qlog (example 3.10)

Average Error: 61.5 → 0.5

Time: 25.4s

Precision: binary64

Cost: 6912

\[-1 < x \land x < 1\]

Math

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

↓

\[-1 - \left(x + {x}^{2} \cdot 0.5\right) \]

FPCore

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

↓

(FPCore (x) :precision binary64 (- -1.0 (+ x (* (pow x 2.0) 0.5))))

C

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

↓

double code(double x) {
	return -1.0 - (x + (pow(x, 2.0) * 0.5));
}

Fortran

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) - (x + ((x ** 2.0d0) * 0.5d0))
end function

Java

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

↓

public static double code(double x) {
	return -1.0 - (x + (Math.pow(x, 2.0) * 0.5));
}

Python

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

↓

def code(x):
	return -1.0 - (x + (math.pow(x, 2.0) * 0.5))

Julia

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

↓

function code(x)
	return Float64(-1.0 - Float64(x + Float64((x ^ 2.0) * 0.5)))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = -1.0 - (x + ((x ^ 2.0) * 0.5));
end

Wolfram

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

↓

code[x_] := N[(-1.0 - N[(x + N[(N[Power[x, 2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	61.5
Target	0.4
Herbie	0.5

\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.4166666666666667 \cdot {x}^{3}\right) \]

Derivation

1. Initial program 61.5

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

2. Taylor expanded in x around 0 0.5

   \[\leadsto \color{blue}{\left(-0.5 \cdot {x}^{2} + -1 \cdot x\right) - 1} \]

3. Simplified 0.5

   \[\leadsto \color{blue}{-0.5 \cdot {x}^{2} + \left(-1 - x\right)} \]
   Proof

   [Start] 0.5	\[ \left(-0.5 \cdot {x}^{2} + -1 \cdot x\right) - 1 \]
   rational.json-simplify-1 [=>] 0.5	\[ \color{blue}{\left(-1 \cdot x + -0.5 \cdot {x}^{2}\right)} - 1 \]
   rational.json-simplify-48 [=>] 0.5	\[ \color{blue}{-0.5 \cdot {x}^{2} + \left(-1 \cdot x - 1\right)} \]
   rational.json-simplify-2 [=>] 0.5	\[ -0.5 \cdot {x}^{2} + \left(\color{blue}{x \cdot -1} - 1\right) \]
   rational.json-simplify-9 [=>] 0.5	\[ -0.5 \cdot {x}^{2} + \left(\color{blue}{\left(-x\right)} - 1\right) \]
   rational.json-simplify-12 [=>] 0.5	\[ -0.5 \cdot {x}^{2} + \left(\color{blue}{\left(0 - x\right)} - 1\right) \]
   rational.json-simplify-42 [=>] 0.5	\[ -0.5 \cdot {x}^{2} + \color{blue}{\left(\left(0 - 1\right) - x\right)} \]
   metadata-eval [=>] 0.5	\[ -0.5 \cdot {x}^{2} + \left(\color{blue}{-1} - x\right) \]

4. Applied egg-rr 0.5

   \[\leadsto \color{blue}{0 - \left(\left(x + 1\right) + {x}^{2} \cdot 0.5\right)} \]

5. Simplified 0.5

   \[\leadsto \color{blue}{-1 - \left(x + {x}^{2} \cdot 0.5\right)} \]
   Proof

   [Start] 0.5	\[ 0 - \left(\left(x + 1\right) + {x}^{2} \cdot 0.5\right) \]
   rational.json-simplify-1 [=>] 0.5	\[ 0 - \color{blue}{\left({x}^{2} \cdot 0.5 + \left(x + 1\right)\right)} \]
   rational.json-simplify-1 [=>] 0.5	\[ 0 - \left({x}^{2} \cdot 0.5 + \color{blue}{\left(1 + x\right)}\right) \]
   rational.json-simplify-41 [=>] 0.5	\[ 0 - \color{blue}{\left(1 + \left(x + {x}^{2} \cdot 0.5\right)\right)} \]
   rational.json-simplify-17 [=>] 0.5	\[ 0 - \color{blue}{\left(\left(x + {x}^{2} \cdot 0.5\right) - -1\right)} \]
   rational.json-simplify-45 [=>] 0.5	\[ \color{blue}{-1 - \left(\left(x + {x}^{2} \cdot 0.5\right) - 0\right)} \]
   rational.json-simplify-5 [=>] 0.5	\[ -1 - \color{blue}{\left(x + {x}^{2} \cdot 0.5\right)} \]

6. Final simplification 0.5

   \[\leadsto -1 - \left(x + {x}^{2} \cdot 0.5\right) \]

Alternatives

Alternative 1
Error	0.5
Cost	6912

\[-0.5 \cdot {x}^{2} + \left(-1 - x\right) \]

Alternative 2
Error	0.7
Cost	192

\[-1 - x \]

Alternative 3
Error	1.3
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023073 
(FPCore (x)
  :name "qlog (example 3.10)"
  :precision binary64
  :pre (and (< -1.0 x) (< x 1.0))

  :herbie-target
  (- (+ (+ (+ 1.0 x) (/ (* x x) 2.0)) (* 0.4166666666666667 (pow x 3.0))))

  (/ (log (- 1.0 x)) (log (+ 1.0 x))))