_* \frac{im}{re} \cdot \tan^{-1}_* \frac{im}{re}}{\tan^{-1}_* \frac{im}{re}}}{\frac{\log 10}{\frac{1}{\log 10}}}} \]
   rational.json-simplify-7 [<=] 9.3	\[ \log 10 \cdot \frac{\frac{\tan^{-1}_* \frac{im}{re} \cdot \tan^{-1}_* \frac{im}{re}}{\color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{1}}}}{\frac{\log 10}{\frac{1}{\log 10}}} \]
   rational.json-simplify-35 [=>] 9.3	\[ \log 10 \cdot \frac{\frac{\tan^{-1}_* \frac{im}{re} \cdot \tan^{-1}_* \frac{im}{re}}{\color{blue}{\frac{\tan^{-1}_* \frac{im}{re} + \tan^{-1}_* \frac{im}{re}}{1 + 1}}}}{\frac{\log 10}{\frac{1}{\log 10}}} \]
   metadata-eval [=>] 9.3	\[ \log 10 \cdot \frac{\frac{\tan^{-1}_* \frac{im}{re} \cdot \tan^{-1}_* \frac{im}{re}}{\frac{\tan^{-1}_* \frac{im}{re} + \tan^{-1}_* \frac{im}{re}}{\color{blue}{2}}}}{\frac{\log 10}{\frac{1}{\log 10}}} \]
   rational.json-simplify-61 [<=] 9.3	\[ \log 10 \cdot \frac{\color{blue}{\frac{2}{\frac{\tan^{-1}_* \frac{im}{re} + \tan^{-1}_* \frac{im}{re}}{\tan^{-1}_* \frac{im}{re} \cdot \tan^{-1}_* \frac{im}{re}}}}}{\frac{\log 10}{\frac{1}{\log 10}}} \]
   metadata-eval [<=] 9.3	\[ \log 10 \cdot \frac{\frac{\color{blue}{1 + 1}}{\frac{\tan^{-1}_* \frac{im}{re} + \tan^{-1}_* \frac{im}{re}}{\tan^{-1}_* \frac{im}{re} \cdot \tan^{-1}_* \frac{im}{re}}}}{\frac{\log 10}{\frac{1}{\log 10}}} \]
   rational.json-simplify-29 [<=] 1.1	\[ \log 10 \cdot \frac{\frac{1 + 1}{\color{blue}{\frac{1}{\tan^{-1}_* \frac{im}{re}} + \frac{1}{\tan^{-1}_* \frac{im}{re}}}}}{\frac{\log 10}{\frac{1}{\log 10}}} \]
   rational.json-simplify-35 [<=] 1.0	\[ \log 10 \cdot \frac{\color{blue}{\frac{1}{\frac{1}{\tan^{-1}_* \frac{im}{re}}}}}{\frac{\log 10}{\frac{1}{\log 10}}} \]
   rational.json-simplify-61 [=>] 0.9	\[ \log 10 \cdot \frac{\color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{\frac{1}{1}}}}{\frac{\log 10}{\frac{1}{\log 10}}} \]
   metadata-eval [=>] 0.9	\[ \log 10 \cdot \frac{\frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{1}}}{\frac{\log 10}{\frac{1}{\log 10}}} \]
   rational.json-simplify-7 [=>] 0.9	\[ \log 10 \cdot \frac{\color{blue}{\tan^{-1}_* \frac{im}{re}}}{\frac{\log 10}{\frac{1}{\log 10}}} \]

4. Applied egg-rr 0.1

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

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.7
Cost	13504

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

Alternative 2
Error	0.6
Cost	13504

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

Alternative 3
Error	0.8
Cost	13056

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

Reproduce

herbie shell --seed 2023066 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10.0)))