qlog (example 3.10)

Average Error: 61.4 → 0.3

Time: 12.2s

Precision: binary64

Cost: 7296

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

Math

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

↓

\[-1 + \left(-0.5 \cdot \left(x \cdot x\right) + \left(-0.4166666666666667 \cdot {x}^{3} - x\right)\right) \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (+ -1.0 (+ (* -0.5 (* x x)) (- (* -0.4166666666666667 (pow x 3.0)) x))))

C

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

↓

double code(double x) {
	return -1.0 + ((-0.5 * (x * x)) + ((-0.4166666666666667 * pow(x, 3.0)) - x));
}

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) + (((-0.5d0) * (x * x)) + (((-0.4166666666666667d0) * (x ** 3.0d0)) - x))
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 + ((-0.5 * (x * x)) + ((-0.4166666666666667 * Math.pow(x, 3.0)) - x));
}

Python

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

↓

def code(x):
	return -1.0 + ((-0.5 * (x * x)) + ((-0.4166666666666667 * math.pow(x, 3.0)) - x))

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(Float64(-0.5 * Float64(x * x)) + Float64(Float64(-0.4166666666666667 * (x ^ 3.0)) - x)))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = -1.0 + ((-0.5 * (x * x)) + ((-0.4166666666666667 * (x ^ 3.0)) - x));
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[(N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.4166666666666667 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	61.4
Target	0.3
Herbie	0.3

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

Derivation

1. Initial program 61.4

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

2. Taylor expanded in x around 0 0.3

   \[\leadsto \color{blue}{\left(-0.5 \cdot {x}^{2} + \left(-1 \cdot x + -0.4166666666666667 \cdot {x}^{3}\right)\right) - 1} \]

3. Applied egg-rr 0.3

   \[\leadsto \left(-0.5 \cdot \color{blue}{\left(x \cdot x\right)} + \left(-1 \cdot x + -0.4166666666666667 \cdot {x}^{3}\right)\right) - 1 \]

4. Final simplification 0.3

   \[\leadsto -1 + \left(-0.5 \cdot \left(x \cdot x\right) + \left(-0.4166666666666667 \cdot {x}^{3} - x\right)\right) \]

Alternatives

Alternative 1
Error	0.4
Cost	7104

\[\mathsf{fma}\left(x \cdot x, -0.5 + x \cdot -0.16666666666666666, -1\right) - x \]

Alternative 2
Error	0.4
Cost	960

\[x \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right) + -0.5 \cdot x\right) + \left(-1 - x\right) \]

Alternative 3
Error	0.4
Cost	576

\[-1 + \left(x \cdot \left(-0.5 \cdot x\right) - x\right) \]

Alternative 4
Error	0.7
Cost	192

\[-1 - x \]

Alternative 5
Error	1.3
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022313 
(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))))