neg log

Average Error: 0.0 → 0.0

Time: 6.9s

Precision: binary64

Cost: 7040

Math

\[-\log \left(\frac{1}{x} - 1\right) \]

↓

\[-\log \left(\frac{0.125}{x} - \left(1 - \frac{0.875}{x}\right)\right) \]

FPCore

(FPCore (x) :precision binary64 (- (log (- (/ 1.0 x) 1.0))))

↓

(FPCore (x) :precision binary64 (- (log (- (/ 0.125 x) (- 1.0 (/ 0.875 x))))))

C

double code(double x) {
	return -log(((1.0 / x) - 1.0));
}

↓

double code(double x) {
	return -log(((0.125 / x) - (1.0 - (0.875 / x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -log(((1.0d0 / x) - 1.0d0))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = -log(((0.125d0 / x) - (1.0d0 - (0.875d0 / x))))
end function

Java

public static double code(double x) {
	return -Math.log(((1.0 / x) - 1.0));
}

↓

public static double code(double x) {
	return -Math.log(((0.125 / x) - (1.0 - (0.875 / x))));
}

Python

def code(x):
	return -math.log(((1.0 / x) - 1.0))

↓

def code(x):
	return -math.log(((0.125 / x) - (1.0 - (0.875 / x))))

Julia

function code(x)
	return Float64(-log(Float64(Float64(1.0 / x) - 1.0)))
end

↓

function code(x)
	return Float64(-log(Float64(Float64(0.125 / x) - Float64(1.0 - Float64(0.875 / x)))))
end

MATLAB

function tmp = code(x)
	tmp = -log(((1.0 / x) - 1.0));
end

↓

function tmp = code(x)
	tmp = -log(((0.125 / x) - (1.0 - (0.875 / x))));
end

Wolfram

code[x_] := (-N[Log[N[(N[(1.0 / x), $MachinePrecision] - 1.0), $MachinePrecision]], $MachinePrecision])

↓

code[x_] := (-N[Log[N[(N[(0.125 / x), $MachinePrecision] - N[(1.0 - N[(0.875 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 0.0

   \[-\log \left(\frac{1}{x} - 1\right) \]

2. Applied egg-rr 0.0

   \[\leadsto -\log \color{blue}{\left(\frac{-0.5 + \frac{1.5}{x}}{2} + \frac{\frac{0.5}{x} - 1.5}{2}\right)} \]

3. Simplified 0.0

   \[\leadsto -\log \color{blue}{\left(\left(-0.25 + \frac{0.75}{x}\right) + \left(\frac{0.25}{x} - 0.75\right)\right)} \]
   Proof

   [Start] 0.0	\[ -\log \left(\frac{-0.5 + \frac{1.5}{x}}{2} + \frac{\frac{0.5}{x} - 1.5}{2}\right) \]
   rational_best-simplify-77 [=>] 0.0	\[ -\log \left(\color{blue}{\left(\frac{-0.5}{2} + \frac{\frac{1.5}{x}}{2}\right)} + \frac{\frac{0.5}{x} - 1.5}{2}\right) \]
   metadata-eval [=>] 0.0	\[ -\log \left(\left(\color{blue}{-0.25} + \frac{\frac{1.5}{x}}{2}\right) + \frac{\frac{0.5}{x} - 1.5}{2}\right) \]
   rational_best-simplify-51 [=>] 0.0	\[ -\log \left(\left(-0.25 + \color{blue}{\frac{\frac{1.5}{2}}{x}}\right) + \frac{\frac{0.5}{x} - 1.5}{2}\right) \]
   metadata-eval [=>] 0.0	\[ -\log \left(\left(-0.25 + \frac{\color{blue}{0.75}}{x}\right) + \frac{\frac{0.5}{x} - 1.5}{2}\right) \]
   rational_best-simplify-79 [=>] 0.0	\[ -\log \left(\left(-0.25 + \frac{0.75}{x}\right) + \color{blue}{\left(\frac{\frac{0.5}{x}}{2} - \frac{1.5}{2}\right)}\right) \]
   rational_best-simplify-51 [=>] 0.0	\[ -\log \left(\left(-0.25 + \frac{0.75}{x}\right) + \left(\color{blue}{\frac{\frac{0.5}{2}}{x}} - \frac{1.5}{2}\right)\right) \]
   metadata-eval [=>] 0.0	\[ -\log \left(\left(-0.25 + \frac{0.75}{x}\right) + \left(\frac{\color{blue}{0.25}}{x} - \frac{1.5}{2}\right)\right) \]
   metadata-eval [=>] 0.0	\[ -\log \left(\left(-0.25 + \frac{0.75}{x}\right) + \left(\frac{0.25}{x} - \color{blue}{0.75}\right)\right) \]

4. Applied egg-rr 0.0

   \[\leadsto -\log \color{blue}{\left(\frac{0.125}{x} - \left(\left(0.25 - \left(\frac{0.125}{x} - \left(0.375 + \frac{-0.75}{x}\right)\right)\right) + 0.375\right)\right)} \]

5. Taylor expanded in x around 0 0.0

   \[\leadsto -\log \left(\frac{0.125}{x} - \color{blue}{\left(1 - 0.875 \cdot \frac{1}{x}\right)}\right) \]

6. Simplified 0.0

   \[\leadsto -\log \left(\frac{0.125}{x} - \color{blue}{\left(1 - \frac{0.875}{x}\right)}\right) \]
   Proof

   [Start] 0.0	\[ -\log \left(\frac{0.125}{x} - \left(1 - 0.875 \cdot \frac{1}{x}\right)\right) \]
   rational_best-simplify-62 [=>] 0.0	\[ -\log \left(\frac{0.125}{x} - \left(1 - \color{blue}{\frac{1 \cdot 0.875}{x}}\right)\right) \]
   metadata-eval [=>] 0.0	\[ -\log \left(\frac{0.125}{x} - \left(1 - \frac{\color{blue}{0.875}}{x}\right)\right) \]

7. Final simplification 0.0

   \[\leadsto -\log \left(\frac{0.125}{x} - \left(1 - \frac{0.875}{x}\right)\right) \]

Alternatives

Alternative 1
Error	0.0
Cost	6784

\[-\log \left(\frac{1}{x} - 1\right) \]

Alternative 2
Error	1.2
Cost	6592

\[-\left(-\log x\right) \]

Alternative 3
Error	64.0
Cost	6528

\[-\log -1 \]

Reproduce

herbie shell --seed 2023101 
(FPCore (x)
  :name "neg log"
  :precision binary64
  (- (log (- (/ 1.0 x) 1.0))))