neg log

Average Accuracy: 99.9% → 99.9%

Time: 2.4s

Precision: binary64

Cost: 6784

Math

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

↓

\[-\mathsf{log1p}\left(\frac{1}{x} + -2\right) \]

FPCore

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

↓

(FPCore (x) :precision binary64 (- (log1p (+ (/ 1.0 x) -2.0))))

C

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

↓

double code(double x) {
	return -log1p(((1.0 / x) + -2.0));
}

Java

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

↓

public static double code(double x) {
	return -Math.log1p(((1.0 / x) + -2.0));
}

Python

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

↓

def code(x):
	return -math.log1p(((1.0 / x) + -2.0))

Julia

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

↓

function code(x)
	return Float64(-log1p(Float64(Float64(1.0 / x) + -2.0)))
end

Wolfram

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

↓

code[x_] := (-N[Log[1 + N[(N[(1.0 / x), $MachinePrecision] + -2.0), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 99.9%

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

2. Applied egg-rr 99.9%

   \[\leadsto -\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-\log x\right) - 1\right)} \]
   Proof

   [Start] 99.9	\[ -\log \left(\frac{1}{x} - 1\right) \]
   log1p-expm1-u [=>] 99.9	\[ -\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{1}{x} - 1\right)\right)\right)} \]
   expm1-udef [=>] 99.9	\[ -\mathsf{log1p}\left(\color{blue}{e^{\log \left(\frac{1}{x} - 1\right)} - 1}\right) \]
   add-exp-log [<=] 99.9	\[ -\mathsf{log1p}\left(\color{blue}{\left(\frac{1}{x} - 1\right)} - 1\right) \]
   add-exp-log [=>] 99.9	\[ -\mathsf{log1p}\left(\left(\color{blue}{e^{\log \left(\frac{1}{x}\right)}} - 1\right) - 1\right) \]
   expm1-def [=>] 99.9	\[ -\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(\frac{1}{x}\right)\right)} - 1\right) \]
   log-rec [=>] 99.9	\[ -\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{-\log x}\right) - 1\right) \]

3. Taylor expanded in x around 0 99.9%

   \[\leadsto -\mathsf{log1p}\left(\color{blue}{e^{-\log x} - 2}\right) \]

4. Simplified 99.9%

   \[\leadsto -\mathsf{log1p}\left(\color{blue}{\frac{1}{x} + -2}\right) \]
   Proof

   [Start] 99.9	\[ -\mathsf{log1p}\left(e^{-\log x} - 2\right) \]
   sub-neg [=>] 99.9	\[ -\mathsf{log1p}\left(\color{blue}{e^{-\log x} + \left(-2\right)}\right) \]
   exp-neg [=>] 99.9	\[ -\mathsf{log1p}\left(\color{blue}{\frac{1}{e^{\log x}}} + \left(-2\right)\right) \]
   rem-exp-log [=>] 99.9	\[ -\mathsf{log1p}\left(\frac{1}{\color{blue}{x}} + \left(-2\right)\right) \]
   metadata-eval [=>] 99.9	\[ -\mathsf{log1p}\left(\frac{1}{x} + \color{blue}{-2}\right) \]

5. Final simplification 99.9%

   \[\leadsto -\mathsf{log1p}\left(\frac{1}{x} + -2\right) \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	6784

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

Alternative 2
Accuracy	99.2%
Cost	6592

\[x + \log x \]

Alternative 3
Accuracy	98.3%
Cost	6464

\[\log x \]

Alternative 4
Accuracy	2.3%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023133 
(FPCore (x)
  :name "neg log"
  :precision binary64
  (- (log (- (/ 1.0 x) 1.0))))