expq2 (section 3.11)

Percentage Accurate: 37.2% → 100.0%

Time: 4.7s

Alternatives: 11

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: 37.2% 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, 0.7× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (pow (expm1 x) -1.0) (exp x)))

C

double code(double x) {
	return pow(expm1(x), -1.0) * exp(x);
}

Java

public static double code(double x) {
	return Math.pow(Math.expm1(x), -1.0) * Math.exp(x);
}

Python

def code(x):
	return math.pow(math.expm1(x), -1.0) * math.exp(x)

Julia

function code(x)
	return Float64((expm1(x) ^ -1.0) * exp(x))
end

Wolfram

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

Derivation

1. Initial program 37.0%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. frac-2neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. metadata-eval N/A

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

   10. lift--.f64 N/A

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

   11. lift-exp.f64 N/A

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

   12. lower-expm1.f64 N/A

       \[\leadsto \frac{-1}{\color{blue}{\mathsf{expm1}\left(x\right)}} \cdot \left(\mathsf{neg}\left(e^{x}\right)\right) \]

   13. lower-neg.f64 100.0

       \[\leadsto \frac{-1}{\mathsf{expm1}\left(x\right)} \cdot \color{blue}{\left(-e^{x}\right)} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(x\right)} \cdot \left(-e^{x}\right)} \]

5. Final simplification 100.0%

   \[\leadsto {\left(\mathsf{expm1}\left(x\right)\right)}^{-1} \cdot e^{x} \]
6. Add Preprocessing

Alternative 2: 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 37.0%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 3: 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(-1.0 / expm1(Float64(-x)))
end

Wolfram

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

Derivation

1. Initial program 37.0%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. neg-sub0 N/A

      \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \left(e^{x} - 1\right)}}{e^{x}}} \]

   8. lift--.f64 N/A

      \[\leadsto \frac{-1}{\frac{0 - \color{blue}{\left(e^{x} - 1\right)}}{e^{x}}} \]

   9. associate-+l- N/A

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

   10. neg-sub0 N/A

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

   11. +-commutative N/A

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

   12. sub-neg N/A

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

   13. div-sub N/A

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

   14. *-inverses N/A

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

   15. lift-exp.f64 N/A

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

   16. rec-exp N/A

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

   17. lower-expm1.f64 N/A

       \[\leadsto \frac{-1}{\color{blue}{\mathsf{expm1}\left(\mathsf{neg}\left(x\right)\right)}} \]

   18. lower-neg.f64 100.0

       \[\leadsto \frac{-1}{\mathsf{expm1}\left(\color{blue}{-x}\right)} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(-x\right)}} \]
5. Add Preprocessing

Alternative 4: 67.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 37.0%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. *-lft-identity N/A

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

   2. associate-/l* N/A

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

   3. associate-*l/ N/A

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

   4. distribute-lft-in N/A

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

   5. *-commutative N/A

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

   6. associate-+r+ N/A

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

   7. distribute-rgt-in N/A

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

   8. associate-/l* N/A

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

   9. *-rgt-identity N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. lft-mult-inverse N/A

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

   13. *-lft-identity N/A

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

   14. +-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. *-rgt-identity N/A

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

   17. associate-/l* N/A

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

   18. +-commutative N/A

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

   19. distribute-lft1-in N/A

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

   20. lower-+.f64 N/A

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

5. Applied rewrites 67.7%

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

6. Final simplification 67.7%

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

Alternative 5: 67.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.0%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. *-rgt-identity N/A

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

   2. associate-/l* N/A

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

   3. +-commutative N/A

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

   4. distribute-lft1-in N/A

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

   5. lower-+.f64 N/A

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

   6. associate-*l* N/A

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

   7. rgt-mult-inverse N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 67.5

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

5. Applied rewrites 67.5%

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

6. Final simplification 67.5%

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

Alternative 6: 67.3% 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 37.0%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 67.5

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

5. Applied rewrites 67.5%

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

6. Final simplification 67.5%

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

Alternative 7: 91.3% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, -0.16666666666666666\right), x, 0.5\right), x, -1\right) \cdot x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  -1.0
  (*
   (fma (fma (fma 0.041666666666666664 x -0.16666666666666666) x 0.5) x -1.0)
   x)))

C

double code(double x) {
	return -1.0 / (fma(fma(fma(0.041666666666666664, x, -0.16666666666666666), x, 0.5), x, -1.0) * x);
}

Julia

function code(x)
	return Float64(-1.0 / Float64(fma(fma(fma(0.041666666666666664, x, -0.16666666666666666), x, 0.5), x, -1.0) * x))
end

Wolfram

code[x_] := N[(-1.0 / N[(N[(N[(N[(0.041666666666666664 * x + -0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + -1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.0%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. neg-sub0 N/A

      \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \left(e^{x} - 1\right)}}{e^{x}}} \]

   8. lift--.f64 N/A

      \[\leadsto \frac{-1}{\frac{0 - \color{blue}{\left(e^{x} - 1\right)}}{e^{x}}} \]

   9. associate-+l- N/A

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

   10. neg-sub0 N/A

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

   11. +-commutative N/A

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

   12. sub-neg N/A

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

   13. div-sub N/A

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

   14. *-inverses N/A

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

   15. lift-exp.f64 N/A

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

   16. rec-exp N/A

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

   17. lower-expm1.f64 N/A

       \[\leadsto \frac{-1}{\color{blue}{\mathsf{expm1}\left(\mathsf{neg}\left(x\right)\right)}} \]

   18. lower-neg.f64 100.0

       \[\leadsto \frac{-1}{\mathsf{expm1}\left(\color{blue}{-x}\right)} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(-x\right)}} \]

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 93.1

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

7. Applied rewrites 93.1%

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

Alternative 8: 88.6% 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 37.0%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. neg-sub0 N/A

      \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \left(e^{x} - 1\right)}}{e^{x}}} \]

   8. lift--.f64 N/A

      \[\leadsto \frac{-1}{\frac{0 - \color{blue}{\left(e^{x} - 1\right)}}{e^{x}}} \]

   9. associate-+l- N/A

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

   10. neg-sub0 N/A

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

   11. +-commutative N/A

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

   12. sub-neg N/A

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

   13. div-sub N/A

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

   14. *-inverses N/A

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

   15. lift-exp.f64 N/A

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

   16. rec-exp N/A

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

   17. lower-expm1.f64 N/A

       \[\leadsto \frac{-1}{\color{blue}{\mathsf{expm1}\left(\mathsf{neg}\left(x\right)\right)}} \]

   18. lower-neg.f64 100.0

       \[\leadsto \frac{-1}{\mathsf{expm1}\left(\color{blue}{-x}\right)} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(-x\right)}} \]

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 91.0

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

7. Applied rewrites 91.0%

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

Alternative 9: 83.0% 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 37.0%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. neg-sub0 N/A

      \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \left(e^{x} - 1\right)}}{e^{x}}} \]

   8. lift--.f64 N/A

      \[\leadsto \frac{-1}{\frac{0 - \color{blue}{\left(e^{x} - 1\right)}}{e^{x}}} \]

   9. associate-+l- N/A

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

   10. neg-sub0 N/A

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

   11. +-commutative N/A

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

   12. sub-neg N/A

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

   13. div-sub N/A

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

   14. *-inverses N/A

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

   15. lift-exp.f64 N/A

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

   16. rec-exp N/A

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

   17. lower-expm1.f64 N/A

       \[\leadsto \frac{-1}{\color{blue}{\mathsf{expm1}\left(\mathsf{neg}\left(x\right)\right)}} \]

   18. lower-neg.f64 100.0

       \[\leadsto \frac{-1}{\mathsf{expm1}\left(\color{blue}{-x}\right)} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(-x\right)}} \]

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f64 84.5

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

7. Applied rewrites 84.5%

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

Alternative 10: 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 37.0%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. *-lft-identity N/A

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

   2. associate-/l* N/A

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

   3. associate-*l/ N/A

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

   4. distribute-lft-in N/A

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

   5. *-commutative N/A

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

   6. associate-+r+ N/A

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

   7. distribute-rgt-in N/A

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

   8. associate-/l* N/A

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

   9. *-rgt-identity N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. lft-mult-inverse N/A

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

   13. *-lft-identity N/A

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

   14. +-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. *-rgt-identity N/A

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

   17. associate-/l* N/A

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

   18. +-commutative N/A

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

   19. distribute-lft1-in N/A

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

   20. lower-+.f64 N/A

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

5. Applied rewrites 67.7%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 3.3%

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

Alternative 11: 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 37.0%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. *-rgt-identity N/A

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

   2. associate-/l* N/A

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

   3. +-commutative N/A

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

   4. distribute-lft1-in N/A

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

   5. lower-+.f64 N/A

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

   6. associate-*l* N/A

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

   7. rgt-mult-inverse N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 67.5

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

5. Applied rewrites 67.5%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 3.1%

   \[\leadsto 0.5 \]
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 2024306 
(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)))