expq2 (section 3.11)

Percentage Accurate: 37.9% → 100.0%

Time: 8.0s

Alternatives: 12

Speedup: 68.3×

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: 37.9% 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, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{expm1}\left(0 - x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -1.0 (expm1 (- 0.0 x))))

C

double code(double x) {
	return -1.0 / expm1((0.0 - x));
}

Java

public static double code(double x) {
	return -1.0 / Math.expm1((0.0 - x));
}

Python

def code(x):
	return -1.0 / math.expm1((0.0 - x))

Julia

function code(x)
	return Float64(-1.0 / expm1(Float64(0.0 - x)))
end

Wolfram

code[x_] := N[(-1.0 / N[(Exp[N[(0.0 - x), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.0%

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

   3. expm1-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

   4. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{e^{x}}{\mathsf{expm1}\left(x\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)\right)}\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\color{blue}{\frac{e^{x} - 1}{e^{x}}}\right)\right)\right) \]

   5. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \color{blue}{\frac{e^{x} - 1}{e^{x}}}\right)\right) \]

   6. div-sub N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \left(\frac{e^{x}}{e^{x}} - \color{blue}{\frac{1}{e^{x}}}\right)\right)\right) \]

   7. *-inverses N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \left(1 - \frac{\color{blue}{1}}{e^{x}}\right)\right)\right) \]

   8. associate--r- N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(0 - 1\right) + \color{blue}{\frac{1}{e^{x}}}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(-1 + \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(\mathsf{neg}\left(1\right)\right) + \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1}{e^{x}} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1}{e^{x}} - \color{blue}{1}\right)\right) \]

   13. rec-exp N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(e^{\mathsf{neg}\left(x\right)} - 1\right)\right) \]

   14. expm1-define N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{expm1}\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   15. rem-log-exp N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{expm1}\left(\log \left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right) \]

   16. rec-exp N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{expm1}\left(\log \left(\frac{1}{e^{x}}\right)\right)\right)\right) \]

   17. expm1-lowering-expm1.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\log \left(\frac{1}{e^{x}}\right)\right)\right) \]

   18. rec-exp N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\log \left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

   19. rem-log-exp N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   20. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\left(0 - x\right)\right)\right) \]

   21. --lowering--.f64 100.0%

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(0 - x\right)}} \]
7. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   2. neg-lowering-neg.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

   \[\leadsto \frac{-1}{\mathsf{expm1}\left(0 - x\right)} \]
10. Add Preprocessing

Alternative 2: 99.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9:\\ \;\;\;\;\frac{e^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.9)
   (/ (exp x) x)
   (+ (/ 1.0 x) (+ 0.5 (* x 0.08333333333333333)))))

C

double code(double x) {
	double tmp;
	if (x <= -3.9) {
		tmp = exp(x) / x;
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.9d0)) then
        tmp = exp(x) / x
    else
        tmp = (1.0d0 / x) + (0.5d0 + (x * 0.08333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.9) {
		tmp = Math.exp(x) / x;
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.9:
		tmp = math.exp(x) / x
	else:
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.9)
		tmp = Float64(exp(x) / x);
	else
		tmp = Float64(Float64(1.0 / x) + Float64(0.5 + Float64(x * 0.08333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.9)
		tmp = exp(x) / x;
	else
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.9], N[(N[Exp[x], $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.89999999999999991

   1. Initial program 100.0%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \color{blue}{x}\right) \]
   6. Step-by-step derivation

   7. Simplified 98.9%

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

   if -3.89999999999999991 < x

   1. Initial program 7.5%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r* N/A

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

      7. lft-mult-inverse N/A

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

      8. *-lft-identity N/A

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

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{\frac{1}{2}} + \frac{1}{12} \cdot x\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{12} \cdot x\right)}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\frac{1}{12}}\right)\right)\right) \]

      13. *-lowering-*.f64 98.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{12}}\right)\right)\right) \]

   7. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 91.4% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) / (x * ((-1.0d0) + (x * (0.5d0 + (x * ((-0.16666666666666666d0) + (x * 0.041666666666666664d0)))))))
end function

Java

public static double code(double x) {
	return -1.0 / (x * (-1.0 + (x * (0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664)))))));
}

Python

def code(x):
	return -1.0 / (x * (-1.0 + (x * (0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664)))))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = -1.0 / (x * (-1.0 + (x * (0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664)))))));
end

Wolfram

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

Derivation

1. Initial program 35.0%

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

   3. expm1-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

   4. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{e^{x}}{\mathsf{expm1}\left(x\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)\right)}\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\color{blue}{\frac{e^{x} - 1}{e^{x}}}\right)\right)\right) \]

   5. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \color{blue}{\frac{e^{x} - 1}{e^{x}}}\right)\right) \]

   6. div-sub N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \left(\frac{e^{x}}{e^{x}} - \color{blue}{\frac{1}{e^{x}}}\right)\right)\right) \]

   7. *-inverses N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \left(1 - \frac{\color{blue}{1}}{e^{x}}\right)\right)\right) \]

   8. associate--r- N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(0 - 1\right) + \color{blue}{\frac{1}{e^{x}}}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(-1 + \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(\mathsf{neg}\left(1\right)\right) + \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1}{e^{x}} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1}{e^{x}} - \color{blue}{1}\right)\right) \]

   13. rec-exp N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(e^{\mathsf{neg}\left(x\right)} - 1\right)\right) \]

   14. expm1-define N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{expm1}\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   15. rem-log-exp N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{expm1}\left(\log \left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right) \]

   16. rec-exp N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{expm1}\left(\log \left(\frac{1}{e^{x}}\right)\right)\right)\right) \]

   17. expm1-lowering-expm1.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\log \left(\frac{1}{e^{x}}\right)\right)\right) \]

   18. rec-exp N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\log \left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

   19. rem-log-exp N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   20. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\left(0 - x\right)\right)\right) \]

   21. --lowering--.f64 100.0%

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(0 - x\right)}} \]

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)\right)}\right) \]
8. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)}\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + -1\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \left(-1 + \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right)}\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot x - \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot x + \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{-1}{6} + \color{blue}{\frac{1}{24} \cdot x}\right)\right)\right)\right)\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \left(x \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 93.6%

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

9. Simplified 93.6%

   \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)}} \]
10. Add Preprocessing

Alternative 4: 89.0% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3:\\ \;\;\;\;\frac{6 + \frac{18}{x}}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.3)
   (/ (+ 6.0 (/ 18.0 x)) (* x (* x x)))
   (+ (/ 1.0 x) (+ 0.5 (* x 0.08333333333333333)))))

C

double code(double x) {
	double tmp;
	if (x <= -3.3) {
		tmp = (6.0 + (18.0 / x)) / (x * (x * x));
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.3d0)) then
        tmp = (6.0d0 + (18.0d0 / x)) / (x * (x * x))
    else
        tmp = (1.0d0 / x) + (0.5d0 + (x * 0.08333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.3) {
		tmp = (6.0 + (18.0 / x)) / (x * (x * x));
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.3:
		tmp = (6.0 + (18.0 / x)) / (x * (x * x))
	else:
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.3)
		tmp = Float64(Float64(6.0 + Float64(18.0 / x)) / Float64(x * Float64(x * x)));
	else
		tmp = Float64(Float64(1.0 / x) + Float64(0.5 + Float64(x * 0.08333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.3)
		tmp = (6.0 + (18.0 / x)) / (x * (x * x));
	else
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.3], N[(N[(6.0 + N[(18.0 / x), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.2999999999999998

   1. Initial program 100.0%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{x}}{\mathsf{expm1}\left(x\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\color{blue}{\frac{e^{x} - 1}{e^{x}}}\right)\right)\right) \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \color{blue}{\frac{e^{x} - 1}{e^{x}}}\right)\right) \]

      6. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \left(\frac{e^{x}}{e^{x}} - \color{blue}{\frac{1}{e^{x}}}\right)\right)\right) \]

      7. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \left(1 - \frac{\color{blue}{1}}{e^{x}}\right)\right)\right) \]

      8. associate--r- N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(0 - 1\right) + \color{blue}{\frac{1}{e^{x}}}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(-1 + \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(\mathsf{neg}\left(1\right)\right) + \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1}{e^{x}} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1}{e^{x}} - \color{blue}{1}\right)\right) \]

      13. rec-exp N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(e^{\mathsf{neg}\left(x\right)} - 1\right)\right) \]

      14. expm1-define N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{expm1}\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. rem-log-exp N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{expm1}\left(\log \left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right) \]

      16. rec-exp N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{expm1}\left(\log \left(\frac{1}{e^{x}}\right)\right)\right)\right) \]

      17. expm1-lowering-expm1.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\log \left(\frac{1}{e^{x}}\right)\right)\right) \]

      18. rec-exp N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\log \left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      19. rem-log-exp N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      20. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\left(0 - x\right)\right)\right) \]

      21. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(0 - x\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right)\right)}\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + -1\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \left(-1 + \color{blue}{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{6} \cdot x\right)}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 70.4%

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right) \]

   9. Simplified 70.4%

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

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{6 + 18 \cdot \frac{1}{x}}{{x}^{3}}} \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\left(6 + 18 \cdot \frac{1}{x}\right), \color{blue}{\left({x}^{3}\right)}\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(6, \left(18 \cdot \frac{1}{x}\right)\right), \left({\color{blue}{x}}^{3}\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(6, \left(\frac{18 \cdot 1}{x}\right)\right), \left({x}^{3}\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(6, \left(\frac{18}{x}\right)\right), \left({x}^{3}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(6, \mathsf{/.f64}\left(18, x\right)\right), \left({x}^{3}\right)\right) \]

       6. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(6, \mathsf{/.f64}\left(18, x\right)\right), \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(6, \mathsf{/.f64}\left(18, x\right)\right), \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(6, \mathsf{/.f64}\left(18, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(6, \mathsf{/.f64}\left(18, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

       10. *-lowering-*.f64 71.5%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(6, \mathsf{/.f64}\left(18, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   12. Simplified 71.5%

       \[\leadsto \color{blue}{\frac{6 + \frac{18}{x}}{x \cdot \left(x \cdot x\right)}} \]

   if -3.2999999999999998 < x

   1. Initial program 7.5%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r* N/A

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

      7. lft-mult-inverse N/A

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

      8. *-lft-identity N/A

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

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{\frac{1}{2}} + \frac{1}{12} \cdot x\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{12} \cdot x\right)}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\frac{1}{12}}\right)\right)\right) \]

      13. *-lowering-*.f64 98.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{12}}\right)\right)\right) \]

   7. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 88.7% accurate, 13.7× speedup

Math

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

FPCore

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

C

double code(double x) {
	return -1.0 / (x / (-1.0 / (1.0 - (x * (0.5 + (x * -0.16666666666666666))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) / (x / ((-1.0d0) / (1.0d0 - (x * (0.5d0 + (x * (-0.16666666666666666d0)))))))
end function

Java

public static double code(double x) {
	return -1.0 / (x / (-1.0 / (1.0 - (x * (0.5 + (x * -0.16666666666666666))))));
}

Python

def code(x):
	return -1.0 / (x / (-1.0 / (1.0 - (x * (0.5 + (x * -0.16666666666666666))))))

Julia

function code(x)
	return Float64(-1.0 / Float64(x / Float64(-1.0 / Float64(1.0 - Float64(x * Float64(0.5 + Float64(x * -0.16666666666666666)))))))
end

MATLAB

function tmp = code(x)
	tmp = -1.0 / (x / (-1.0 / (1.0 - (x * (0.5 + (x * -0.16666666666666666))))));
end

Wolfram

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

Derivation

1. Initial program 35.0%

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

   3. expm1-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

   4. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{e^{x}}{\mathsf{expm1}\left(x\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)\right)}\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\color{blue}{\frac{e^{x} - 1}{e^{x}}}\right)\right)\right) \]

   5. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \color{blue}{\frac{e^{x} - 1}{e^{x}}}\right)\right) \]

   6. div-sub N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \left(\frac{e^{x}}{e^{x}} - \color{blue}{\frac{1}{e^{x}}}\right)\right)\right) \]

   7. *-inverses N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \left(1 - \frac{\color{blue}{1}}{e^{x}}\right)\right)\right) \]

   8. associate--r- N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(0 - 1\right) + \color{blue}{\frac{1}{e^{x}}}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(-1 + \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(\mathsf{neg}\left(1\right)\right) + \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1}{e^{x}} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1}{e^{x}} - \color{blue}{1}\right)\right) \]

   13. rec-exp N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(e^{\mathsf{neg}\left(x\right)} - 1\right)\right) \]

   14. expm1-define N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{expm1}\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   15. rem-log-exp N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{expm1}\left(\log \left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right) \]

   16. rec-exp N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{expm1}\left(\log \left(\frac{1}{e^{x}}\right)\right)\right)\right) \]

   17. expm1-lowering-expm1.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\log \left(\frac{1}{e^{x}}\right)\right)\right) \]

   18. rec-exp N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\log \left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

   19. rem-log-exp N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   20. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\left(0 - x\right)\right)\right) \]

   21. --lowering--.f64 100.0%

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(0 - x\right)}} \]

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right)\right)}\right) \]
8. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right)}\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + -1\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \left(-1 + \color{blue}{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}\right)\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{6} \cdot x\right)}\right)\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 90.2%

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right) \]

9. Simplified 90.2%

   \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}} \]
10. Step-by-step derivation

    1. /-rgt-identity N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{x}{1} \cdot \left(\color{blue}{-1} + x \cdot \left(\frac{1}{2} + x \cdot \frac{-1}{6}\right)\right)\right)\right) \]

    2. associate-/r/ N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{x}{\color{blue}{\frac{1}{-1 + x \cdot \left(\frac{1}{2} + x \cdot \frac{-1}{6}\right)}}}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{1}{-1 + x \cdot \left(\frac{1}{2} + x \cdot \frac{-1}{6}\right)}\right)}\right)\right) \]

    4. frac-2neg N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(x, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{\mathsf{neg}\left(\left(-1 + x \cdot \left(\frac{1}{2} + x \cdot \frac{-1}{6}\right)\right)\right)}}\right)\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(x, \left(\frac{-1}{\mathsf{neg}\left(\color{blue}{\left(-1 + x \cdot \left(\frac{1}{2} + x \cdot \frac{-1}{6}\right)\right)}\right)}\right)\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{\left(\mathsf{neg}\left(\left(-1 + x \cdot \left(\frac{1}{2} + x \cdot \frac{-1}{6}\right)\right)\right)\right)}\right)\right)\right) \]

    7. distribute-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(-1, \left(\left(\mathsf{neg}\left(-1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(x \cdot \left(\frac{1}{2} + x \cdot \frac{-1}{6}\right)\right)\right)}\right)\right)\right)\right) \]

    8. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(-1, \left(1 + \left(\mathsf{neg}\left(\color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \frac{-1}{6}\right)}\right)\right)\right)\right)\right)\right) \]

    9. unsub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(-1, \left(1 - \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \frac{-1}{6}\right)}\right)\right)\right)\right) \]

    10. --lowering--.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \frac{-1}{6}\right)\right)}\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + x \cdot \frac{-1}{6}\right)}\right)\right)\right)\right)\right) \]

    12. +-lowering-+.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(x \cdot \frac{-1}{6}\right)}\right)\right)\right)\right)\right)\right) \]

    13. *-lowering-*.f64 90.2%

        \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

11. Applied egg-rr 90.2%

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

Alternative 6: 89.0% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;\frac{6}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.2)
   (/ 6.0 (* x (* x x)))
   (+ (/ 1.0 x) (+ 0.5 (* x 0.08333333333333333)))))

C

double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = 6.0 / (x * (x * x));
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4.2d0)) then
        tmp = 6.0d0 / (x * (x * x))
    else
        tmp = (1.0d0 / x) + (0.5d0 + (x * 0.08333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = 6.0 / (x * (x * x));
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.2:
		tmp = 6.0 / (x * (x * x))
	else:
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.2)
		tmp = Float64(6.0 / Float64(x * Float64(x * x)));
	else
		tmp = Float64(Float64(1.0 / x) + Float64(0.5 + Float64(x * 0.08333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.2)
		tmp = 6.0 / (x * (x * x));
	else
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.2], N[(6.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.20000000000000018

   1. Initial program 100.0%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{x}}{\mathsf{expm1}\left(x\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\color{blue}{\frac{e^{x} - 1}{e^{x}}}\right)\right)\right) \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \color{blue}{\frac{e^{x} - 1}{e^{x}}}\right)\right) \]

      6. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \left(\frac{e^{x}}{e^{x}} - \color{blue}{\frac{1}{e^{x}}}\right)\right)\right) \]

      7. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \left(1 - \frac{\color{blue}{1}}{e^{x}}\right)\right)\right) \]

      8. associate--r- N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(0 - 1\right) + \color{blue}{\frac{1}{e^{x}}}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(-1 + \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(\mathsf{neg}\left(1\right)\right) + \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1}{e^{x}} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1}{e^{x}} - \color{blue}{1}\right)\right) \]

      13. rec-exp N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(e^{\mathsf{neg}\left(x\right)} - 1\right)\right) \]

      14. expm1-define N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{expm1}\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. rem-log-exp N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{expm1}\left(\log \left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right) \]

      16. rec-exp N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{expm1}\left(\log \left(\frac{1}{e^{x}}\right)\right)\right)\right) \]

      17. expm1-lowering-expm1.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\log \left(\frac{1}{e^{x}}\right)\right)\right) \]

      18. rec-exp N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\log \left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      19. rem-log-exp N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      20. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\left(0 - x\right)\right)\right) \]

      21. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(0 - x\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right)\right)}\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + -1\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \left(-1 + \color{blue}{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{6} \cdot x\right)}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 70.4%

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right) \]

   9. Simplified 70.4%

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

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{6}{{x}^{3}}} \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left({x}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(6, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

       6. *-lowering-*.f64 71.5%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   12. Simplified 71.5%

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

   if -4.20000000000000018 < x

   1. Initial program 7.5%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r* N/A

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

      7. lft-mult-inverse N/A

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

      8. *-lft-identity N/A

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

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{\frac{1}{2}} + \frac{1}{12} \cdot x\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{12} \cdot x\right)}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\frac{1}{12}}\right)\right)\right) \]

      13. *-lowering-*.f64 98.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{12}}\right)\right)\right) \]

   7. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 88.7% accurate, 15.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	return -1.0 / (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) / (x * ((-1.0d0) + (x * (0.5d0 + (x * (-0.16666666666666666d0))))))
end function

Java

public static double code(double x) {
	return -1.0 / (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))));
}

Python

def code(x):
	return -1.0 / (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))

Julia

function code(x)
	return Float64(-1.0 / Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * -0.16666666666666666))))))
end

MATLAB

function tmp = code(x)
	tmp = -1.0 / (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))));
end

Wolfram

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

Derivation

1. Initial program 35.0%

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

   3. expm1-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

   4. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{e^{x}}{\mathsf{expm1}\left(x\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)\right)}\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\color{blue}{\frac{e^{x} - 1}{e^{x}}}\right)\right)\right) \]

   5. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \color{blue}{\frac{e^{x} - 1}{e^{x}}}\right)\right) \]

   6. div-sub N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \left(\frac{e^{x}}{e^{x}} - \color{blue}{\frac{1}{e^{x}}}\right)\right)\right) \]

   7. *-inverses N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \left(1 - \frac{\color{blue}{1}}{e^{x}}\right)\right)\right) \]

   8. associate--r- N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(0 - 1\right) + \color{blue}{\frac{1}{e^{x}}}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(-1 + \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(\mathsf{neg}\left(1\right)\right) + \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1}{e^{x}} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1}{e^{x}} - \color{blue}{1}\right)\right) \]

   13. rec-exp N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(e^{\mathsf{neg}\left(x\right)} - 1\right)\right) \]

   14. expm1-define N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{expm1}\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   15. rem-log-exp N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{expm1}\left(\log \left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right) \]

   16. rec-exp N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{expm1}\left(\log \left(\frac{1}{e^{x}}\right)\right)\right)\right) \]

   17. expm1-lowering-expm1.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\log \left(\frac{1}{e^{x}}\right)\right)\right) \]

   18. rec-exp N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\log \left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

   19. rem-log-exp N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   20. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\left(0 - x\right)\right)\right) \]

   21. --lowering--.f64 100.0%

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(0 - x\right)}} \]

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right)\right)}\right) \]
8. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right)}\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + -1\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \left(-1 + \color{blue}{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}\right)\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{6} \cdot x\right)}\right)\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 90.2%

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right) \]

9. Simplified 90.2%

   \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}} \]
10. Add Preprocessing

Alternative 8: 88.5% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85:\\ \;\;\;\;\frac{6}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.85) (/ 6.0 (* x (* x x))) (+ (/ 1.0 x) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= -1.85) {
		tmp = 6.0 / (x * (x * x));
	} else {
		tmp = (1.0 / x) + 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.85d0)) then
        tmp = 6.0d0 / (x * (x * x))
    else
        tmp = (1.0d0 / x) + 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.85) {
		tmp = 6.0 / (x * (x * x));
	} else {
		tmp = (1.0 / x) + 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.85:
		tmp = 6.0 / (x * (x * x))
	else:
		tmp = (1.0 / x) + 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.85)
		tmp = Float64(6.0 / Float64(x * Float64(x * x)));
	else
		tmp = Float64(Float64(1.0 / x) + 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.85)
		tmp = 6.0 / (x * (x * x));
	else
		tmp = (1.0 / x) + 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.85], N[(6.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8500000000000001

   1. Initial program 100.0%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{x}}{\mathsf{expm1}\left(x\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\color{blue}{\frac{e^{x} - 1}{e^{x}}}\right)\right)\right) \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \color{blue}{\frac{e^{x} - 1}{e^{x}}}\right)\right) \]

      6. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \left(\frac{e^{x}}{e^{x}} - \color{blue}{\frac{1}{e^{x}}}\right)\right)\right) \]

      7. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \left(1 - \frac{\color{blue}{1}}{e^{x}}\right)\right)\right) \]

      8. associate--r- N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(0 - 1\right) + \color{blue}{\frac{1}{e^{x}}}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(-1 + \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(\mathsf{neg}\left(1\right)\right) + \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1}{e^{x}} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1}{e^{x}} - \color{blue}{1}\right)\right) \]

      13. rec-exp N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(e^{\mathsf{neg}\left(x\right)} - 1\right)\right) \]

      14. expm1-define N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{expm1}\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. rem-log-exp N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{expm1}\left(\log \left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right) \]

      16. rec-exp N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{expm1}\left(\log \left(\frac{1}{e^{x}}\right)\right)\right)\right) \]

      17. expm1-lowering-expm1.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\log \left(\frac{1}{e^{x}}\right)\right)\right) \]

      18. rec-exp N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\log \left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      19. rem-log-exp N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      20. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\left(0 - x\right)\right)\right) \]

      21. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(0 - x\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right)\right)}\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + -1\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \left(-1 + \color{blue}{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{6} \cdot x\right)}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 70.4%

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right) \]

   9. Simplified 70.4%

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

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{6}{{x}^{3}}} \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left({x}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(6, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

       6. *-lowering-*.f64 71.5%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   12. Simplified 71.5%

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

   if -1.8500000000000001 < x

   1. Initial program 7.5%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. rgt-mult-inverse N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\frac{1}{2}}\right) \]

      9. /-lowering-/.f64 98.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \frac{1}{2}\right) \]

   7. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 83.4% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.8) (/ -2.0 (* x x)) (+ (/ 1.0 x) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= -1.8) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = (1.0 / x) + 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.8d0)) then
        tmp = (-2.0d0) / (x * x)
    else
        tmp = (1.0d0 / x) + 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.8) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = (1.0 / x) + 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.8:
		tmp = -2.0 / (x * x)
	else:
		tmp = (1.0 / x) + 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.8)
		tmp = Float64(-2.0 / Float64(x * x));
	else
		tmp = Float64(Float64(1.0 / x) + 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.8)
		tmp = -2.0 / (x * x);
	else
		tmp = (1.0 / x) + 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.8], N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.80000000000000004

   1. Initial program 100.0%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{x}}{\mathsf{expm1}\left(x\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\color{blue}{\frac{e^{x} - 1}{e^{x}}}\right)\right)\right) \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \color{blue}{\frac{e^{x} - 1}{e^{x}}}\right)\right) \]

      6. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \left(\frac{e^{x}}{e^{x}} - \color{blue}{\frac{1}{e^{x}}}\right)\right)\right) \]

      7. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(0 - \left(1 - \frac{\color{blue}{1}}{e^{x}}\right)\right)\right) \]

      8. associate--r- N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(0 - 1\right) + \color{blue}{\frac{1}{e^{x}}}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(-1 + \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(\mathsf{neg}\left(1\right)\right) + \frac{\color{blue}{1}}{e^{x}}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1}{e^{x}} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1}{e^{x}} - \color{blue}{1}\right)\right) \]

      13. rec-exp N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(e^{\mathsf{neg}\left(x\right)} - 1\right)\right) \]

      14. expm1-define N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{expm1}\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. rem-log-exp N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{expm1}\left(\log \left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right) \]

      16. rec-exp N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{expm1}\left(\log \left(\frac{1}{e^{x}}\right)\right)\right)\right) \]

      17. expm1-lowering-expm1.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\log \left(\frac{1}{e^{x}}\right)\right)\right) \]

      18. rec-exp N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\log \left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      19. rem-log-exp N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      20. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\left(0 - x\right)\right)\right) \]

      21. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(0 - x\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot x - 1\right)\right)}\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot x - 1\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot x + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot x + -1\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot x\right), \color{blue}{-1}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(x \cdot \frac{1}{2}\right), -1\right)\right)\right) \]

      6. *-lowering-*.f64 57.7%

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), -1\right)\right)\right) \]

   9. Simplified 57.7%

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

   10. Taylor expanded in x around inf

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

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

          \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{\left({x}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \left(x \cdot \color{blue}{x}\right)\right) \]

       3. *-lowering-*.f64 57.7%

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   12. Simplified 57.7%

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

   if -1.80000000000000004 < x

   1. Initial program 7.5%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

      4. expm1-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. rgt-mult-inverse N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\frac{1}{2}}\right) \]

      9. /-lowering-/.f64 98.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \frac{1}{2}\right) \]

   7. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 66.7% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return (1.0 / x) + 0.5;
}

Python

def code(x):
	return (1.0 / x) + 0.5

Julia

function code(x)
	return Float64(Float64(1.0 / x) + 0.5)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.0%

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

   3. expm1-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

   4. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

   1. *-lft-identity N/A

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

   2. associate-*l/ N/A

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

   3. distribute-rgt-in N/A

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

   4. associate-*l* N/A

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

   5. rgt-mult-inverse N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity N/A

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

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\frac{1}{2}}\right) \]

   9. /-lowering-/.f64 70.0%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \frac{1}{2}\right) \]

7. Simplified 70.0%

   \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]
8. Add Preprocessing

Alternative 11: 66.7% accurate, 68.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return 1.0 / x;
}

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.0%

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

   3. expm1-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

   4. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 69.5%

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{x}\right) \]

7. Simplified 69.5%

   \[\leadsto \color{blue}{\frac{1}{x}} \]
8. Add Preprocessing

Alternative 12: 3.3% accurate, 205.0× 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 35.0%

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

   3. expm1-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{expm1}\left(x\right)\right)\right) \]

   4. expm1-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

   1. *-lft-identity N/A

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

   2. associate-*l/ N/A

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

   3. distribute-rgt-in N/A

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

   4. associate-*l* N/A

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

   5. rgt-mult-inverse N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity N/A

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

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\frac{1}{2}}\right) \]

   9. /-lowering-/.f64 70.0%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \frac{1}{2}\right) \]

7. Simplified 70.0%

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

8. Taylor expanded in x around inf

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

10. Simplified 3.4%

    \[\leadsto \color{blue}{0.5} \]
11. Add Preprocessing

Developer Target 1: 100.0% accurate, 2.0× 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 2024164 
(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)))