qlog (example 3.10)

Average Accuracy: 4.0% → 99.3%

Time: 9.8s

Precision: binary64

Cost: 576

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

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (+ -1.0 (- (* -0.5 (* x x)) 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)) - 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)) - 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)) - 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)) - 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)) - 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)) - 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] - x), $MachinePrecision]), $MachinePrecision]

Target

Original	4.0%
Target	99.4%
Herbie	99.3%

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

Derivation

1. Initial program 4.0%

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

2. Simplified 100.0%

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

   [Start] 4.0	\[ \frac{\log \left(1 - x\right)}{\log \left(1 + x\right)} \]
   sub-neg [=>] 4.0	\[ \frac{\log \color{blue}{\left(1 + \left(-x\right)\right)}}{\log \left(1 + x\right)} \]
   log1p-def [=>] 3.1	\[ \frac{\color{blue}{\mathsf{log1p}\left(-x\right)}}{\log \left(1 + x\right)} \]
   log1p-def [=>] 100.0	\[ \frac{\mathsf{log1p}\left(-x\right)}{\color{blue}{\mathsf{log1p}\left(x\right)}} \]

3. Taylor expanded in x around 0 99.3%

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

4. Simplified 99.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, -1 - x\right)} \]
   Proof

   [Start] 99.3	\[ \left(-0.5 \cdot {x}^{2} + -1 \cdot x\right) - 1 \]
   associate--l+ [=>] 99.3	\[ \color{blue}{-0.5 \cdot {x}^{2} + \left(-1 \cdot x - 1\right)} \]
   *-commutative [=>] 99.3	\[ \color{blue}{{x}^{2} \cdot -0.5} + \left(-1 \cdot x - 1\right) \]
   unpow2 [=>] 99.3	\[ \color{blue}{\left(x \cdot x\right)} \cdot -0.5 + \left(-1 \cdot x - 1\right) \]
   associate-*l* [=>] 99.3	\[ \color{blue}{x \cdot \left(x \cdot -0.5\right)} + \left(-1 \cdot x - 1\right) \]
   fma-def [=>] 99.3	\[ \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, -1 \cdot x - 1\right)} \]
   sub-neg [=>] 99.3	\[ \mathsf{fma}\left(x, x \cdot -0.5, \color{blue}{-1 \cdot x + \left(-1\right)}\right) \]
   metadata-eval [=>] 99.3	\[ \mathsf{fma}\left(x, x \cdot -0.5, -1 \cdot x + \color{blue}{-1}\right) \]
   +-commutative [=>] 99.3	\[ \mathsf{fma}\left(x, x \cdot -0.5, \color{blue}{-1 + -1 \cdot x}\right) \]
   mul-1-neg [=>] 99.3	\[ \mathsf{fma}\left(x, x \cdot -0.5, -1 + \color{blue}{\left(-x\right)}\right) \]
   unsub-neg [=>] 99.3	\[ \mathsf{fma}\left(x, x \cdot -0.5, \color{blue}{-1 - x}\right) \]

5. Applied egg-rr 99.3%

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

   [Start] 99.3	\[ \mathsf{fma}\left(x, x \cdot -0.5, -1 - x\right) \]
   fma-udef [=>] 99.3	\[ \color{blue}{x \cdot \left(x \cdot -0.5\right) + \left(-1 - x\right)} \]
   +-commutative [=>] 99.3	\[ \color{blue}{\left(-1 - x\right) + x \cdot \left(x \cdot -0.5\right)} \]

6. Applied egg-rr 99.3%

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

   [Start] 99.3	\[ \left(-1 - x\right) + x \cdot \left(x \cdot -0.5\right) \]
   associate-+l- [=>] 99.3	\[ \color{blue}{-1 - \left(x - x \cdot \left(x \cdot -0.5\right)\right)} \]
   associate-*r* [=>] 99.3	\[ -1 - \left(x - \color{blue}{\left(x \cdot x\right) \cdot -0.5}\right) \]
   *-commutative [=>] 99.3	\[ -1 - \left(x - \color{blue}{-0.5 \cdot \left(x \cdot x\right)}\right) \]

7. Final simplification 99.3%

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

Alternatives

Alternative 1
Accuracy	99.3%
Cost	576

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

Alternative 2
Accuracy	98.9%
Cost	192

\[-1 - x \]

Alternative 3
Accuracy	97.9%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023142 
(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))))