neg log

Percentage Accurate: 100.0% → 100.0%

Time: 9.6s

Alternatives: 8

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ -\log \left(\frac{1}{x} - 1\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -\log \left(\frac{1}{x} - 1\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -\log \left(\frac{1}{x} + -1\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[-\log \left(\frac{1}{x} - 1\right) \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto -\log \left(\frac{1}{x} + -1\right) \]
4. Add Preprocessing

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot x, 0.5, x + \log x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma (* x x) 0.5 (+ x (log x))))

C

double code(double x) {
	return fma((x * x), 0.5, (x + log(x)));
}

Julia

function code(x)
	return fma(Float64(x * x), 0.5, Float64(x + log(x)))
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * 0.5 + N[(x + N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[-\log \left(\frac{1}{x} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) - -1 \cdot \log x} \]
4. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \log x} \]

   2. metadata-eval N/A

      \[\leadsto x \cdot \left(1 + \frac{1}{2} \cdot x\right) + \color{blue}{1} \cdot \log x \]

   3. *-lft-identity N/A

      \[\leadsto x \cdot \left(1 + \frac{1}{2} \cdot x\right) + \color{blue}{\log x} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + \frac{1}{2} \cdot x, \log x\right)} \]

   5. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot x + 1}, \log x\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}} + 1, \log x\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1\right)}, \log x\right) \]

   8. lower-log.f64 99.4

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

5. Simplified 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), \log x\right)} \]
6. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \frac{1}{2}\right) + x \cdot 1\right)} + \log x \]

   2. *-rgt-identity N/A

      \[\leadsto \left(x \cdot \left(x \cdot \frac{1}{2}\right) + \color{blue}{x}\right) + \log x \]

   3. lift-log.f64 N/A

      \[\leadsto \left(x \cdot \left(x \cdot \frac{1}{2}\right) + x\right) + \color{blue}{\log x} \]

   4. associate-+l+ N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{1}{2}\right) + \left(x + \log x\right)} \]

   5. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \frac{1}{2}} + \left(x + \log x\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{2} + \left(x + \log x\right) \]

   7. lift-+.f64 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \frac{1}{2} + \color{blue}{\left(x + \log x\right)} \]

   8. lower-fma.f64 99.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, x + \log x\right)} \]

7. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, x + \log x\right)} \]
8. Add Preprocessing

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), \log x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x (fma x 0.5 1.0) (log x)))

C

double code(double x) {
	return fma(x, fma(x, 0.5, 1.0), log(x));
}

Julia

function code(x)
	return fma(x, fma(x, 0.5, 1.0), log(x))
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[-\log \left(\frac{1}{x} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) - -1 \cdot \log x} \]
4. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \log x} \]

   2. metadata-eval N/A

      \[\leadsto x \cdot \left(1 + \frac{1}{2} \cdot x\right) + \color{blue}{1} \cdot \log x \]

   3. *-lft-identity N/A

      \[\leadsto x \cdot \left(1 + \frac{1}{2} \cdot x\right) + \color{blue}{\log x} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + \frac{1}{2} \cdot x, \log x\right)} \]

   5. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot x + 1}, \log x\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}} + 1, \log x\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1\right)}, \log x\right) \]

   8. lower-log.f64 99.4

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

5. Simplified 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), \log x\right)} \]
6. Add Preprocessing

Alternative 4: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ x + \log x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[-\log \left(\frac{1}{x} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x - -1 \cdot \log x} \]
4. Step-by-step derivation

   1. sub-neg N/A

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

   2. mul-1-neg N/A

      \[\leadsto x + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log x\right)\right)}\right)\right) \]

   3. remove-double-neg N/A

      \[\leadsto x + \color{blue}{\log x} \]

   4. lower-+.f64 N/A

      \[\leadsto \color{blue}{x + \log x} \]

   5. lower-log.f64 99.2

      \[\leadsto x + \color{blue}{\log x} \]

5. Simplified 99.2%

   \[\leadsto \color{blue}{x + \log x} \]
6. Add Preprocessing

Alternative 5: 98.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \log x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return log(x)
end

MATLAB

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

Wolfram

code[x_] := N[Log[x], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[-\log \left(\frac{1}{x} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\log x} \]
4. Step-by-step derivation

   1. lower-log.f64 98.0

      \[\leadsto \color{blue}{\log x} \]

5. Simplified 98.0%

   \[\leadsto \color{blue}{\log x} \]
6. Add Preprocessing

Alternative 6: 9.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{x} \leq 10^{+153}:\\ \;\;\;\;\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.25 - x \cdot x\right) \cdot \frac{2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, x \cdot 0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ 1.0 x) 1e+153)
   (* (- (* (* x (* x (* x x))) 0.25) (* x x)) (/ 2.0 (* x x)))
   (- (fma x (* x 0.5) x))))

C

double code(double x) {
	double tmp;
	if ((1.0 / x) <= 1e+153) {
		tmp = (((x * (x * (x * x))) * 0.25) - (x * x)) * (2.0 / (x * x));
	} else {
		tmp = -fma(x, (x * 0.5), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 / x) <= 1e+153)
		tmp = Float64(Float64(Float64(Float64(x * Float64(x * Float64(x * x))) * 0.25) - Float64(x * x)) * Float64(2.0 / Float64(x * x)));
	else
		tmp = Float64(-fma(x, Float64(x * 0.5), x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 / x), $MachinePrecision], 1e+153], N[(N[(N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision] - N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(x * N[(x * 0.5), $MachinePrecision] + x), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) x) < 1e153

   1. Initial program 100.0%

      \[-\log \left(\frac{1}{x} - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) - -1 \cdot \log x} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \log x} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(1 + \frac{1}{2} \cdot x\right) + \color{blue}{1} \cdot \log x \]

      3. *-lft-identity N/A

         \[\leadsto x \cdot \left(1 + \frac{1}{2} \cdot x\right) + \color{blue}{\log x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + \frac{1}{2} \cdot x, \log x\right)} \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot x + 1}, \log x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}} + 1, \log x\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1\right)}, \log x\right) \]

      8. lower-log.f64 98.8

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

   5. Simplified 98.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{x}\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{1}{2} + {x}^{2} \cdot \frac{1}{x}} \]

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. rgt-mult-inverse N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} \cdot x, x\right)} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}}, x\right) \]

      11. lower-*.f64 2.0

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

   8. Simplified 2.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 0.5, x\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

   10. Applied egg-rr 2.0%

       \[\leadsto \color{blue}{\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.25 - x \cdot x\right) \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot 0.5, -x\right)}} \]

   11. Taylor expanded in x around inf

       \[\leadsto \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{4} - x \cdot x\right) \cdot \color{blue}{\frac{2}{{x}^{2}}} \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{4} - x \cdot x\right) \cdot \color{blue}{\frac{2}{{x}^{2}}} \]

       2. unpow2 N/A

          \[\leadsto \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{4} - x \cdot x\right) \cdot \frac{2}{\color{blue}{x \cdot x}} \]

       3. lower-*.f64 14.7

          \[\leadsto \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.25 - x \cdot x\right) \cdot \frac{2}{\color{blue}{x \cdot x}} \]

   13. Simplified 14.7%

       \[\leadsto \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.25 - x \cdot x\right) \cdot \color{blue}{\frac{2}{x \cdot x}} \]

   if 1e153 < (/.f64 #s(literal 1 binary64) x)

   1. Initial program 100.0%

      \[-\log \left(\frac{1}{x} - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) - -1 \cdot \log x} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \log x} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(1 + \frac{1}{2} \cdot x\right) + \color{blue}{1} \cdot \log x \]

      3. *-lft-identity N/A

         \[\leadsto x \cdot \left(1 + \frac{1}{2} \cdot x\right) + \color{blue}{\log x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + \frac{1}{2} \cdot x, \log x\right)} \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot x + 1}, \log x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}} + 1, \log x\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1\right)}, \log x\right) \]

      8. lower-log.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{x}\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{1}{2} + {x}^{2} \cdot \frac{1}{x}} \]

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. rgt-mult-inverse N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} \cdot x, x\right)} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}}, x\right) \]

      11. lower-*.f64 2.6

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

   8. Simplified 2.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 0.5, x\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

   10. Applied egg-rr 3.1%

       \[\leadsto \color{blue}{\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.25 - x \cdot x\right) \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot 0.5, -x\right)}} \]

   12. Applied egg-rr 3.8%

       \[\leadsto \color{blue}{-\mathsf{fma}\left(x, x \cdot 0.5, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 5.3% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ -\mathsf{fma}\left(x, x \cdot 0.5, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (fma x (* x 0.5) x)))

C

double code(double x) {
	return -fma(x, (x * 0.5), x);
}

Julia

function code(x)
	return Float64(-fma(x, Float64(x * 0.5), x))
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[-\log \left(\frac{1}{x} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) - -1 \cdot \log x} \]
4. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \log x} \]

   2. metadata-eval N/A

      \[\leadsto x \cdot \left(1 + \frac{1}{2} \cdot x\right) + \color{blue}{1} \cdot \log x \]

   3. *-lft-identity N/A

      \[\leadsto x \cdot \left(1 + \frac{1}{2} \cdot x\right) + \color{blue}{\log x} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + \frac{1}{2} \cdot x, \log x\right)} \]

   5. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot x + 1}, \log x\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}} + 1, \log x\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1\right)}, \log x\right) \]

   8. lower-log.f64 99.4

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

5. Simplified 99.4%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{x}\right)} \]
7. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto \color{blue}{{x}^{2} \cdot \frac{1}{2} + {x}^{2} \cdot \frac{1}{x}} \]

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. rgt-mult-inverse N/A

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

   8. *-rgt-identity N/A

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

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} \cdot x, x\right)} \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}}, x\right) \]

   11. lower-*.f64 2.3

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

8. Simplified 2.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 0.5, x\right)} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. flip-+ N/A

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

   4. div-inv N/A

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

   5. lower-*.f64 N/A

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

10. Applied egg-rr 2.5%

    \[\leadsto \color{blue}{\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.25 - x \cdot x\right) \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot 0.5, -x\right)}} \]

12. Applied egg-rr 5.2%

    \[\leadsto \color{blue}{-\mathsf{fma}\left(x, x \cdot 0.5, x\right)} \]
13. Add Preprocessing

Alternative 8: 2.7% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (* x 0.5)))

C

double code(double x) {
	return x * (x * 0.5);
}

Fortran

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

Java

public static double code(double x) {
	return x * (x * 0.5);
}

Python

def code(x):
	return x * (x * 0.5)

Julia

function code(x)
	return Float64(x * Float64(x * 0.5))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * 0.5);
end

Wolfram

code[x_] := N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[-\log \left(\frac{1}{x} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) - -1 \cdot \log x} \]
4. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \log x} \]

   2. metadata-eval N/A

      \[\leadsto x \cdot \left(1 + \frac{1}{2} \cdot x\right) + \color{blue}{1} \cdot \log x \]

   3. *-lft-identity N/A

      \[\leadsto x \cdot \left(1 + \frac{1}{2} \cdot x\right) + \color{blue}{\log x} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + \frac{1}{2} \cdot x, \log x\right)} \]

   5. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot x + 1}, \log x\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}} + 1, \log x\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1\right)}, \log x\right) \]

   8. lower-log.f64 99.4

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

5. Simplified 99.4%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2}} \]
7. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot x\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x\right) \cdot x} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} \cdot x\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} \cdot x\right)} \]

   5. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \frac{1}{2}\right)} \]

   6. lower-*.f64 2.8

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

8. Simplified 2.8%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot 0.5\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(FPCore (x)
  :name "neg log"
  :precision binary64
  (- (log (- (/ 1.0 x) 1.0))))