ln(1 + x)

Percentage Accurate: 38.2% → 100.0%

Time: 9.6s

Alternatives: 6

Speedup: 10.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log1p x))

C

double code(double x) {
	return log1p(x);
}

Java

public static double code(double x) {
	return Math.log1p(x);
}

Python

def code(x):
	return math.log1p(x)

Julia

function code(x)
	return log1p(x)
end

Wolfram

code[x_] := N[Log[1 + x], $MachinePrecision]

Derivation

1. Initial program 43.6%

   \[\log \left(1 + x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. accelerator-lowering-log1p.f64 100.0

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(x\right)} \]
5. Add Preprocessing

Alternative 2: 71.5% accurate, 4.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ x 1.0) 2.0) (fma (* x x) -0.5 x) 2.0))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 9.1%

      \[\log \left(1 + x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + \frac{-1}{2} \cdot x, 0\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{-1}{2} \cdot x + 1}, 0\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{2}} + 1, 0\right) \]

      5. accelerator-lowering-fma.f64 98.9

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.5, 1\right)}, 0\right) \]

   5. Simplified 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.5, 1\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 98.9

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

   7. Applied egg-rr 98.9%

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

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

   1. Initial program 100.0%

      \[\log \left(1 + x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right) + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + x \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right), 0\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right) + 1}, 0\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}, 1\right)}, 0\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right), 0\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) + \color{blue}{\frac{-1}{2}}, 1\right), 0\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{3} + \frac{-1}{4} \cdot x, \frac{-1}{2}\right)}, 1\right), 0\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{-1}{4} \cdot x + \frac{1}{3}}, \frac{-1}{2}\right), 1\right), 0\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{4}} + \frac{1}{3}, \frac{-1}{2}\right), 1\right), 0\right) \]

      10. accelerator-lowering-fma.f64 0.7

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.25, 0.3333333333333333\right)}, -0.5\right), 1\right), 0\right) \]

   5. Simplified 0.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, 0.3333333333333333\right), -0.5\right), 1\right), 0\right)} \]
   6. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) - 0 \cdot 0}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right) - 0}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) - \color{blue}{0}}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right) - 0} \]

      3. --rgt-identity N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) - 0}{\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)}} \]

      4. --rgt-identity N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right)}}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)} \]

      5. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)}{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right)}}} \]

   7. Applied egg-rr 0.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, 0.3333333333333333\right), -0.5\right), 1\right), 0\right)}}} \]

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 14.4

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

   10. Simplified 14.4%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{2} \]
   12. Step-by-step derivation

   13. Simplified 14.4%

       \[\leadsto \color{blue}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + 1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.5% accurate, 4.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ x 1.0) 2.0) (* x (fma x -0.5 1.0)) 2.0))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 9.1%

      \[\log \left(1 + x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + \frac{-1}{2} \cdot x, 0\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{-1}{2} \cdot x + 1}, 0\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{2}} + 1, 0\right) \]

      5. accelerator-lowering-fma.f64 98.9

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.5, 1\right)}, 0\right) \]

   5. Simplified 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.5, 1\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. accelerator-lowering-fma.f64 98.9

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

   7. Applied egg-rr 98.9%

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

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

   1. Initial program 100.0%

      \[\log \left(1 + x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right) + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + x \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right), 0\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right) + 1}, 0\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}, 1\right)}, 0\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right), 0\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) + \color{blue}{\frac{-1}{2}}, 1\right), 0\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{3} + \frac{-1}{4} \cdot x, \frac{-1}{2}\right)}, 1\right), 0\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{-1}{4} \cdot x + \frac{1}{3}}, \frac{-1}{2}\right), 1\right), 0\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{4}} + \frac{1}{3}, \frac{-1}{2}\right), 1\right), 0\right) \]

      10. accelerator-lowering-fma.f64 0.7

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.25, 0.3333333333333333\right)}, -0.5\right), 1\right), 0\right) \]

   5. Simplified 0.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, 0.3333333333333333\right), -0.5\right), 1\right), 0\right)} \]
   6. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) - 0 \cdot 0}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right) - 0}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) - \color{blue}{0}}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right) - 0} \]

      3. --rgt-identity N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) - 0}{\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)}} \]

      4. --rgt-identity N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right)}}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)} \]

      5. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)}{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right)}}} \]

   7. Applied egg-rr 0.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, 0.3333333333333333\right), -0.5\right), 1\right), 0\right)}}} \]

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 14.4

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

   10. Simplified 14.4%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{2} \]
   12. Step-by-step derivation

   13. Simplified 14.4%

       \[\leadsto \color{blue}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + 1 \leq 2:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.6% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(x / fma(x, 0.5, 1.0))
end

Wolfram

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

Derivation

1. Initial program 43.6%

   \[\log \left(1 + x\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)} \]
4. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right) + 0} \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + x \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right), 0\right)} \]

   3. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right) + 1}, 0\right) \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}, 1\right)}, 0\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right), 0\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) + \color{blue}{\frac{-1}{2}}, 1\right), 0\right) \]

   7. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{3} + \frac{-1}{4} \cdot x, \frac{-1}{2}\right)}, 1\right), 0\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{-1}{4} \cdot x + \frac{1}{3}}, \frac{-1}{2}\right), 1\right), 0\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{4}} + \frac{1}{3}, \frac{-1}{2}\right), 1\right), 0\right) \]

   10. accelerator-lowering-fma.f64 62.0

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.25, 0.3333333333333333\right)}, -0.5\right), 1\right), 0\right) \]

5. Simplified 62.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, 0.3333333333333333\right), -0.5\right), 1\right), 0\right)} \]
6. Step-by-step derivation

   1. flip-+ N/A

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) - 0 \cdot 0}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right) - 0}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) - \color{blue}{0}}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right) - 0} \]

   3. --rgt-identity N/A

      \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) - 0}{\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)}} \]

   4. --rgt-identity N/A

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right)}}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)} \]

   5. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)}{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right)}}} \]

7. Applied egg-rr 61.9%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, 0.3333333333333333\right), -0.5\right), 1\right), 0\right)}}} \]

8. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. accelerator-lowering-fma.f64 66.7

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

10. Simplified 66.7%

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

    1. clear-num N/A

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

    2. /-lowering-/.f64 N/A

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

    3. accelerator-lowering-fma.f64 66.9

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

12. Applied egg-rr 66.9%

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

Alternative 5: 70.8% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + 1 \leq 2:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (+ x 1.0) 2.0) x 2.0))

C

double code(double x) {
	double tmp;
	if ((x + 1.0) <= 2.0) {
		tmp = x;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x + 1.0d0) <= 2.0d0) then
        tmp = x
    else
        tmp = 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x + 1.0) <= 2.0) {
		tmp = x;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x + 1.0) <= 2.0:
		tmp = x
	else:
		tmp = 2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x + 1.0) <= 2.0)
		tmp = x;
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x + 1.0) <= 2.0)
		tmp = x;
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x + 1.0), $MachinePrecision], 2.0], x, 2.0]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 9.1%

      \[\log \left(1 + x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 97.6%

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

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

   1. Initial program 100.0%

      \[\log \left(1 + x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right) + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + x \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right), 0\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right) + 1}, 0\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}, 1\right)}, 0\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right), 0\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) + \color{blue}{\frac{-1}{2}}, 1\right), 0\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{3} + \frac{-1}{4} \cdot x, \frac{-1}{2}\right)}, 1\right), 0\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{-1}{4} \cdot x + \frac{1}{3}}, \frac{-1}{2}\right), 1\right), 0\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{4}} + \frac{1}{3}, \frac{-1}{2}\right), 1\right), 0\right) \]

      10. accelerator-lowering-fma.f64 0.7

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.25, 0.3333333333333333\right)}, -0.5\right), 1\right), 0\right) \]

   5. Simplified 0.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, 0.3333333333333333\right), -0.5\right), 1\right), 0\right)} \]
   6. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) - 0 \cdot 0}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right) - 0}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) - \color{blue}{0}}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right) - 0} \]

      3. --rgt-identity N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) - 0}{\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)}} \]

      4. --rgt-identity N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right)}}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)} \]

      5. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)}{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right)}}} \]

   7. Applied egg-rr 0.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, 0.3333333333333333\right), -0.5\right), 1\right), 0\right)}}} \]

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 14.4

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

   10. Simplified 14.4%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{2} \]
   12. Step-by-step derivation

   13. Simplified 14.4%

       \[\leadsto \color{blue}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + 1 \leq 2:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 7.3% accurate, 104.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0
end function

Java

public static double code(double x) {
	return 2.0;
}

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

function tmp = code(x)
	tmp = 2.0;
end

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 43.6%

   \[\log \left(1 + x\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)} \]
4. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right) + 0} \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + x \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right), 0\right)} \]

   3. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right) + 1}, 0\right) \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}, 1\right)}, 0\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right), 0\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) + \color{blue}{\frac{-1}{2}}, 1\right), 0\right) \]

   7. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{3} + \frac{-1}{4} \cdot x, \frac{-1}{2}\right)}, 1\right), 0\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{-1}{4} \cdot x + \frac{1}{3}}, \frac{-1}{2}\right), 1\right), 0\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{4}} + \frac{1}{3}, \frac{-1}{2}\right), 1\right), 0\right) \]

   10. accelerator-lowering-fma.f64 62.0

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.25, 0.3333333333333333\right)}, -0.5\right), 1\right), 0\right) \]

5. Simplified 62.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, 0.3333333333333333\right), -0.5\right), 1\right), 0\right)} \]
6. Step-by-step derivation

   1. flip-+ N/A

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) - 0 \cdot 0}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right) - 0}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) - \color{blue}{0}}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right) - 0} \]

   3. --rgt-identity N/A

      \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) - 0}{\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)}} \]

   4. --rgt-identity N/A

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right)}}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)} \]

   5. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)}{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{4} + \frac{1}{3}\right) + \frac{-1}{2}\right) + 1\right)\right)}}} \]

7. Applied egg-rr 61.9%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, 0.3333333333333333\right), -0.5\right), 1\right), 0\right)}}} \]

8. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. accelerator-lowering-fma.f64 66.7

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

10. Simplified 66.7%

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

11. Taylor expanded in x around inf

    \[\leadsto \color{blue}{2} \]
12. Step-by-step derivation

13. Simplified 7.9%

    \[\leadsto \color{blue}{2} \]
14. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((1.0 + x) == 1.0) {
		tmp = x;
	} else {
		tmp = (x * log((1.0 + x))) / ((1.0 + x) - 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((1.0d0 + x) == 1.0d0) then
        tmp = x
    else
        tmp = (x * log((1.0d0 + x))) / ((1.0d0 + x) - 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((1.0 + x) == 1.0) {
		tmp = x;
	} else {
		tmp = (x * Math.log((1.0 + x))) / ((1.0 + x) - 1.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (1.0 + x) == 1.0:
		tmp = x
	else:
		tmp = (x * math.log((1.0 + x))) / ((1.0 + x) - 1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 + x) == 1.0)
		tmp = x;
	else
		tmp = Float64(Float64(x * log(Float64(1.0 + x))) / Float64(Float64(1.0 + x) - 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((1.0 + x) == 1.0)
		tmp = x;
	else
		tmp = (x * log((1.0 + x))) / ((1.0 + x) - 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

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

Reproduce

herbie shell --seed 2024196 
(FPCore (x)
  :name "ln(1 + x)"
  :precision binary64

  :alt
  (! :herbie-platform default (if (== (+ 1 x) 1) x (/ (* x (log (+ 1 x))) (- (+ 1 x) 1))))

  (log (+ 1.0 x)))