expq2 (section 3.11)

Percentage Accurate: 36.9% → 100.0%

Time: 6.2s

Alternatives: 12

Speedup: 9.3×

Specification

\[710 > x\]

Math

\[\begin{array}{l} \\ \frac{e^{x}}{e^{x} - 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 36.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x}}{e^{x} - 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x}}{\mathsf{expm1}\left(x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (exp x) (expm1 x)))

C

double code(double x) {
	return exp(x) / expm1(x);
}

Java

public static double code(double x) {
	return Math.exp(x) / Math.expm1(x);
}

Python

def code(x):
	return math.exp(x) / math.expm1(x)

Julia

function code(x)
	return Float64(exp(x) / expm1(x))
end

Wolfram

code[x_] := N[(N[Exp[x], $MachinePrecision] / N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.3%

   \[\frac{e^{x}}{e^{x} - 1} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{e^{x}}{\color{blue}{e^{x} - 1}} \]

   2. lift-exp.f64 N/A

      \[\leadsto \frac{e^{x}}{\color{blue}{e^{x}} - 1} \]

   3. lower-expm1.f64 100.0

      \[\leadsto \frac{e^{x}}{\color{blue}{\mathsf{expm1}\left(x\right)}} \]

4. Applied rewrites 100.0%

   \[\leadsto \frac{e^{x}}{\color{blue}{\mathsf{expm1}\left(x\right)}} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} \leq 0:\\ \;\;\;\;\frac{e^{x}}{\left(1 + x\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, {x}^{-1} - -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp x) 0.0)
   (/ (exp x) (- (+ 1.0 x) 1.0))
   (fma
    (fma (* x x) -0.001388888888888889 0.08333333333333333)
    x
    (- (pow x -1.0) -0.5))))

C

double code(double x) {
	double tmp;
	if (exp(x) <= 0.0) {
		tmp = exp(x) / ((1.0 + x) - 1.0);
	} else {
		tmp = fma(fma((x * x), -0.001388888888888889, 0.08333333333333333), x, (pow(x, -1.0) - -0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (exp(x) <= 0.0)
		tmp = Float64(exp(x) / Float64(Float64(1.0 + x) - 1.0));
	else
		tmp = fma(fma(Float64(x * x), -0.001388888888888889, 0.08333333333333333), x, Float64((x ^ -1.0) - -0.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Exp[x], $MachinePrecision], 0.0], N[(N[Exp[x], $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.08333333333333333), $MachinePrecision] * x + N[(N[Power[x, -1.0], $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 x) < 0.0

   1. Initial program 100.0%

      \[\frac{e^{x}}{e^{x} - 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 x)

   1. Initial program 6.4%

      \[\frac{e^{x}}{e^{x} - 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{e^{x}}{\color{blue}{e^{x} - 1}} \]

      2. lift-exp.f64 N/A

         \[\leadsto \frac{e^{x}}{\color{blue}{e^{x}} - 1} \]

      3. lower-expm1.f64 100.0

         \[\leadsto \frac{e^{x}}{\color{blue}{\mathsf{expm1}\left(x\right)}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \frac{e^{x}}{\color{blue}{\mathsf{expm1}\left(x\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{1 + \color{blue}{\left(x \cdot \frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{1 + \left(\color{blue}{\frac{1}{2} \cdot x} + x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)}{x} \]

      3. associate-+r+ N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot x\right) + x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}}{x} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right) + \left(1 + \frac{1}{2} \cdot x\right)}}{x} \]

      5. div-add N/A

         \[\leadsto \color{blue}{\frac{x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x} + \frac{1 + \frac{1}{2} \cdot x}{x}} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x\right) \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)}}{x} + \frac{1 + \frac{1}{2} \cdot x}{x} \]

      7. unpow2 N/A

         \[\leadsto \frac{\color{blue}{{x}^{2}} \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)}{x} + \frac{1 + \frac{1}{2} \cdot x}{x} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2}}}{x} + \frac{1 + \frac{1}{2} \cdot x}{x} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot \frac{{x}^{2}}{x}} + \frac{1 + \frac{1}{2} \cdot x}{x} \]

      10. unpow2 N/A

          \[\leadsto \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot \frac{\color{blue}{x \cdot x}}{x} + \frac{1 + \frac{1}{2} \cdot x}{x} \]

      11. associate-/l* N/A

          \[\leadsto \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot \color{blue}{\left(x \cdot \frac{x}{x}\right)} + \frac{1 + \frac{1}{2} \cdot x}{x} \]

      12. *-inverses N/A

          \[\leadsto \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot \left(x \cdot \color{blue}{1}\right) + \frac{1 + \frac{1}{2} \cdot x}{x} \]

      13. *-rgt-identity N/A

          \[\leadsto \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot \color{blue}{x} + \frac{1 + \frac{1}{2} \cdot x}{x} \]

      14. div-add N/A

          \[\leadsto \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x + \color{blue}{\left(\frac{1}{x} + \frac{\frac{1}{2} \cdot x}{x}\right)} \]

      15. associate-/l* N/A

          \[\leadsto \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x + \left(\frac{1}{x} + \color{blue}{\frac{1}{2} \cdot \frac{x}{x}}\right) \]

      16. *-inverses N/A

          \[\leadsto \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x + \left(\frac{1}{x} + \frac{1}{2} \cdot \color{blue}{1}\right) \]

      17. metadata-eval N/A

          \[\leadsto \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x + \left(\frac{1}{x} + \color{blue}{\frac{1}{2}}\right) \]

      18. +-commutative N/A

          \[\leadsto \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x + \color{blue}{\left(\frac{1}{2} + \frac{1}{x}\right)} \]

   7. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, \frac{1}{x} - -0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \leq 0:\\ \;\;\;\;\frac{e^{x}}{\left(1 + x\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, {x}^{-1} - -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.3%

   \[\frac{e^{x}}{e^{x} - 1} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 99.1

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

5. Applied rewrites 99.1%

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

Alternative 4: 83.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -700:\\ \;\;\;\;\frac{1}{\left(0.5 \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1} - \mathsf{fma}\left(-0.08333333333333333, x, -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -700.0)
   (/ 1.0 (* (* 0.5 x) x))
   (- (pow x -1.0) (fma -0.08333333333333333 x -0.5))))

C

double code(double x) {
	double tmp;
	if (x <= -700.0) {
		tmp = 1.0 / ((0.5 * x) * x);
	} else {
		tmp = pow(x, -1.0) - fma(-0.08333333333333333, x, -0.5);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -700.0)
		tmp = Float64(1.0 / Float64(Float64(0.5 * x) * x));
	else
		tmp = Float64((x ^ -1.0) - fma(-0.08333333333333333, x, -0.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -700.0], N[(1.0 / N[(N[(0.5 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -1.0], $MachinePrecision] - N[(-0.08333333333333333 * x + -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -700

   1. Initial program 100.0%

      \[\frac{e^{x}}{e^{x} - 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.7%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 53.7%

       \[\leadsto \frac{1}{\left(0.5 \cdot x\right) \cdot x} \]

   if -700 < x

   1. Initial program 6.4%

      \[\frac{e^{x}}{e^{x} - 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.3%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) \cdot x}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) \cdot x}} \]

       3. +-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) + 1\right)} \cdot x} \]

       4. *-commutative N/A

          \[\leadsto \frac{1}{\left(\color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) \cdot x} + 1\right) \cdot x} \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right), x, 1\right)} \cdot x} \]

       6. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2}}, x, 1\right) \cdot x} \]

       7. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{6} + \frac{1}{24} \cdot x\right) \cdot x} + \frac{1}{2}, x, 1\right) \cdot x} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{1}{24} \cdot x, x, \frac{1}{2}\right)}, x, 1\right) \cdot x} \]

       9. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot x + \frac{1}{6}}, x, \frac{1}{2}\right), x, 1\right) \cdot x} \]

       10. lower-fma.f64 97.4

           \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right)}, x, 0.5\right), x, 1\right) \cdot x} \]

   11. Applied rewrites 97.4%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x}} \]

   12. Taylor expanded in x around 0

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

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

          \[\leadsto \frac{\color{blue}{1 - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}}{x} \]

       2. div-sub N/A

          \[\leadsto \color{blue}{\frac{1}{x} - \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]

       3. distribute-lft-neg-out N/A

          \[\leadsto \frac{1}{x} - \frac{\color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right)}}{x} \]

       4. distribute-frac-neg N/A

          \[\leadsto \frac{1}{x} - \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}\right)\right)} \]

       5. *-commutative N/A

          \[\leadsto \frac{1}{x} - \left(\mathsf{neg}\left(\frac{\color{blue}{\left(\frac{1}{2} + \frac{1}{12} \cdot x\right) \cdot x}}{x}\right)\right) \]

       6. associate-/l* N/A

          \[\leadsto \frac{1}{x} - \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{12} \cdot x\right) \cdot \frac{x}{x}}\right)\right) \]

       7. *-rgt-identity N/A

          \[\leadsto \frac{1}{x} - \left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{12} \cdot x\right) \cdot \frac{\color{blue}{x \cdot 1}}{x}\right)\right) \]

       8. associate-*r/ N/A

          \[\leadsto \frac{1}{x} - \left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{12} \cdot x\right) \cdot \color{blue}{\left(x \cdot \frac{1}{x}\right)}\right)\right) \]

       9. rgt-mult-inverse N/A

          \[\leadsto \frac{1}{x} - \left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{12} \cdot x\right) \cdot \color{blue}{1}\right)\right) \]

       10. *-rgt-identity N/A

           \[\leadsto \frac{1}{x} - \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}\right)\right) \]

       11. lower--.f64 N/A

           \[\leadsto \color{blue}{\frac{1}{x} - \left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right)\right)} \]

       12. lower-/.f64 N/A

           \[\leadsto \color{blue}{\frac{1}{x}} - \left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right)\right) \]

       13. +-commutative N/A

           \[\leadsto \frac{1}{x} - \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{12} \cdot x + \frac{1}{2}\right)}\right)\right) \]

       14. distribute-neg-in N/A

           \[\leadsto \frac{1}{x} - \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{12} \cdot x\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \]

       15. distribute-lft-neg-in N/A

           \[\leadsto \frac{1}{x} - \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{12}\right)\right) \cdot x} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

       16. metadata-eval N/A

           \[\leadsto \frac{1}{x} - \left(\left(\mathsf{neg}\left(\frac{1}{12}\right)\right) \cdot x + \color{blue}{\frac{-1}{2}}\right) \]

       17. lower-fma.f64 N/A

           \[\leadsto \frac{1}{x} - \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{1}{12}\right), x, \frac{-1}{2}\right)} \]

       18. metadata-eval 99.2

           \[\leadsto \frac{1}{x} - \mathsf{fma}\left(\color{blue}{-0.08333333333333333}, x, -0.5\right) \]

   14. Applied rewrites 99.2%

       \[\leadsto \color{blue}{\frac{1}{x} - \mathsf{fma}\left(-0.08333333333333333, x, -0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -700:\\ \;\;\;\;\frac{1}{\left(0.5 \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1} - \mathsf{fma}\left(-0.08333333333333333, x, -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ {x}^{-1} - \mathsf{fma}\left(-0.08333333333333333, x, -0.5\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (pow x -1.0) (fma -0.08333333333333333 x -0.5)))

C

double code(double x) {
	return pow(x, -1.0) - fma(-0.08333333333333333, x, -0.5);
}

Julia

function code(x)
	return Float64((x ^ -1.0) - fma(-0.08333333333333333, x, -0.5))
end

Wolfram

code[x_] := N[(N[Power[x, -1.0], $MachinePrecision] - N[(-0.08333333333333333 * x + -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.3%

   \[\frac{e^{x}}{e^{x} - 1} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 82.0%

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

9. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{1}{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) \cdot x}} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) \cdot x}} \]

    3. +-commutative N/A

       \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) + 1\right)} \cdot x} \]

    4. *-commutative N/A

       \[\leadsto \frac{1}{\left(\color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) \cdot x} + 1\right) \cdot x} \]

    5. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right), x, 1\right)} \cdot x} \]

    6. +-commutative N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2}}, x, 1\right) \cdot x} \]

    7. *-commutative N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{6} + \frac{1}{24} \cdot x\right) \cdot x} + \frac{1}{2}, x, 1\right) \cdot x} \]

    8. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{1}{24} \cdot x, x, \frac{1}{2}\right)}, x, 1\right) \cdot x} \]

    9. +-commutative N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot x + \frac{1}{6}}, x, \frac{1}{2}\right), x, 1\right) \cdot x} \]

    10. lower-fma.f64 89.4

        \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right)}, x, 0.5\right), x, 1\right) \cdot x} \]

11. Applied rewrites 89.4%

    \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x}} \]

12. Taylor expanded in x around 0

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

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

       \[\leadsto \frac{\color{blue}{1 - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}}{x} \]

    2. div-sub N/A

       \[\leadsto \color{blue}{\frac{1}{x} - \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]

    3. distribute-lft-neg-out N/A

       \[\leadsto \frac{1}{x} - \frac{\color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right)}}{x} \]

    4. distribute-frac-neg N/A

       \[\leadsto \frac{1}{x} - \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}\right)\right)} \]

    5. *-commutative N/A

       \[\leadsto \frac{1}{x} - \left(\mathsf{neg}\left(\frac{\color{blue}{\left(\frac{1}{2} + \frac{1}{12} \cdot x\right) \cdot x}}{x}\right)\right) \]

    6. associate-/l* N/A

       \[\leadsto \frac{1}{x} - \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{12} \cdot x\right) \cdot \frac{x}{x}}\right)\right) \]

    7. *-rgt-identity N/A

       \[\leadsto \frac{1}{x} - \left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{12} \cdot x\right) \cdot \frac{\color{blue}{x \cdot 1}}{x}\right)\right) \]

    8. associate-*r/ N/A

       \[\leadsto \frac{1}{x} - \left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{12} \cdot x\right) \cdot \color{blue}{\left(x \cdot \frac{1}{x}\right)}\right)\right) \]

    9. rgt-mult-inverse N/A

       \[\leadsto \frac{1}{x} - \left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{12} \cdot x\right) \cdot \color{blue}{1}\right)\right) \]

    10. *-rgt-identity N/A

        \[\leadsto \frac{1}{x} - \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}\right)\right) \]

    11. lower--.f64 N/A

        \[\leadsto \color{blue}{\frac{1}{x} - \left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right)\right)} \]

    12. lower-/.f64 N/A

        \[\leadsto \color{blue}{\frac{1}{x}} - \left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right)\right) \]

    13. +-commutative N/A

        \[\leadsto \frac{1}{x} - \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{12} \cdot x + \frac{1}{2}\right)}\right)\right) \]

    14. distribute-neg-in N/A

        \[\leadsto \frac{1}{x} - \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{12} \cdot x\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \]

    15. distribute-lft-neg-in N/A

        \[\leadsto \frac{1}{x} - \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{12}\right)\right) \cdot x} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

    16. metadata-eval N/A

        \[\leadsto \frac{1}{x} - \left(\left(\mathsf{neg}\left(\frac{1}{12}\right)\right) \cdot x + \color{blue}{\frac{-1}{2}}\right) \]

    17. lower-fma.f64 N/A

        \[\leadsto \frac{1}{x} - \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{1}{12}\right), x, \frac{-1}{2}\right)} \]

    18. metadata-eval 65.1

        \[\leadsto \frac{1}{x} - \mathsf{fma}\left(\color{blue}{-0.08333333333333333}, x, -0.5\right) \]

14. Applied rewrites 65.1%

    \[\leadsto \color{blue}{\frac{1}{x} - \mathsf{fma}\left(-0.08333333333333333, x, -0.5\right)} \]

15. Final simplification 65.1%

    \[\leadsto {x}^{-1} - \mathsf{fma}\left(-0.08333333333333333, x, -0.5\right) \]
16. Add Preprocessing

Alternative 6: 67.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {x}^{-1} - -0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (pow x -1.0) -0.5))

C

double code(double x) {
	return pow(x, -1.0) - -0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** (-1.0d0)) - (-0.5d0)
end function

Java

public static double code(double x) {
	return Math.pow(x, -1.0) - -0.5;
}

Python

def code(x):
	return math.pow(x, -1.0) - -0.5

Julia

function code(x)
	return Float64((x ^ -1.0) - -0.5)
end

MATLAB

function tmp = code(x)
	tmp = (x ^ -1.0) - -0.5;
end

Wolfram

code[x_] := N[(N[Power[x, -1.0], $MachinePrecision] - -0.5), $MachinePrecision]

Derivation

1. Initial program 39.3%

   \[\frac{e^{x}}{e^{x} - 1} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

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

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

   2. div-sub N/A

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

   3. associate-/l* N/A

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

   4. *-inverses N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower--.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval 65.1

       \[\leadsto \frac{1}{x} - \color{blue}{-0.5} \]

5. Applied rewrites 65.1%

   \[\leadsto \color{blue}{\frac{1}{x} - -0.5} \]

6. Final simplification 65.1%

   \[\leadsto {x}^{-1} - -0.5 \]
7. Add Preprocessing

Alternative 7: 67.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ {x}^{-1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow x -1.0))

C

double code(double x) {
	return pow(x, -1.0);
}

Fortran

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

Java

public static double code(double x) {
	return Math.pow(x, -1.0);
}

Python

def code(x):
	return math.pow(x, -1.0)

Julia

function code(x)
	return x ^ -1.0
end

MATLAB

function tmp = code(x)
	tmp = x ^ -1.0;
end

Wolfram

code[x_] := N[Power[x, -1.0], $MachinePrecision]

Derivation

1. Initial program 39.3%

   \[\frac{e^{x}}{e^{x} - 1} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. lower-/.f64 65.0

      \[\leadsto \color{blue}{\frac{1}{x}} \]

5. Applied rewrites 65.0%

   \[\leadsto \color{blue}{\frac{1}{x}} \]

6. Final simplification 65.0%

   \[\leadsto {x}^{-1} \]
7. Add Preprocessing

Alternative 8: 90.3% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(\left(0.041666666666666664 \cdot x\right) \cdot x, x, 1\right) \cdot x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (* (fma (* (* 0.041666666666666664 x) x) x 1.0) x)))

C

double code(double x) {
	return 1.0 / (fma(((0.041666666666666664 * x) * x), x, 1.0) * x);
}

Julia

function code(x)
	return Float64(1.0 / Float64(fma(Float64(Float64(0.041666666666666664 * x) * x), x, 1.0) * x))
end

Wolfram

code[x_] := N[(1.0 / N[(N[(N[(N[(0.041666666666666664 * x), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.3%

   \[\frac{e^{x}}{e^{x} - 1} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 82.0%

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

9. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{1}{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) \cdot x}} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) \cdot x}} \]

    3. +-commutative N/A

       \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) + 1\right)} \cdot x} \]

    4. *-commutative N/A

       \[\leadsto \frac{1}{\left(\color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) \cdot x} + 1\right) \cdot x} \]

    5. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right), x, 1\right)} \cdot x} \]

    6. +-commutative N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2}}, x, 1\right) \cdot x} \]

    7. *-commutative N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{6} + \frac{1}{24} \cdot x\right) \cdot x} + \frac{1}{2}, x, 1\right) \cdot x} \]

    8. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{1}{24} \cdot x, x, \frac{1}{2}\right)}, x, 1\right) \cdot x} \]

    9. +-commutative N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot x + \frac{1}{6}}, x, \frac{1}{2}\right), x, 1\right) \cdot x} \]

    10. lower-fma.f64 89.4

        \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right)}, x, 0.5\right), x, 1\right) \cdot x} \]

11. Applied rewrites 89.4%

    \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x}} \]

12. Taylor expanded in x around inf

    \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2}, x, 1\right) \cdot x} \]
13. Step-by-step derivation

14. Applied rewrites 89.4%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\left(0.041666666666666664 \cdot x\right) \cdot x, x, 1\right) \cdot x} \]
15. Add Preprocessing

Alternative 9: 87.6% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x)))

C

double code(double x) {
	return 1.0 / (fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x);
}

Julia

function code(x)
	return Float64(1.0 / Float64(fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x))
end

Wolfram

code[x_] := N[(1.0 / N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.3%

   \[\frac{e^{x}}{e^{x} - 1} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 82.0%

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

9. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)}} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{1}{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) \cdot x}} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) \cdot x}} \]

    3. +-commutative N/A

       \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1\right)} \cdot x} \]

    4. *-commutative N/A

       \[\leadsto \frac{1}{\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot x} + 1\right) \cdot x} \]

    5. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot x, x, 1\right)} \cdot x} \]

    6. +-commutative N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, x, 1\right) \cdot x} \]

    7. lower-fma.f64 87.2

       \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, x, 0.5\right)}, x, 1\right) \cdot x} \]

11. Applied rewrites 87.2%

    \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x}} \]
12. Add Preprocessing

Alternative 10: 82.4% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.3%

   \[\frac{e^{x}}{e^{x} - 1} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 82.0%

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

Alternative 11: 3.4% accurate, 35.8× speedup

Math

\[\begin{array}{l} \\ 0.08333333333333333 \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.08333333333333333 x))

C

double code(double x) {
	return 0.08333333333333333 * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.08333333333333333d0 * x
end function

Java

public static double code(double x) {
	return 0.08333333333333333 * x;
}

Python

def code(x):
	return 0.08333333333333333 * x

Julia

function code(x)
	return Float64(0.08333333333333333 * x)
end

MATLAB

function tmp = code(x)
	tmp = 0.08333333333333333 * x;
end

Wolfram

code[x_] := N[(0.08333333333333333 * x), $MachinePrecision]

Derivation

1. Initial program 39.3%

   \[\frac{e^{x}}{e^{x} - 1} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. distribute-lft-in N/A

      \[\leadsto \frac{1 + \color{blue}{\left(x \cdot \frac{1}{2} + x \cdot \left(\frac{1}{12} \cdot x\right)\right)}}{x} \]

   2. *-commutative N/A

      \[\leadsto \frac{1 + \left(\color{blue}{\frac{1}{2} \cdot x} + x \cdot \left(\frac{1}{12} \cdot x\right)\right)}{x} \]

   3. associate-+r+ N/A

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot x\right) + x \cdot \left(\frac{1}{12} \cdot x\right)}}{x} \]

   4. div-add N/A

      \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot x}{x} + \frac{x \cdot \left(\frac{1}{12} \cdot x\right)}{x}} \]

   5. *-commutative N/A

      \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \frac{\color{blue}{\left(\frac{1}{12} \cdot x\right) \cdot x}}{x} \]

   6. associate-/l* N/A

      \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \color{blue}{\left(\frac{1}{12} \cdot x\right) \cdot \frac{x}{x}} \]

   7. *-inverses N/A

      \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \left(\frac{1}{12} \cdot x\right) \cdot \color{blue}{1} \]

   8. *-rgt-identity N/A

      \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \color{blue}{\frac{1}{12} \cdot x} \]

   9. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{12} \cdot x + \frac{1 + \frac{1}{2} \cdot x}{x}} \]

   10. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{12}, x, \frac{1 + \frac{1}{2} \cdot x}{x}\right)} \]

   11. fp-cancel-sign-sub-inv N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{\color{blue}{1 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot x}}{x}\right) \]

   12. div-sub N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \color{blue}{\frac{1}{x} - \frac{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot x}{x}}\right) \]

   13. associate-/l* N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{x}{x}}\right) \]

   14. *-inverses N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \color{blue}{1}\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \color{blue}{\frac{-1}{2}} \cdot 1\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \color{blue}{\frac{-1}{2}}\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

   18. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \color{blue}{\frac{1}{x} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

   19. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \color{blue}{\frac{1}{x}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

   20. metadata-eval 65.1

       \[\leadsto \mathsf{fma}\left(0.08333333333333333, x, \frac{1}{x} - \color{blue}{-0.5}\right) \]

5. Applied rewrites 65.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.08333333333333333, x, \frac{1}{x} - -0.5\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto \frac{1}{12} \cdot \color{blue}{x} \]
7. Step-by-step derivation

8. Applied rewrites 3.4%

   \[\leadsto 0.08333333333333333 \cdot \color{blue}{x} \]
9. Add Preprocessing

Alternative 12: 3.2% accurate, 215.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

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

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 39.3%

   \[\frac{e^{x}}{e^{x} - 1} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

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

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

   2. div-sub N/A

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

   3. associate-/l* N/A

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

   4. *-inverses N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower--.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval 65.1

       \[\leadsto \frac{1}{x} - \color{blue}{-0.5} \]

5. Applied rewrites 65.1%

   \[\leadsto \color{blue}{\frac{1}{x} - -0.5} \]

6. Taylor expanded in x around inf

   \[\leadsto \frac{1}{2} \]
7. Step-by-step derivation

8. Applied rewrites 3.3%

   \[\leadsto 0.5 \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{expm1}\left(-x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0) (expm1 (- x))))

C

double code(double x) {
	return -1.0 / expm1(-x);
}

Java

public static double code(double x) {
	return -1.0 / Math.expm1(-x);
}

Python

def code(x):
	return -1.0 / math.expm1(-x)

Julia

function code(x)
	return Float64(Float64(-1.0) / expm1(Float64(-x)))
end

Wolfram

code[x_] := N[((-1.0) / N[(Exp[(-x)] - 1), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024327 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64
  :pre (> 710.0 x)

  :alt
  (! :herbie-platform default (/ (- 1) (expm1 (- x))))

  (/ (exp x) (- (exp x) 1.0)))