expq2 (section 3.11)

Percentage Accurate: 37.3% → 100.0%

Time: 9.2s

Alternatives: 15

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: 37.3% 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 36.6%

   \[\frac{e^{x}}{e^{x} - 1} \]
2. Add Preprocessing
3. 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. distribute-neg-frac N/A

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

   6. neg-sub0 N/A

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

   7. associate-+l- N/A

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

   8. neg-sub0 N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. div-sub N/A

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

   12. rec-exp N/A

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

   13. *-inverses N/A

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

   14. accelerator-lowering-expm1.f64 N/A

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

   15. neg-lowering-neg.f64 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 98.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.6%

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

3. Taylor expanded in x around 0

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

5. Simplified 98.9%

   \[\leadsto \frac{e^{x}}{\color{blue}{x}} \]
6. Add Preprocessing

Alternative 3: 96.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\\ t_1 := x \cdot t\_0\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;x \leq -5 \cdot 10^{+51}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot \left(1 - x \cdot \left(t\_0 \cdot t\_1\right)\right)}{-1 - x \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\frac{x \cdot \left(1 - x \cdot \left(\left(x \cdot \left(t\_0 \cdot t\_0\right)\right) \cdot t\_2\right)\right)}{1 + t\_2}}{-1 - t\_1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* x (+ -0.16666666666666666 (* x 0.041666666666666664)))))
        (t_1 (* x t_0))
        (t_2 (* t_1 t_1)))
   (if (<= x -5e+51)
     (/ -1.0 (/ (* x (- 1.0 (* x (* t_0 t_1)))) (- -1.0 (* x 0.5))))
     (/
      -1.0
      (/
       (/ (* x (- 1.0 (* x (* (* x (* t_0 t_0)) t_2)))) (+ 1.0 t_2))
       (- -1.0 t_1))))))

C

double code(double x) {
	double t_0 = 0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664)));
	double t_1 = x * t_0;
	double t_2 = t_1 * t_1;
	double tmp;
	if (x <= -5e+51) {
		tmp = -1.0 / ((x * (1.0 - (x * (t_0 * t_1)))) / (-1.0 - (x * 0.5)));
	} else {
		tmp = -1.0 / (((x * (1.0 - (x * ((x * (t_0 * t_0)) * t_2)))) / (1.0 + t_2)) / (-1.0 - t_1));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 + (x * ((-0.16666666666666666d0) + (x * 0.041666666666666664d0)))
    t_1 = x * t_0
    t_2 = t_1 * t_1
    if (x <= (-5d+51)) then
        tmp = (-1.0d0) / ((x * (1.0d0 - (x * (t_0 * t_1)))) / ((-1.0d0) - (x * 0.5d0)))
    else
        tmp = (-1.0d0) / (((x * (1.0d0 - (x * ((x * (t_0 * t_0)) * t_2)))) / (1.0d0 + t_2)) / ((-1.0d0) - t_1))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664)));
	double t_1 = x * t_0;
	double t_2 = t_1 * t_1;
	double tmp;
	if (x <= -5e+51) {
		tmp = -1.0 / ((x * (1.0 - (x * (t_0 * t_1)))) / (-1.0 - (x * 0.5)));
	} else {
		tmp = -1.0 / (((x * (1.0 - (x * ((x * (t_0 * t_0)) * t_2)))) / (1.0 + t_2)) / (-1.0 - t_1));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664)))
	t_1 = x * t_0
	t_2 = t_1 * t_1
	tmp = 0
	if x <= -5e+51:
		tmp = -1.0 / ((x * (1.0 - (x * (t_0 * t_1)))) / (-1.0 - (x * 0.5)))
	else:
		tmp = -1.0 / (((x * (1.0 - (x * ((x * (t_0 * t_0)) * t_2)))) / (1.0 + t_2)) / (-1.0 - t_1))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 + Float64(x * Float64(-0.16666666666666666 + Float64(x * 0.041666666666666664))))
	t_1 = Float64(x * t_0)
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (x <= -5e+51)
		tmp = Float64(-1.0 / Float64(Float64(x * Float64(1.0 - Float64(x * Float64(t_0 * t_1)))) / Float64(-1.0 - Float64(x * 0.5))));
	else
		tmp = Float64(-1.0 / Float64(Float64(Float64(x * Float64(1.0 - Float64(x * Float64(Float64(x * Float64(t_0 * t_0)) * t_2)))) / Float64(1.0 + t_2)) / Float64(-1.0 - t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664)));
	t_1 = x * t_0;
	t_2 = t_1 * t_1;
	tmp = 0.0;
	if (x <= -5e+51)
		tmp = -1.0 / ((x * (1.0 - (x * (t_0 * t_1)))) / (-1.0 - (x * 0.5)));
	else
		tmp = -1.0 / (((x * (1.0 - (x * ((x * (t_0 * t_0)) * t_2)))) / (1.0 + t_2)) / (-1.0 - t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 + N[(x * N[(-0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, If[LessEqual[x, -5e+51], N[(-1.0 / N[(N[(x * N[(1.0 - N[(x * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(N[(x * N[(1.0 - N[(x * N[(N[(x * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -5e51

   1. Initial program 100.0%

      \[\frac{e^{x}}{e^{x} - 1} \]
   2. Add Preprocessing
   3. 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. distribute-neg-frac N/A

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

      6. neg-sub0 N/A

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

      7. associate-+l- N/A

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

      8. neg-sub0 N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. rec-exp N/A

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

      13. *-inverses N/A

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

      14. accelerator-lowering-expm1.f64 N/A

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

      15. neg-lowering-neg.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)\right)}\right) \]
   6. 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 90.8%

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

   7. Simplified 90.8%

      \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   9. Applied egg-rr 18.0%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 100.0%

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

   12. Simplified 100.0%

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

   if -5e51 < x

   1. Initial program 16.7%

      \[\frac{e^{x}}{e^{x} - 1} \]
   2. Add Preprocessing
   3. 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. distribute-neg-frac N/A

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

      6. neg-sub0 N/A

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

      7. associate-+l- N/A

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

      8. neg-sub0 N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. rec-exp N/A

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

      13. *-inverses N/A

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

      14. accelerator-lowering-expm1.f64 N/A

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

      15. neg-lowering-neg.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)\right)}\right) \]
   6. 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 88.1%

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

   7. Simplified 88.1%

      \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   9. Applied egg-rr 88.6%

      \[\leadsto \frac{-1}{\color{blue}{\frac{\left(1 - x \cdot \left(\left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right)\right) \cdot x}{-1 - x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)}}} \]
   10. Step-by-step derivation

       1. flip-- N/A

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

       2. associate-*l/ N/A

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

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

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

   11. Applied egg-rr 95.0%

       \[\leadsto \frac{-1}{\frac{\color{blue}{\frac{\left(1 - x \cdot \left(\left(x \cdot \left(\left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right) \cdot \left(\left(x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right) \cdot \left(x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right)\right)\right) \cdot x}{1 + \left(x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right) \cdot \left(x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)}}}{-1 - x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+51}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot \left(1 - x \cdot \left(\left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right)\right)}{-1 - x \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\frac{x \cdot \left(1 - x \cdot \left(\left(x \cdot \left(\left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right) \cdot \left(\left(x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right) \cdot \left(x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right)\right)\right)}{1 + \left(x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right) \cdot \left(x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)}}{-1 - x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\\ t_1 := x \cdot t\_0\\ t_2 := x \cdot \left(t\_0 \cdot t\_1\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{+51}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot \left(1 - t\_2\right)}{-1 - x \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot \left(-1 + t\_1 \cdot t\_2\right)}{1 + t\_1 \cdot \left(t\_1 - -1\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* x (+ -0.16666666666666666 (* x 0.041666666666666664)))))
        (t_1 (* x t_0))
        (t_2 (* x (* t_0 t_1))))
   (if (<= x -5e+51)
     (/ -1.0 (/ (* x (- 1.0 t_2)) (- -1.0 (* x 0.5))))
     (/ -1.0 (/ (* x (+ -1.0 (* t_1 t_2))) (+ 1.0 (* t_1 (- t_1 -1.0))))))))

C

double code(double x) {
	double t_0 = 0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664)));
	double t_1 = x * t_0;
	double t_2 = x * (t_0 * t_1);
	double tmp;
	if (x <= -5e+51) {
		tmp = -1.0 / ((x * (1.0 - t_2)) / (-1.0 - (x * 0.5)));
	} else {
		tmp = -1.0 / ((x * (-1.0 + (t_1 * t_2))) / (1.0 + (t_1 * (t_1 - -1.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 + (x * ((-0.16666666666666666d0) + (x * 0.041666666666666664d0)))
    t_1 = x * t_0
    t_2 = x * (t_0 * t_1)
    if (x <= (-5d+51)) then
        tmp = (-1.0d0) / ((x * (1.0d0 - t_2)) / ((-1.0d0) - (x * 0.5d0)))
    else
        tmp = (-1.0d0) / ((x * ((-1.0d0) + (t_1 * t_2))) / (1.0d0 + (t_1 * (t_1 - (-1.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664)));
	double t_1 = x * t_0;
	double t_2 = x * (t_0 * t_1);
	double tmp;
	if (x <= -5e+51) {
		tmp = -1.0 / ((x * (1.0 - t_2)) / (-1.0 - (x * 0.5)));
	} else {
		tmp = -1.0 / ((x * (-1.0 + (t_1 * t_2))) / (1.0 + (t_1 * (t_1 - -1.0))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664)))
	t_1 = x * t_0
	t_2 = x * (t_0 * t_1)
	tmp = 0
	if x <= -5e+51:
		tmp = -1.0 / ((x * (1.0 - t_2)) / (-1.0 - (x * 0.5)))
	else:
		tmp = -1.0 / ((x * (-1.0 + (t_1 * t_2))) / (1.0 + (t_1 * (t_1 - -1.0))))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 + Float64(x * Float64(-0.16666666666666666 + Float64(x * 0.041666666666666664))))
	t_1 = Float64(x * t_0)
	t_2 = Float64(x * Float64(t_0 * t_1))
	tmp = 0.0
	if (x <= -5e+51)
		tmp = Float64(-1.0 / Float64(Float64(x * Float64(1.0 - t_2)) / Float64(-1.0 - Float64(x * 0.5))));
	else
		tmp = Float64(-1.0 / Float64(Float64(x * Float64(-1.0 + Float64(t_1 * t_2))) / Float64(1.0 + Float64(t_1 * Float64(t_1 - -1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664)));
	t_1 = x * t_0;
	t_2 = x * (t_0 * t_1);
	tmp = 0.0;
	if (x <= -5e+51)
		tmp = -1.0 / ((x * (1.0 - t_2)) / (-1.0 - (x * 0.5)));
	else
		tmp = -1.0 / ((x * (-1.0 + (t_1 * t_2))) / (1.0 + (t_1 * (t_1 - -1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 + N[(x * N[(-0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5e+51], N[(-1.0 / N[(N[(x * N[(1.0 - t$95$2), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(x * N[(-1.0 + N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$1 * N[(t$95$1 - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -5e51

   1. Initial program 100.0%

      \[\frac{e^{x}}{e^{x} - 1} \]
   2. Add Preprocessing
   3. 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. distribute-neg-frac N/A

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

      6. neg-sub0 N/A

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

      7. associate-+l- N/A

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

      8. neg-sub0 N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. rec-exp N/A

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

      13. *-inverses N/A

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

      14. accelerator-lowering-expm1.f64 N/A

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

      15. neg-lowering-neg.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)\right)}\right) \]
   6. 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 90.8%

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

   7. Simplified 90.8%

      \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   9. Applied egg-rr 18.0%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 100.0%

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

   12. Simplified 100.0%

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

   if -5e51 < x

   1. Initial program 16.7%

      \[\frac{e^{x}}{e^{x} - 1} \]
   2. Add Preprocessing
   3. 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. distribute-neg-frac N/A

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

      6. neg-sub0 N/A

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

      7. associate-+l- N/A

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

      8. neg-sub0 N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. rec-exp N/A

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

      13. *-inverses N/A

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

      14. accelerator-lowering-expm1.f64 N/A

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

      15. neg-lowering-neg.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)\right)}\right) \]
   6. 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 88.1%

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

   7. Simplified 88.1%

      \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. flip3-+ N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{{-1}^{3} + {\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{-1}{6} + x \cdot \frac{1}{24}\right)\right)\right)}^{3}}{-1 \cdot -1 + \left(\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{-1}{6} + x \cdot \frac{1}{24}\right)\right)\right) \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{-1}{6} + x \cdot \frac{1}{24}\right)\right)\right) - -1 \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{-1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right)} \cdot x\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\left({-1}^{3} + {\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{-1}{6} + x \cdot \frac{1}{24}\right)\right)\right)}^{3}\right) \cdot x}{\color{blue}{-1 \cdot -1 + \left(\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{-1}{6} + x \cdot \frac{1}{24}\right)\right)\right) \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{-1}{6} + x \cdot \frac{1}{24}\right)\right)\right) - -1 \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{-1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\left(\left({-1}^{3} + {\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{-1}{6} + x \cdot \frac{1}{24}\right)\right)\right)}^{3}\right) \cdot x\right), \color{blue}{\left(-1 \cdot -1 + \left(\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{-1}{6} + x \cdot \frac{1}{24}\right)\right)\right) \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{-1}{6} + x \cdot \frac{1}{24}\right)\right)\right) - -1 \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{-1}{6} + x \cdot \frac{1}{24}\right)\right)\right)\right)\right)}\right)\right) \]

   9. Applied egg-rr 93.0%

      \[\leadsto \frac{-1}{\color{blue}{\frac{\left(-1 + \left(x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right) \cdot \left(x \cdot \left(\left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right)\right)\right) \cdot x}{1 + \left(x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right) \cdot \left(x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) - -1\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+51}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot \left(1 - x \cdot \left(\left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right)\right)}{-1 - x \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot \left(-1 + \left(x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right) \cdot \left(x \cdot \left(\left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right)\right)\right)}{1 + \left(x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right) \cdot \left(x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) - -1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.7% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\\ \frac{-1}{\frac{x \cdot \left(1 - x \cdot \left(t\_0 \cdot \left(x \cdot t\_0\right)\right)\right)}{-1 - x \cdot 0.5}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (+ 0.5 (* x (+ -0.16666666666666666 (* x 0.041666666666666664))))))
   (/ -1.0 (/ (* x (- 1.0 (* x (* t_0 (* x t_0))))) (- -1.0 (* x 0.5))))))

C

double code(double x) {
	double t_0 = 0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664)));
	return -1.0 / ((x * (1.0 - (x * (t_0 * (x * t_0))))) / (-1.0 - (x * 0.5)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = 0.5d0 + (x * ((-0.16666666666666666d0) + (x * 0.041666666666666664d0)))
    code = (-1.0d0) / ((x * (1.0d0 - (x * (t_0 * (x * t_0))))) / ((-1.0d0) - (x * 0.5d0)))
end function

Java

public static double code(double x) {
	double t_0 = 0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664)));
	return -1.0 / ((x * (1.0 - (x * (t_0 * (x * t_0))))) / (-1.0 - (x * 0.5)));
}

Python

def code(x):
	t_0 = 0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664)))
	return -1.0 / ((x * (1.0 - (x * (t_0 * (x * t_0))))) / (-1.0 - (x * 0.5)))

Julia

function code(x)
	t_0 = Float64(0.5 + Float64(x * Float64(-0.16666666666666666 + Float64(x * 0.041666666666666664))))
	return Float64(-1.0 / Float64(Float64(x * Float64(1.0 - Float64(x * Float64(t_0 * Float64(x * t_0))))) / Float64(-1.0 - Float64(x * 0.5))))
end

MATLAB

function tmp = code(x)
	t_0 = 0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664)));
	tmp = -1.0 / ((x * (1.0 - (x * (t_0 * (x * t_0))))) / (-1.0 - (x * 0.5)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 + N[(x * N[(-0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(-1.0 / N[(N[(x * N[(1.0 - N[(x * N[(t$95$0 * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 36.6%

   \[\frac{e^{x}}{e^{x} - 1} \]
2. Add Preprocessing
3. 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. distribute-neg-frac N/A

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

   6. neg-sub0 N/A

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

   7. associate-+l- N/A

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

   8. neg-sub0 N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. div-sub N/A

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

   12. rec-exp N/A

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

   13. *-inverses N/A

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

   14. accelerator-lowering-expm1.f64 N/A

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

   15. neg-lowering-neg.f64 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)\right)}\right) \]
6. 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 88.8%

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

7. Simplified 88.8%

   \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)}} \]
8. Step-by-step derivation

   1. *-commutative N/A

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

   2. flip-+ N/A

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

   3. associate-*l/ N/A

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

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

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

9. Applied egg-rr 71.8%

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

10. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. *-lowering-*.f64 91.1%

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

12. Simplified 91.1%

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

13. Final simplification 91.1%

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

Alternative 6: 91.6% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.6%

   \[\frac{e^{x}}{e^{x} - 1} \]
2. Add Preprocessing
3. 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. distribute-neg-frac N/A

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

   6. neg-sub0 N/A

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

   7. associate-+l- N/A

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

   8. neg-sub0 N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. div-sub N/A

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

   12. rec-exp N/A

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

   13. *-inverses N/A

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

   14. accelerator-lowering-expm1.f64 N/A

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

   15. neg-lowering-neg.f64 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)\right)}\right) \]
6. 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 88.8%

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

7. Simplified 88.8%

   \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)}} \]
8. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. *-lowering-*.f64 88.8%

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

9. Applied egg-rr 88.8%

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

Alternative 7: 91.7% 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 36.6%

   \[\frac{e^{x}}{e^{x} - 1} \]
2. Add Preprocessing
3. 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. distribute-neg-frac N/A

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

   6. neg-sub0 N/A

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

   7. associate-+l- N/A

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

   8. neg-sub0 N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. div-sub N/A

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

   12. rec-exp N/A

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

   13. *-inverses N/A

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

   14. accelerator-lowering-expm1.f64 N/A

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

   15. neg-lowering-neg.f64 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)\right)}\right) \]
6. 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 88.8%

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

7. Simplified 88.8%

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

Alternative 8: 91.9% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot \left(0.5 + x \cdot 0.08333333333333333\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.0)
   (/ -24.0 (* x (* x (* x x))))
   (/ (+ 1.0 (* x (+ 0.5 (* x 0.08333333333333333)))) x)))

C

double code(double x) {
	double tmp;
	if (x <= -4.0) {
		tmp = -24.0 / (x * (x * (x * x)));
	} else {
		tmp = (1.0 + (x * (0.5 + (x * 0.08333333333333333)))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4.0d0)) then
        tmp = (-24.0d0) / (x * (x * (x * x)))
    else
        tmp = (1.0d0 + (x * (0.5d0 + (x * 0.08333333333333333d0)))) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4.0) {
		tmp = -24.0 / (x * (x * (x * x)));
	} else {
		tmp = (1.0 + (x * (0.5 + (x * 0.08333333333333333)))) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.0:
		tmp = -24.0 / (x * (x * (x * x)))
	else:
		tmp = (1.0 + (x * (0.5 + (x * 0.08333333333333333)))) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.0)
		tmp = Float64(-24.0 / Float64(x * Float64(x * Float64(x * x))));
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.08333333333333333)))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.0)
		tmp = -24.0 / (x * (x * (x * x)));
	else
		tmp = (1.0 + (x * (0.5 + (x * 0.08333333333333333)))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.0], N[(-24.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x * N[(0.5 + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4

   1. Initial program 100.0%

      \[\frac{e^{x}}{e^{x} - 1} \]
   2. Add Preprocessing
   3. 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. distribute-neg-frac N/A

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

      6. neg-sub0 N/A

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

      7. associate-+l- N/A

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

      8. neg-sub0 N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. rec-exp N/A

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

      13. *-inverses N/A

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

      14. accelerator-lowering-expm1.f64 N/A

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

      15. neg-lowering-neg.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)\right)}\right) \]
   6. 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 67.1%

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

   7. Simplified 67.1%

      \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. *-lowering-*.f64 67.1%

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

   9. Applied egg-rr 67.1%

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

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-24}{{x}^{4}}} \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(-24, \color{blue}{\left({x}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-24, \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right) \]

       3. pow-plus N/A

          \[\leadsto \mathsf{/.f64}\left(-24, \left({x}^{3} \cdot \color{blue}{x}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-24, \left(x \cdot \color{blue}{{x}^{3}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(-24, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

       6. cube-mult N/A

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

       7. unpow2 N/A

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 67.1%

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

   12. Simplified 67.1%

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

   if -4 < x

   1. Initial program 5.6%

      \[\frac{e^{x}}{e^{x} - 1} \]
   2. Add Preprocessing
   3. 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. distribute-neg-frac N/A

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

      6. neg-sub0 N/A

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

      7. associate-+l- N/A

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

      8. neg-sub0 N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. rec-exp N/A

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

      13. *-inverses N/A

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

      14. accelerator-lowering-expm1.f64 N/A

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

      15. neg-lowering-neg.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 99.4%

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

Alternative 9: 91.9% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.0)
   (/ -24.0 (* x (* x (* x x))))
   (+ (/ 1.0 x) (+ 0.5 (* x 0.08333333333333333)))))

C

double code(double x) {
	double tmp;
	if (x <= -4.0) {
		tmp = -24.0 / (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 <= (-4.0d0)) then
        tmp = (-24.0d0) / (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 <= -4.0) {
		tmp = -24.0 / (x * (x * (x * x)));
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.0:
		tmp = -24.0 / (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 <= -4.0)
		tmp = Float64(-24.0 / Float64(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 <= -4.0)
		tmp = -24.0 / (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, -4.0], N[(-24.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $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

   1. Initial program 100.0%

      \[\frac{e^{x}}{e^{x} - 1} \]
   2. Add Preprocessing
   3. 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. distribute-neg-frac N/A

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

      6. neg-sub0 N/A

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

      7. associate-+l- N/A

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

      8. neg-sub0 N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. rec-exp N/A

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

      13. *-inverses N/A

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

      14. accelerator-lowering-expm1.f64 N/A

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

      15. neg-lowering-neg.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)\right)}\right) \]
   6. 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 67.1%

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

   7. Simplified 67.1%

      \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. *-lowering-*.f64 67.1%

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

   9. Applied egg-rr 67.1%

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

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-24}{{x}^{4}}} \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(-24, \color{blue}{\left({x}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-24, \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right) \]

       3. pow-plus N/A

          \[\leadsto \mathsf{/.f64}\left(-24, \left({x}^{3} \cdot \color{blue}{x}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-24, \left(x \cdot \color{blue}{{x}^{3}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(-24, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

       6. cube-mult N/A

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

       7. unpow2 N/A

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 67.1%

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

   12. Simplified 67.1%

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

   if -4 < x

   1. Initial program 5.6%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      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. +-lowering-+.f64 N/A

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

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

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-in N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lft-mult-inverse N/A

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

      15. *-lft-identity N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. associate-*l* N/A

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

      20. rgt-mult-inverse N/A

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

      21. metadata-eval 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.1% 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. Add Preprocessing
   3. 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. distribute-neg-frac N/A

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

      6. neg-sub0 N/A

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

      7. associate-+l- N/A

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

      8. neg-sub0 N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. rec-exp N/A

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

      13. *-inverses N/A

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

      14. accelerator-lowering-expm1.f64 N/A

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

      15. neg-lowering-neg.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)\right)}\right) \]
   6. 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 67.1%

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

   7. Simplified 67.1%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3} \cdot \left(\frac{1}{24} - \frac{1}{6} \cdot \frac{1}{x}\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. sub-neg N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f64 67.1%

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

   10. Simplified 67.1%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{6}{{x}^{3}}} \]
   12. 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 61.4%

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

   13. Simplified 61.4%

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

   if -4.20000000000000018 < x

   1. Initial program 5.6%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      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. +-lowering-+.f64 N/A

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

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

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-in N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lft-mult-inverse N/A

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

      15. *-lft-identity N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. associate-*l* N/A

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

      20. rgt-mult-inverse N/A

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

      21. metadata-eval 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 86.9%

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

Alternative 11: 90.7% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.6%

   \[\frac{e^{x}}{e^{x} - 1} \]
2. Add Preprocessing
3. 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. distribute-neg-frac N/A

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

   6. neg-sub0 N/A

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

   7. associate-+l- N/A

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

   8. neg-sub0 N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. div-sub N/A

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

   12. rec-exp N/A

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

   13. *-inverses N/A

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

   14. accelerator-lowering-expm1.f64 N/A

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

   15. neg-lowering-neg.f64 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)\right)}\right) \]
6. 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 88.8%

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

7. Simplified 88.8%

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

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)}\right)\right)\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left({x}^{3} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

   2. cube-mult N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

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

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

   7. *-commutative N/A

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

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

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 88.1%

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

10. Simplified 88.1%

    \[\leadsto \frac{-1}{x \cdot \left(-1 + \color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)}\right)} \]

11. Final simplification 88.1%

    \[\leadsto \frac{-1}{x \cdot \left(-1 + x \cdot \left(0.041666666666666664 \cdot \left(x \cdot x\right)\right)\right)} \]
12. Add Preprocessing

Alternative 12: 88.7% accurate, 17.1× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -1.85) {
		tmp = 6.0 / (x * (x * x));
	} else {
		tmp = (1.0 + (x * 0.5)) / x;
	}
	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)) / x
    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)) / x;
	}
	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)) / x
	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 + Float64(x * 0.5)) / x);
	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)) / x;
	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 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] / x), $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. Add Preprocessing
   3. 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. distribute-neg-frac N/A

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

      6. neg-sub0 N/A

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

      7. associate-+l- N/A

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

      8. neg-sub0 N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. rec-exp N/A

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

      13. *-inverses N/A

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

      14. accelerator-lowering-expm1.f64 N/A

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

      15. neg-lowering-neg.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)\right)}\right) \]
   6. 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 67.1%

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

   7. Simplified 67.1%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3} \cdot \left(\frac{1}{24} - \frac{1}{6} \cdot \frac{1}{x}\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. sub-neg N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f64 67.1%

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

   10. Simplified 67.1%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{6}{{x}^{3}}} \]
   12. 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 61.4%

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

   13. Simplified 61.4%

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

   if -1.8500000000000001 < x

   1. Initial program 5.6%

      \[\frac{e^{x}}{e^{x} - 1} \]
   2. Add Preprocessing
   3. 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. distribute-neg-frac N/A

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

      6. neg-sub0 N/A

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

      7. associate-+l- N/A

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

      8. neg-sub0 N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. rec-exp N/A

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

      13. *-inverses N/A

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

      14. accelerator-lowering-expm1.f64 N/A

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

      15. neg-lowering-neg.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 99.4%

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

   8. Taylor expanded in x around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 99.0%

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

   10. Simplified 99.0%

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

Alternative 13: 88.7% accurate, 17.1× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -1.85) {
		tmp = 6.0 / (x * (x * x));
	} else {
		tmp = 0.5 + (1.0 / x);
	}
	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 = 0.5d0 + (1.0d0 / x)
    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 = 0.5 + (1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.85:
		tmp = 6.0 / (x * (x * x))
	else:
		tmp = 0.5 + (1.0 / x)
	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(0.5 + Float64(1.0 / x));
	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 = 0.5 + (1.0 / x);
	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[(0.5 + N[(1.0 / x), $MachinePrecision]), $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. Add Preprocessing
   3. 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. distribute-neg-frac N/A

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

      6. neg-sub0 N/A

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

      7. associate-+l- N/A

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

      8. neg-sub0 N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. rec-exp N/A

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

      13. *-inverses N/A

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

      14. accelerator-lowering-expm1.f64 N/A

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

      15. neg-lowering-neg.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)\right)}\right) \]
   6. 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 67.1%

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

   7. Simplified 67.1%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3} \cdot \left(\frac{1}{24} - \frac{1}{6} \cdot \frac{1}{x}\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. sub-neg N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f64 67.1%

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

   10. Simplified 67.1%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{6}{{x}^{3}}} \]
   12. 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 61.4%

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

   13. Simplified 61.4%

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

   if -1.8500000000000001 < x

   1. Initial program 5.6%

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

   3. Taylor expanded in x around 0

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

      1. *-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. *-lft-identity N/A

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

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

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

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

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

      7. associate-*l* N/A

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

      8. rgt-mult-inverse N/A

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

      9. metadata-eval 99.0%

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

   5. Simplified 99.0%

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

4. Final simplification 86.7%

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

Alternative 14: 67.1% 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 36.6%

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

3. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 67.8%

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

5. Simplified 67.8%

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

Alternative 15: 3.2% 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 36.6%

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

3. Taylor expanded in x around 0

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

   1. *-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. *-lft-identity N/A

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

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

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

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

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

   7. associate-*l* N/A

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

   8. rgt-mult-inverse N/A

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

   9. metadata-eval 67.5%

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

5. Simplified 67.5%

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

6. Taylor expanded in x around inf

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

8. Simplified 3.2%

   \[\leadsto \color{blue}{0.5} \]
9. 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 2024191 
(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)))