ln(1 + x)

Average Error: 39.0 → 0.2

Time: 10.9s

Precision: binary64

Cost: 20484

Math

\[\log \left(1 + x\right) \]

↓

\[\begin{array}{l} \mathbf{if}\;1 + x \leq 2:\\ \;\;\;\;\left(x + -0.25 \cdot {x}^{4}\right) + \left(-0.5 \cdot {x}^{2} + 0.3333333333333333 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= (+ 1.0 x) 2.0)
   (+
    (+ x (* -0.25 (pow x 4.0)))
    (+ (* -0.5 (pow x 2.0)) (* 0.3333333333333333 (pow x 3.0))))
   (log (+ 1.0 x))))

C

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

↓

double code(double x) {
	double tmp;
	if ((1.0 + x) <= 2.0) {
		tmp = (x + (-0.25 * pow(x, 4.0))) + ((-0.5 * pow(x, 2.0)) + (0.3333333333333333 * pow(x, 3.0)));
	} else {
		tmp = log((1.0 + x));
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((1.0d0 + x) <= 2.0d0) then
        tmp = (x + ((-0.25d0) * (x ** 4.0d0))) + (((-0.5d0) * (x ** 2.0d0)) + (0.3333333333333333d0 * (x ** 3.0d0)))
    else
        tmp = log((1.0d0 + x))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x) {
	double tmp;
	if ((1.0 + x) <= 2.0) {
		tmp = (x + (-0.25 * Math.pow(x, 4.0))) + ((-0.5 * Math.pow(x, 2.0)) + (0.3333333333333333 * Math.pow(x, 3.0)));
	} else {
		tmp = Math.log((1.0 + x));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if (1.0 + x) <= 2.0:
		tmp = (x + (-0.25 * math.pow(x, 4.0))) + ((-0.5 * math.pow(x, 2.0)) + (0.3333333333333333 * math.pow(x, 3.0)))
	else:
		tmp = math.log((1.0 + x))
	return tmp

Julia

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

↓

function code(x)
	tmp = 0.0
	if (Float64(1.0 + x) <= 2.0)
		tmp = Float64(Float64(x + Float64(-0.25 * (x ^ 4.0))) + Float64(Float64(-0.5 * (x ^ 2.0)) + Float64(0.3333333333333333 * (x ^ 3.0))));
	else
		tmp = log(Float64(1.0 + x));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if ((1.0 + x) <= 2.0)
		tmp = (x + (-0.25 * (x ^ 4.0))) + ((-0.5 * (x ^ 2.0)) + (0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = log((1.0 + x));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_] := If[LessEqual[N[(1.0 + x), $MachinePrecision], 2.0], N[(N[(x + N[(-0.25 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]]

Target

Original	39.0
Target	0.3
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;1 + x = 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \log \left(1 + x\right)}{\left(1 + x\right) - 1}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if (+.f64 1 x) < 2

   1. Initial program 58.6

      \[\log \left(1 + x\right) \]

   2. Applied egg-rr 58.6

      \[\leadsto \color{blue}{\left(0 - \left(-1 - \log \left(1 + x\right)\right)\right) - 1} \]

   3. Taylor expanded in x around 0 0.3

      \[\leadsto \color{blue}{-0.5 \cdot {x}^{2} + \left(0.3333333333333333 \cdot {x}^{3} + \left(-0.25 \cdot {x}^{4} + x\right)\right)} \]

   4. Simplified 0.3

      \[\leadsto \color{blue}{\left(x + -0.25 \cdot {x}^{4}\right) + \left(-0.5 \cdot {x}^{2} + 0.3333333333333333 \cdot {x}^{3}\right)} \]
      Proof

      [Start] 0.3	\[ -0.5 \cdot {x}^{2} + \left(0.3333333333333333 \cdot {x}^{3} + \left(-0.25 \cdot {x}^{4} + x\right)\right) \]
      rational.json-simplify-41 [<=] 0.3	\[ \color{blue}{\left(-0.25 \cdot {x}^{4} + x\right) + \left(-0.5 \cdot {x}^{2} + 0.3333333333333333 \cdot {x}^{3}\right)} \]
      rational.json-simplify-1 [=>] 0.3	\[ \color{blue}{\left(x + -0.25 \cdot {x}^{4}\right)} + \left(-0.5 \cdot {x}^{2} + 0.3333333333333333 \cdot {x}^{3}\right) \]

   if 2 < (+.f64 1 x)

   1. Initial program 0

      \[\log \left(1 + x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + x \leq 2:\\ \;\;\;\;\left(x + -0.25 \cdot {x}^{4}\right) + \left(-0.5 \cdot {x}^{2} + 0.3333333333333333 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	20484

\[\begin{array}{l} \mathbf{if}\;1 + x \leq 2:\\ \;\;\;\;-0.5 \cdot {x}^{2} + \left(0.3333333333333333 \cdot {x}^{3} + \left(-0.25 \cdot {x}^{4} + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x\right)\\ \end{array} \]

Alternative 2
Error	0.2
Cost	13764

\[\begin{array}{l} \mathbf{if}\;1 + x \leq 1.0005:\\ \;\;\;\;-0.5 \cdot {x}^{2} + \left(0.3333333333333333 \cdot {x}^{3} + x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x\right)\\ \end{array} \]

Alternative 3
Error	0.3
Cost	7044

\[\begin{array}{l} \mathbf{if}\;1 + x \leq 1.0005:\\ \;\;\;\;-0.5 \cdot {x}^{2} + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x\right)\\ \end{array} \]

Alternative 4
Error	0.7
Cost	6852

\[\begin{array}{l} \mathbf{if}\;1 + x \leq 1.00000002:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x\right)\\ \end{array} \]

Alternative 5
Error	21.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023077 
(FPCore (x)
  :name "ln(1 + x)"
  :precision binary64

  :herbie-target
  (if (== (+ 1.0 x) 1.0) x (/ (* x (log (+ 1.0 x))) (- (+ 1.0 x) 1.0)))

  (log (+ 1.0 x)))