expq2 (section 3.11)

Percentage Accurate: 37.3% → 100.0%

Time: 11.6s

Alternatives: 16

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. Step-by-step derivation

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

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

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

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

   3. accelerator-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. 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%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

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

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

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

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

   3. accelerator-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

7. Simplified 98.9%

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

Alternative 3: 96.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\\ t_1 := x \cdot \left(x \cdot x\right)\\ t_2 := x \cdot \left(\left(x \cdot x\right) \cdot 7.233796296296296 \cdot 10^{-5}\right)\\ \mathbf{if}\;x \leq -2 \cdot 10^{+77}:\\ \;\;\;\;\frac{-24}{x \cdot t\_1}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{+40}:\\ \;\;\;\;\frac{-1}{\frac{t\_0 \cdot t\_0 - x \cdot x}{x + t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{-1 + x \cdot \left(0.5 + \frac{x \cdot \left(-9.92290301275212 \cdot 10^{-8} + t\_1 \cdot \left(\left(t\_1 \cdot t\_1\right) \cdot 3.785287098980759 \cdot 10^{-13}\right)\right)}{\left(2.143347050754458 \cdot 10^{-5} + t\_2 \cdot \left(t\_2 + 0.004629629629629629\right)\right) \cdot \left(0.027777777777777776 + 0.041666666666666664 \cdot \left(x \cdot \left(x \cdot 0.041666666666666664 + 0.16666666666666666\right)\right)\right)}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (*
          (* x x)
          (+ 0.5 (* x (+ -0.16666666666666666 (* x 0.041666666666666664))))))
        (t_1 (* x (* x x)))
        (t_2 (* x (* (* x x) 7.233796296296296e-5))))
   (if (<= x -2e+77)
     (/ -24.0 (* x t_1))
     (if (<= x -1e+40)
       (/ -1.0 (/ (- (* t_0 t_0) (* x x)) (+ x t_0)))
       (/
        (/
         -1.0
         (+
          -1.0
          (*
           x
           (+
            0.5
            (/
             (*
              x
              (+
               -9.92290301275212e-8
               (* t_1 (* (* t_1 t_1) 3.785287098980759e-13))))
             (*
              (+ 2.143347050754458e-5 (* t_2 (+ t_2 0.004629629629629629)))
              (+
               0.027777777777777776
               (*
                0.041666666666666664
                (*
                 x
                 (+ (* x 0.041666666666666664) 0.16666666666666666))))))))))
        x)))))

C

double code(double x) {
	double t_0 = (x * x) * (0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664))));
	double t_1 = x * (x * x);
	double t_2 = x * ((x * x) * 7.233796296296296e-5);
	double tmp;
	if (x <= -2e+77) {
		tmp = -24.0 / (x * t_1);
	} else if (x <= -1e+40) {
		tmp = -1.0 / (((t_0 * t_0) - (x * x)) / (x + t_0));
	} else {
		tmp = (-1.0 / (-1.0 + (x * (0.5 + ((x * (-9.92290301275212e-8 + (t_1 * ((t_1 * t_1) * 3.785287098980759e-13)))) / ((2.143347050754458e-5 + (t_2 * (t_2 + 0.004629629629629629))) * (0.027777777777777776 + (0.041666666666666664 * (x * ((x * 0.041666666666666664) + 0.16666666666666666)))))))))) / x;
	}
	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 = (x * x) * (0.5d0 + (x * ((-0.16666666666666666d0) + (x * 0.041666666666666664d0))))
    t_1 = x * (x * x)
    t_2 = x * ((x * x) * 7.233796296296296d-5)
    if (x <= (-2d+77)) then
        tmp = (-24.0d0) / (x * t_1)
    else if (x <= (-1d+40)) then
        tmp = (-1.0d0) / (((t_0 * t_0) - (x * x)) / (x + t_0))
    else
        tmp = ((-1.0d0) / ((-1.0d0) + (x * (0.5d0 + ((x * ((-9.92290301275212d-8) + (t_1 * ((t_1 * t_1) * 3.785287098980759d-13)))) / ((2.143347050754458d-5 + (t_2 * (t_2 + 0.004629629629629629d0))) * (0.027777777777777776d0 + (0.041666666666666664d0 * (x * ((x * 0.041666666666666664d0) + 0.16666666666666666d0)))))))))) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * (0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664))));
	double t_1 = x * (x * x);
	double t_2 = x * ((x * x) * 7.233796296296296e-5);
	double tmp;
	if (x <= -2e+77) {
		tmp = -24.0 / (x * t_1);
	} else if (x <= -1e+40) {
		tmp = -1.0 / (((t_0 * t_0) - (x * x)) / (x + t_0));
	} else {
		tmp = (-1.0 / (-1.0 + (x * (0.5 + ((x * (-9.92290301275212e-8 + (t_1 * ((t_1 * t_1) * 3.785287098980759e-13)))) / ((2.143347050754458e-5 + (t_2 * (t_2 + 0.004629629629629629))) * (0.027777777777777776 + (0.041666666666666664 * (x * ((x * 0.041666666666666664) + 0.16666666666666666)))))))))) / x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) * (0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664))))
	t_1 = x * (x * x)
	t_2 = x * ((x * x) * 7.233796296296296e-5)
	tmp = 0
	if x <= -2e+77:
		tmp = -24.0 / (x * t_1)
	elif x <= -1e+40:
		tmp = -1.0 / (((t_0 * t_0) - (x * x)) / (x + t_0))
	else:
		tmp = (-1.0 / (-1.0 + (x * (0.5 + ((x * (-9.92290301275212e-8 + (t_1 * ((t_1 * t_1) * 3.785287098980759e-13)))) / ((2.143347050754458e-5 + (t_2 * (t_2 + 0.004629629629629629))) * (0.027777777777777776 + (0.041666666666666664 * (x * ((x * 0.041666666666666664) + 0.16666666666666666)))))))))) / x
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(-0.16666666666666666 + Float64(x * 0.041666666666666664)))))
	t_1 = Float64(x * Float64(x * x))
	t_2 = Float64(x * Float64(Float64(x * x) * 7.233796296296296e-5))
	tmp = 0.0
	if (x <= -2e+77)
		tmp = Float64(-24.0 / Float64(x * t_1));
	elseif (x <= -1e+40)
		tmp = Float64(-1.0 / Float64(Float64(Float64(t_0 * t_0) - Float64(x * x)) / Float64(x + t_0)));
	else
		tmp = Float64(Float64(-1.0 / Float64(-1.0 + Float64(x * Float64(0.5 + Float64(Float64(x * Float64(-9.92290301275212e-8 + Float64(t_1 * Float64(Float64(t_1 * t_1) * 3.785287098980759e-13)))) / Float64(Float64(2.143347050754458e-5 + Float64(t_2 * Float64(t_2 + 0.004629629629629629))) * Float64(0.027777777777777776 + Float64(0.041666666666666664 * Float64(x * Float64(Float64(x * 0.041666666666666664) + 0.16666666666666666)))))))))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) * (0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664))));
	t_1 = x * (x * x);
	t_2 = x * ((x * x) * 7.233796296296296e-5);
	tmp = 0.0;
	if (x <= -2e+77)
		tmp = -24.0 / (x * t_1);
	elseif (x <= -1e+40)
		tmp = -1.0 / (((t_0 * t_0) - (x * x)) / (x + t_0));
	else
		tmp = (-1.0 / (-1.0 + (x * (0.5 + ((x * (-9.92290301275212e-8 + (t_1 * ((t_1 * t_1) * 3.785287098980759e-13)))) / ((2.143347050754458e-5 + (t_2 * (t_2 + 0.004629629629629629))) * (0.027777777777777776 + (0.041666666666666664 * (x * ((x * 0.041666666666666664) + 0.16666666666666666)))))))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(-0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(N[(x * x), $MachinePrecision] * 7.233796296296296e-5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2e+77], N[(-24.0 / N[(x * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1e+40], N[(-1.0 / N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(x + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(-1.0 + N[(x * N[(0.5 + N[(N[(x * N[(-9.92290301275212e-8 + N[(t$95$1 * N[(N[(t$95$1 * t$95$1), $MachinePrecision] * 3.785287098980759e-13), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.143347050754458e-5 + N[(t$95$2 * N[(t$95$2 + 0.004629629629629629), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.027777777777777776 + N[(0.041666666666666664 * N[(x * N[(N[(x * 0.041666666666666664), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.99999999999999997e77

   1. Initial program 100.0%

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. 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%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

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

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 100.0%

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

   9. Simplified 100.0%

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

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

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 100.0%

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

   12. Simplified 100.0%

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

   if -1.99999999999999997e77 < x < -1.00000000000000003e40

   1. Initial program 100.0%

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. 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%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

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

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 5.9%

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

   9. Simplified 5.9%

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

       1. distribute-rgt-in N/A

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

       2. neg-mul-1 N/A

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

       3. flip-+ N/A

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

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

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

   11. Applied egg-rr 100.0%

       \[\leadsto \frac{-1}{\color{blue}{\frac{x \cdot x - \left(\left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(0 - x\right) - \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(x \cdot x\right)}}} \]

   if -1.00000000000000003e40 < x

   1. Initial program 14.5%

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. 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%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

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

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 90.3%

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

   9. Simplified 90.3%

      \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)}} \]
   10. 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 90.3%

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

   11. Applied egg-rr 90.3%

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

   13. Applied egg-rr 92.8%

       \[\leadsto \frac{\frac{-1}{-1 + x \cdot \left(0.5 + \color{blue}{\frac{\left(-9.92290301275212 \cdot 10^{-8} + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 3.785287098980759 \cdot 10^{-13}\right)\right) \cdot x}{\left(2.143347050754458 \cdot 10^{-5} + \left(x \cdot \left(\left(x \cdot x\right) \cdot 7.233796296296296 \cdot 10^{-5}\right)\right) \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 7.233796296296296 \cdot 10^{-5}\right) + 0.004629629629629629\right)\right) \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot 0.041666666666666664 + 0.16666666666666666\right)\right) + 0.027777777777777776\right)}}\right)}}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+77}:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{+40}:\\ \;\;\;\;\frac{-1}{\frac{\left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right) - x \cdot x}{x + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{-1 + x \cdot \left(0.5 + \frac{x \cdot \left(-9.92290301275212 \cdot 10^{-8} + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 3.785287098980759 \cdot 10^{-13}\right)\right)}{\left(2.143347050754458 \cdot 10^{-5} + \left(x \cdot \left(\left(x \cdot x\right) \cdot 7.233796296296296 \cdot 10^{-5}\right)\right) \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 7.233796296296296 \cdot 10^{-5}\right) + 0.004629629629629629\right)\right) \cdot \left(0.027777777777777776 + 0.041666666666666664 \cdot \left(x \cdot \left(x \cdot 0.041666666666666664 + 0.16666666666666666\right)\right)\right)}\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;x \leq -6.8 \cdot 10^{+51}:\\ \;\;\;\;\frac{-1}{x \cdot \left(-1 + x \cdot \left(0.5 + \frac{x \cdot \left(-0.004629629629629629 + \left(x \cdot \left(x \cdot x\right)\right) \cdot 7.233796296296296 \cdot 10^{-5}\right)}{0.027777777777777776 + x \cdot 0.006944444444444444}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot \left(-1 + t\_1 \cdot \left(\left(x \cdot x\right) \cdot \left(t\_0 \cdot t\_0\right)\right)\right)}{t\_1 \cdot \left(t\_1 + 1\right) + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* x (+ -0.16666666666666666 (* x 0.041666666666666664)))))
        (t_1 (* x t_0)))
   (if (<= x -6.8e+51)
     (/
      -1.0
      (*
       x
       (+
        -1.0
        (*
         x
         (+
          0.5
          (/
           (*
            x
            (+ -0.004629629629629629 (* (* x (* x x)) 7.233796296296296e-5)))
           (+ 0.027777777777777776 (* x 0.006944444444444444))))))))
     (/
      -1.0
      (/
       (* x (+ -1.0 (* t_1 (* (* x x) (* t_0 t_0)))))
       (+ (* t_1 (+ t_1 1.0)) 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 tmp;
	if (x <= -6.8e+51) {
		tmp = -1.0 / (x * (-1.0 + (x * (0.5 + ((x * (-0.004629629629629629 + ((x * (x * x)) * 7.233796296296296e-5))) / (0.027777777777777776 + (x * 0.006944444444444444)))))));
	} else {
		tmp = -1.0 / ((x * (-1.0 + (t_1 * ((x * x) * (t_0 * t_0))))) / ((t_1 * (t_1 + 1.0)) + 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) :: tmp
    t_0 = 0.5d0 + (x * ((-0.16666666666666666d0) + (x * 0.041666666666666664d0)))
    t_1 = x * t_0
    if (x <= (-6.8d+51)) then
        tmp = (-1.0d0) / (x * ((-1.0d0) + (x * (0.5d0 + ((x * ((-0.004629629629629629d0) + ((x * (x * x)) * 7.233796296296296d-5))) / (0.027777777777777776d0 + (x * 0.006944444444444444d0)))))))
    else
        tmp = (-1.0d0) / ((x * ((-1.0d0) + (t_1 * ((x * x) * (t_0 * t_0))))) / ((t_1 * (t_1 + 1.0d0)) + 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 tmp;
	if (x <= -6.8e+51) {
		tmp = -1.0 / (x * (-1.0 + (x * (0.5 + ((x * (-0.004629629629629629 + ((x * (x * x)) * 7.233796296296296e-5))) / (0.027777777777777776 + (x * 0.006944444444444444)))))));
	} else {
		tmp = -1.0 / ((x * (-1.0 + (t_1 * ((x * x) * (t_0 * t_0))))) / ((t_1 * (t_1 + 1.0)) + 1.0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664)))
	t_1 = x * t_0
	tmp = 0
	if x <= -6.8e+51:
		tmp = -1.0 / (x * (-1.0 + (x * (0.5 + ((x * (-0.004629629629629629 + ((x * (x * x)) * 7.233796296296296e-5))) / (0.027777777777777776 + (x * 0.006944444444444444)))))))
	else:
		tmp = -1.0 / ((x * (-1.0 + (t_1 * ((x * x) * (t_0 * t_0))))) / ((t_1 * (t_1 + 1.0)) + 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)
	tmp = 0.0
	if (x <= -6.8e+51)
		tmp = Float64(-1.0 / Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(Float64(x * Float64(-0.004629629629629629 + Float64(Float64(x * Float64(x * x)) * 7.233796296296296e-5))) / Float64(0.027777777777777776 + Float64(x * 0.006944444444444444))))))));
	else
		tmp = Float64(-1.0 / Float64(Float64(x * Float64(-1.0 + Float64(t_1 * Float64(Float64(x * x) * Float64(t_0 * t_0))))) / Float64(Float64(t_1 * Float64(t_1 + 1.0)) + 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;
	tmp = 0.0;
	if (x <= -6.8e+51)
		tmp = -1.0 / (x * (-1.0 + (x * (0.5 + ((x * (-0.004629629629629629 + ((x * (x * x)) * 7.233796296296296e-5))) / (0.027777777777777776 + (x * 0.006944444444444444)))))));
	else
		tmp = -1.0 / ((x * (-1.0 + (t_1 * ((x * x) * (t_0 * t_0))))) / ((t_1 * (t_1 + 1.0)) + 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]}, If[LessEqual[x, -6.8e+51], N[(-1.0 / N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(N[(x * N[(-0.004629629629629629 + N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * 7.233796296296296e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.027777777777777776 + N[(x * 0.006944444444444444), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(x * N[(-1.0 + N[(t$95$1 * N[(N[(x * x), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.79999999999999969e51

   1. Initial program 100.0%

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. 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%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

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

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 90.8%

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

   9. Simplified 90.8%

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

       1. *-commutative N/A

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

       2. flip3-+ N/A

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

       3. associate-*l/ N/A

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

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

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

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

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

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

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

       7. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \left({\left(x \cdot \frac{1}{24}\right)}^{3}\right)\right), x\right), \left(\frac{-1}{6} \cdot \frac{-1}{6} + \left(\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) - \frac{-1}{6} \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       8. unpow-prod-down N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \left({x}^{3} \cdot {\frac{1}{24}}^{3}\right)\right), x\right), \left(\frac{-1}{6} \cdot \frac{-1}{6} + \left(\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) - \frac{-1}{6} \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       9. cube-unmult N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot {\frac{1}{24}}^{3}\right)\right), x\right), \left(\frac{-1}{6} \cdot \frac{-1}{6} + \left(\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) - \frac{-1}{6} \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), \left({\frac{1}{24}}^{3}\right)\right)\right), x\right), \left(\frac{-1}{6} \cdot \frac{-1}{6} + \left(\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) - \frac{-1}{6} \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), \left({\frac{1}{24}}^{3}\right)\right)\right), x\right), \left(\frac{-1}{6} \cdot \frac{-1}{6} + \left(\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) - \frac{-1}{6} \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left({\frac{1}{24}}^{3}\right)\right)\right), x\right), \left(\frac{-1}{6} \cdot \frac{-1}{6} + \left(\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) - \frac{-1}{6} \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{1}{13824}\right)\right), x\right), \left(\frac{-1}{6} \cdot \frac{-1}{6} + \left(\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) - \frac{-1}{6} \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\left(\frac{-1}{6} \cdot \frac{-1}{6}\right), \color{blue}{\left(\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) - \frac{-1}{6} \cdot \left(x \cdot \frac{1}{24}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{36}, \left(\color{blue}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right)} - \frac{-1}{6} \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       16. distribute-rgt-out-- N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{36}, \left(\left(x \cdot \frac{1}{24}\right) \cdot \color{blue}{\left(x \cdot \frac{1}{24} - \frac{-1}{6}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\left(x \cdot \frac{1}{24}\right), \color{blue}{\left(x \cdot \frac{1}{24} - \frac{-1}{6}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{24}\right), \left(\color{blue}{x \cdot \frac{1}{24}} - \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       19. --lowering--.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{24}\right), \mathsf{\_.f64}\left(\left(x \cdot \frac{1}{24}\right), \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 28.5%

       \[\leadsto \frac{-1}{x \cdot \left(-1 + x \cdot \left(0.5 + \color{blue}{\frac{\left(-0.004629629629629629 + \left(x \cdot \left(x \cdot x\right)\right) \cdot 7.233796296296296 \cdot 10^{-5}\right) \cdot x}{0.027777777777777776 + \left(x \cdot 0.041666666666666664\right) \cdot \left(x \cdot 0.041666666666666664 - -0.16666666666666666\right)}}\right)\right)} \]

   12. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{36}, \color{blue}{\left(\frac{1}{144} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{36}, \left(x \cdot \color{blue}{\frac{1}{144}}\right)\right)\right)\right)\right)\right)\right)\right) \]

       2. *-lowering-*.f64 94.0%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{144}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   14. Simplified 94.0%

       \[\leadsto \frac{-1}{x \cdot \left(-1 + x \cdot \left(0.5 + \frac{\left(-0.004629629629629629 + \left(x \cdot \left(x \cdot x\right)\right) \cdot 7.233796296296296 \cdot 10^{-5}\right) \cdot x}{0.027777777777777776 + \color{blue}{x \cdot 0.006944444444444444}}\right)\right)} \]

   if -6.79999999999999969e51 < x

   1. Initial program 16.7%

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. 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%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

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

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 88.1%

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

   9. Simplified 88.1%

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

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

   11. Applied egg-rr 93.0%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+51}:\\ \;\;\;\;\frac{-1}{x \cdot \left(-1 + x \cdot \left(0.5 + \frac{x \cdot \left(-0.004629629629629629 + \left(x \cdot \left(x \cdot x\right)\right) \cdot 7.233796296296296 \cdot 10^{-5}\right)}{0.027777777777777776 + x \cdot 0.006944444444444444}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot \left(-1 + \left(x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right)\right)}{\left(x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right) \cdot \left(x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) + 1\right) + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\\ \mathbf{if}\;x \leq -2 \cdot 10^{+77}:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\frac{t\_0 \cdot t\_0 - x \cdot x}{x + t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5 + 1}{x} + x \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (*
          (* x x)
          (+ 0.5 (* x (+ -0.16666666666666666 (* x 0.041666666666666664)))))))
   (if (<= x -2e+77)
     (/ -24.0 (* x (* x (* x x))))
     (if (<= x -5e-7)
       (/ -1.0 (/ (- (* t_0 t_0) (* x x)) (+ x t_0)))
       (+
        (/ (+ (* x 0.5) 1.0) x)
        (* x (+ 0.08333333333333333 (* (* x x) -0.001388888888888889))))))))

C

double code(double x) {
	double t_0 = (x * x) * (0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664))));
	double tmp;
	if (x <= -2e+77) {
		tmp = -24.0 / (x * (x * (x * x)));
	} else if (x <= -5e-7) {
		tmp = -1.0 / (((t_0 * t_0) - (x * x)) / (x + t_0));
	} else {
		tmp = (((x * 0.5) + 1.0) / x) + (x * (0.08333333333333333 + ((x * x) * -0.001388888888888889)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * x) * (0.5d0 + (x * ((-0.16666666666666666d0) + (x * 0.041666666666666664d0))))
    if (x <= (-2d+77)) then
        tmp = (-24.0d0) / (x * (x * (x * x)))
    else if (x <= (-5d-7)) then
        tmp = (-1.0d0) / (((t_0 * t_0) - (x * x)) / (x + t_0))
    else
        tmp = (((x * 0.5d0) + 1.0d0) / x) + (x * (0.08333333333333333d0 + ((x * x) * (-0.001388888888888889d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * (0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664))));
	double tmp;
	if (x <= -2e+77) {
		tmp = -24.0 / (x * (x * (x * x)));
	} else if (x <= -5e-7) {
		tmp = -1.0 / (((t_0 * t_0) - (x * x)) / (x + t_0));
	} else {
		tmp = (((x * 0.5) + 1.0) / x) + (x * (0.08333333333333333 + ((x * x) * -0.001388888888888889)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) * (0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664))))
	tmp = 0
	if x <= -2e+77:
		tmp = -24.0 / (x * (x * (x * x)))
	elif x <= -5e-7:
		tmp = -1.0 / (((t_0 * t_0) - (x * x)) / (x + t_0))
	else:
		tmp = (((x * 0.5) + 1.0) / x) + (x * (0.08333333333333333 + ((x * x) * -0.001388888888888889)))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(-0.16666666666666666 + Float64(x * 0.041666666666666664)))))
	tmp = 0.0
	if (x <= -2e+77)
		tmp = Float64(-24.0 / Float64(x * Float64(x * Float64(x * x))));
	elseif (x <= -5e-7)
		tmp = Float64(-1.0 / Float64(Float64(Float64(t_0 * t_0) - Float64(x * x)) / Float64(x + t_0)));
	else
		tmp = Float64(Float64(Float64(Float64(x * 0.5) + 1.0) / x) + Float64(x * Float64(0.08333333333333333 + Float64(Float64(x * x) * -0.001388888888888889))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) * (0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664))));
	tmp = 0.0;
	if (x <= -2e+77)
		tmp = -24.0 / (x * (x * (x * x)));
	elseif (x <= -5e-7)
		tmp = -1.0 / (((t_0 * t_0) - (x * x)) / (x + t_0));
	else
		tmp = (((x * 0.5) + 1.0) / x) + (x * (0.08333333333333333 + ((x * x) * -0.001388888888888889)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(-0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2e+77], N[(-24.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e-7], N[(-1.0 / N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(x + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] + N[(x * N[(0.08333333333333333 + N[(N[(x * x), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.99999999999999997e77

   1. Initial program 100.0%

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. 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%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

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

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 100.0%

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

   9. Simplified 100.0%

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

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

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 100.0%

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

   12. Simplified 100.0%

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

   if -1.99999999999999997e77 < x < -4.99999999999999977e-7

   1. Initial program 99.1%

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. 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%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

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

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 7.8%

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

   9. Simplified 7.8%

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

       1. distribute-rgt-in N/A

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

       2. neg-mul-1 N/A

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

       3. flip-+ N/A

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

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

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

   11. Applied egg-rr 45.6%

       \[\leadsto \frac{-1}{\color{blue}{\frac{x \cdot x - \left(\left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(0 - x\right) - \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(x \cdot x\right)}}} \]

   if -4.99999999999999977e-7 < x

   1. Initial program 5.2%

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

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

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{1 + \frac{1}{2} \cdot x}{x}\right), \color{blue}{\left(\frac{1}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)}\right) \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{\left(\frac{1}{x} + 0.5\right) + x \cdot \left(0.08333333333333333 + -0.001388888888888889 \cdot \left(x \cdot x\right)\right)} \]
   8. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{{\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}}{\frac{1}{x} \cdot \frac{1}{x} + \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{1}{x} \cdot \frac{1}{2}\right)}\right), \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({\left(\frac{1}{x}\right)}^{3} + {\frac{1}{2}}^{3}\right), \left(\frac{1}{x} \cdot \frac{1}{x} + \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{1}{x} \cdot \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({\left(\frac{1}{x}\right)}^{3}\right), \left({\frac{1}{2}}^{3}\right)\right), \left(\frac{1}{x} \cdot \frac{1}{x} + \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{1}{x} \cdot \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

      4. cube-div N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{{1}^{3}}{{x}^{3}}\right), \left({\frac{1}{2}}^{3}\right)\right), \left(\frac{1}{x} \cdot \frac{1}{x} + \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{1}{x} \cdot \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{{x}^{3}}\right), \left({\frac{1}{2}}^{3}\right)\right), \left(\frac{1}{x} \cdot \frac{1}{x} + \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{1}{x} \cdot \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{3}\right)\right), \left({\frac{1}{2}}^{3}\right)\right), \left(\frac{1}{x} \cdot \frac{1}{x} + \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{1}{x} \cdot \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

      7. cube-unmult N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot \left(x \cdot x\right)\right)\right), \left({\frac{1}{2}}^{3}\right)\right), \left(\frac{1}{x} \cdot \frac{1}{x} + \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{1}{x} \cdot \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right), \left({\frac{1}{2}}^{3}\right)\right), \left(\frac{1}{x} \cdot \frac{1}{x} + \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{1}{x} \cdot \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left({\frac{1}{2}}^{3}\right)\right), \left(\frac{1}{x} \cdot \frac{1}{x} + \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{1}{x} \cdot \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \frac{1}{8}\right), \left(\frac{1}{x} \cdot \frac{1}{x} + \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{1}{x} \cdot \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \frac{1}{8}\right), \mathsf{+.f64}\left(\left(\frac{1}{x} \cdot \frac{1}{x}\right), \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{1}{x} \cdot \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

      12. frac-times N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \frac{1}{8}\right), \mathsf{+.f64}\left(\left(\frac{1 \cdot 1}{x \cdot x}\right), \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{1}{x} \cdot \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \frac{1}{8}\right), \mathsf{+.f64}\left(\left(\frac{1}{x \cdot x}\right), \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{1}{x} \cdot \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \frac{1}{8}\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{1}{x} \cdot \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \frac{1}{8}\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{1}{x} \cdot \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \frac{1}{8}\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{2}\right), \left(\frac{1}{x} \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \frac{1}{8}\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \left(\frac{1}{x} \cdot \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

      18. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \frac{1}{8}\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \left(\frac{1 \cdot \frac{1}{2}}{x}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \frac{1}{8}\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \left(\frac{\frac{1}{2}}{x}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

      20. /-lowering-/.f64 24.4%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \frac{1}{8}\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 24.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot \left(x \cdot x\right)} + 0.125}{\frac{1}{x \cdot x} + \left(0.25 - \frac{0.5}{x}\right)}} + x \cdot \left(0.08333333333333333 + -0.001388888888888889 \cdot \left(x \cdot x\right)\right) \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\frac{1 + \frac{1}{2} \cdot x}{x}\right)}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(1 + \frac{1}{2} \cdot x\right), x\right), \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot x\right)\right), x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \frac{1}{2}\right)\right), x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

       4. *-lowering-*.f64 99.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right), x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

   12. Simplified 99.6%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+77}:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\frac{\left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right) - x \cdot x}{x + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5 + 1}{x} + x \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.7% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\\ \mathbf{if}\;x \leq -1 \cdot 10^{+106}:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot \left(1 - \left(x \cdot x\right) \cdot \left(t\_0 \cdot t\_0\right)\right)}{-1 - x \cdot t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (+ 0.5 (* x (+ -0.16666666666666666 (* x 0.041666666666666664))))))
   (if (<= x -1e+106)
     (/ -24.0 (* x (* x (* x x))))
     (/ -1.0 (/ (* x (- 1.0 (* (* x x) (* t_0 t_0)))) (- -1.0 (* x t_0)))))))

C

double code(double x) {
	double t_0 = 0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664)));
	double tmp;
	if (x <= -1e+106) {
		tmp = -24.0 / (x * (x * (x * x)));
	} else {
		tmp = -1.0 / ((x * (1.0 - ((x * x) * (t_0 * t_0)))) / (-1.0 - (x * t_0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 + (x * ((-0.16666666666666666d0) + (x * 0.041666666666666664d0)))
    if (x <= (-1d+106)) then
        tmp = (-24.0d0) / (x * (x * (x * x)))
    else
        tmp = (-1.0d0) / ((x * (1.0d0 - ((x * x) * (t_0 * t_0)))) / ((-1.0d0) - (x * t_0)))
    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 tmp;
	if (x <= -1e+106) {
		tmp = -24.0 / (x * (x * (x * x)));
	} else {
		tmp = -1.0 / ((x * (1.0 - ((x * x) * (t_0 * t_0)))) / (-1.0 - (x * t_0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664)))
	tmp = 0
	if x <= -1e+106:
		tmp = -24.0 / (x * (x * (x * x)))
	else:
		tmp = -1.0 / ((x * (1.0 - ((x * x) * (t_0 * t_0)))) / (-1.0 - (x * t_0)))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 + Float64(x * Float64(-0.16666666666666666 + Float64(x * 0.041666666666666664))))
	tmp = 0.0
	if (x <= -1e+106)
		tmp = Float64(-24.0 / Float64(x * Float64(x * Float64(x * x))));
	else
		tmp = Float64(-1.0 / Float64(Float64(x * Float64(1.0 - Float64(Float64(x * x) * Float64(t_0 * t_0)))) / Float64(-1.0 - Float64(x * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664)));
	tmp = 0.0;
	if (x <= -1e+106)
		tmp = -24.0 / (x * (x * (x * x)));
	else
		tmp = -1.0 / ((x * (1.0 - ((x * x) * (t_0 * t_0)))) / (-1.0 - (x * t_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]}, If[LessEqual[x, -1e+106], N[(-24.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(x * N[(1.0 - N[(N[(x * x), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.00000000000000009e106

   1. Initial program 100.0%

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. 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%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

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

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 100.0%

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

   9. Simplified 100.0%

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

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

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 100.0%

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

   12. Simplified 100.0%

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

   if -1.00000000000000009e106 < x

   1. Initial program 21.2%

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. 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%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

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

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 86.0%

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

   9. Simplified 86.0%

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

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

   11. Applied egg-rr 89.2%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+106}:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot \left(1 - \left(x \cdot x\right) \cdot \left(\left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right)}{-1 - x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.2% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{x \cdot \left(-1 + x \cdot \left(0.5 + \frac{x \cdot \left(-0.004629629629629629 + \left(x \cdot \left(x \cdot x\right)\right) \cdot 7.233796296296296 \cdot 10^{-5}\right)}{0.027777777777777776 + x \cdot 0.006944444444444444}\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  -1.0
  (*
   x
   (+
    -1.0
    (*
     x
     (+
      0.5
      (/
       (* x (+ -0.004629629629629629 (* (* x (* x x)) 7.233796296296296e-5)))
       (+ 0.027777777777777776 (* x 0.006944444444444444)))))))))

C

double code(double x) {
	return -1.0 / (x * (-1.0 + (x * (0.5 + ((x * (-0.004629629629629629 + ((x * (x * x)) * 7.233796296296296e-5))) / (0.027777777777777776 + (x * 0.006944444444444444)))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) / (x * ((-1.0d0) + (x * (0.5d0 + ((x * ((-0.004629629629629629d0) + ((x * (x * x)) * 7.233796296296296d-5))) / (0.027777777777777776d0 + (x * 0.006944444444444444d0)))))))
end function

Java

public static double code(double x) {
	return -1.0 / (x * (-1.0 + (x * (0.5 + ((x * (-0.004629629629629629 + ((x * (x * x)) * 7.233796296296296e-5))) / (0.027777777777777776 + (x * 0.006944444444444444)))))));
}

Python

def code(x):
	return -1.0 / (x * (-1.0 + (x * (0.5 + ((x * (-0.004629629629629629 + ((x * (x * x)) * 7.233796296296296e-5))) / (0.027777777777777776 + (x * 0.006944444444444444)))))))

Julia

function code(x)
	return Float64(-1.0 / Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(Float64(x * Float64(-0.004629629629629629 + Float64(Float64(x * Float64(x * x)) * 7.233796296296296e-5))) / Float64(0.027777777777777776 + Float64(x * 0.006944444444444444))))))))
end

MATLAB

function tmp = code(x)
	tmp = -1.0 / (x * (-1.0 + (x * (0.5 + ((x * (-0.004629629629629629 + ((x * (x * x)) * 7.233796296296296e-5))) / (0.027777777777777776 + (x * 0.006944444444444444)))))));
end

Wolfram

code[x_] := N[(-1.0 / N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(N[(x * N[(-0.004629629629629629 + N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * 7.233796296296296e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.027777777777777776 + N[(x * 0.006944444444444444), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.6%

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

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

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

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

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

   3. accelerator-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. 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%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

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

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

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

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

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

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

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

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. +-commutative N/A

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

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

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

   13. *-commutative N/A

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

   14. *-lowering-*.f64 88.8%

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

9. Simplified 88.8%

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

    1. *-commutative N/A

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

    2. flip3-+ N/A

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

    3. associate-*l/ N/A

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

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

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

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

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

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

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

    7. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \left({\left(x \cdot \frac{1}{24}\right)}^{3}\right)\right), x\right), \left(\frac{-1}{6} \cdot \frac{-1}{6} + \left(\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) - \frac{-1}{6} \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    8. unpow-prod-down N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \left({x}^{3} \cdot {\frac{1}{24}}^{3}\right)\right), x\right), \left(\frac{-1}{6} \cdot \frac{-1}{6} + \left(\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) - \frac{-1}{6} \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    9. cube-unmult N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot {\frac{1}{24}}^{3}\right)\right), x\right), \left(\frac{-1}{6} \cdot \frac{-1}{6} + \left(\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) - \frac{-1}{6} \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), \left({\frac{1}{24}}^{3}\right)\right)\right), x\right), \left(\frac{-1}{6} \cdot \frac{-1}{6} + \left(\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) - \frac{-1}{6} \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), \left({\frac{1}{24}}^{3}\right)\right)\right), x\right), \left(\frac{-1}{6} \cdot \frac{-1}{6} + \left(\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) - \frac{-1}{6} \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left({\frac{1}{24}}^{3}\right)\right)\right), x\right), \left(\frac{-1}{6} \cdot \frac{-1}{6} + \left(\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) - \frac{-1}{6} \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    13. metadata-eval N/A

        \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{1}{13824}\right)\right), x\right), \left(\frac{-1}{6} \cdot \frac{-1}{6} + \left(\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) - \frac{-1}{6} \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\left(\frac{-1}{6} \cdot \frac{-1}{6}\right), \color{blue}{\left(\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) - \frac{-1}{6} \cdot \left(x \cdot \frac{1}{24}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

    15. metadata-eval N/A

        \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{36}, \left(\color{blue}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right)} - \frac{-1}{6} \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    16. distribute-rgt-out-- N/A

        \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{36}, \left(\left(x \cdot \frac{1}{24}\right) \cdot \color{blue}{\left(x \cdot \frac{1}{24} - \frac{-1}{6}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\left(x \cdot \frac{1}{24}\right), \color{blue}{\left(x \cdot \frac{1}{24} - \frac{-1}{6}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{24}\right), \left(\color{blue}{x \cdot \frac{1}{24}} - \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    19. --lowering--.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{24}\right), \mathsf{\_.f64}\left(\left(x \cdot \frac{1}{24}\right), \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

11. Applied egg-rr 73.9%

    \[\leadsto \frac{-1}{x \cdot \left(-1 + x \cdot \left(0.5 + \color{blue}{\frac{\left(-0.004629629629629629 + \left(x \cdot \left(x \cdot x\right)\right) \cdot 7.233796296296296 \cdot 10^{-5}\right) \cdot x}{0.027777777777777776 + \left(x \cdot 0.041666666666666664\right) \cdot \left(x \cdot 0.041666666666666664 - -0.16666666666666666\right)}}\right)\right)} \]

12. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{36}, \color{blue}{\left(\frac{1}{144} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]
13. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{36}, \left(x \cdot \color{blue}{\frac{1}{144}}\right)\right)\right)\right)\right)\right)\right)\right) \]

    2. *-lowering-*.f64 89.6%

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{1}{13824}\right)\right), x\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{144}}\right)\right)\right)\right)\right)\right)\right)\right) \]

14. Simplified 89.6%

    \[\leadsto \frac{-1}{x \cdot \left(-1 + x \cdot \left(0.5 + \frac{\left(-0.004629629629629629 + \left(x \cdot \left(x \cdot x\right)\right) \cdot 7.233796296296296 \cdot 10^{-5}\right) \cdot x}{0.027777777777777776 + \color{blue}{x \cdot 0.006944444444444444}}\right)\right)} \]

15. Final simplification 89.6%

    \[\leadsto \frac{-1}{x \cdot \left(-1 + x \cdot \left(0.5 + \frac{x \cdot \left(-0.004629629629629629 + \left(x \cdot \left(x \cdot x\right)\right) \cdot 7.233796296296296 \cdot 10^{-5}\right)}{0.027777777777777776 + x \cdot 0.006944444444444444}\right)\right)} \]
16. Add Preprocessing

Alternative 8: 91.9% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.95:\\ \;\;\;\;\frac{-24 + \frac{-96}{x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot -0.001388888888888889\right) + \left(0.5 + \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.95)
   (/ (+ -24.0 (/ -96.0 x)) (* x (* x (* x x))))
   (+
    (* x (+ 0.08333333333333333 (* (* x x) -0.001388888888888889)))
    (+ 0.5 (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -3.95) {
		tmp = (-24.0 + (-96.0 / x)) / (x * (x * (x * x)));
	} else {
		tmp = (x * (0.08333333333333333 + ((x * x) * -0.001388888888888889))) + (0.5 + (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.95d0)) then
        tmp = ((-24.0d0) + ((-96.0d0) / x)) / (x * (x * (x * x)))
    else
        tmp = (x * (0.08333333333333333d0 + ((x * x) * (-0.001388888888888889d0)))) + (0.5d0 + (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.95) {
		tmp = (-24.0 + (-96.0 / x)) / (x * (x * (x * x)));
	} else {
		tmp = (x * (0.08333333333333333 + ((x * x) * -0.001388888888888889))) + (0.5 + (1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.95:
		tmp = (-24.0 + (-96.0 / x)) / (x * (x * (x * x)))
	else:
		tmp = (x * (0.08333333333333333 + ((x * x) * -0.001388888888888889))) + (0.5 + (1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.95)
		tmp = Float64(Float64(-24.0 + Float64(-96.0 / x)) / Float64(x * Float64(x * Float64(x * x))));
	else
		tmp = Float64(Float64(x * Float64(0.08333333333333333 + Float64(Float64(x * x) * -0.001388888888888889))) + Float64(0.5 + Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.95)
		tmp = (-24.0 + (-96.0 / x)) / (x * (x * (x * x)));
	else
		tmp = (x * (0.08333333333333333 + ((x * x) * -0.001388888888888889))) + (0.5 + (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.95], N[(N[(-24.0 + N[(-96.0 / x), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(0.08333333333333333 + N[(N[(x * x), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.9500000000000002

   1. Initial program 100.0%

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. 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%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

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

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 67.1%

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

   9. Simplified 67.1%

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

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{-1 \cdot \frac{24 + 96 \cdot \frac{1}{x}}{{x}^{4}}} \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{-1 \cdot \left(24 + 96 \cdot \frac{1}{x}\right)}{\color{blue}{{x}^{4}}} \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(24 + 96 \cdot \frac{1}{x}\right)\right), \color{blue}{\left({x}^{4}\right)}\right) \]

       3. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot 24 + -1 \cdot \left(96 \cdot \frac{1}{x}\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-24 + -1 \cdot \left(96 \cdot \frac{1}{x}\right)\right), \left({x}^{4}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \left(-1 \cdot \left(96 \cdot \frac{1}{x}\right)\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

       6. neg-mul-1 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \left(\mathsf{neg}\left(96 \cdot \frac{1}{x}\right)\right)\right), \left({x}^{4}\right)\right) \]

       7. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \left(\mathsf{neg}\left(\frac{96 \cdot 1}{x}\right)\right)\right), \left({x}^{4}\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \left(\mathsf{neg}\left(\frac{96}{x}\right)\right)\right), \left({x}^{4}\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \left(\frac{\mathsf{neg}\left(96\right)}{x}\right)\right), \left({x}^{4}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(96\right)\right), x\right)\right), \left({x}^{4}\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \left({x}^{4}\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right) \]

       13. pow-plus N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \left({x}^{3} \cdot \color{blue}{x}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \mathsf{*.f64}\left(\left({x}^{3}\right), \color{blue}{x}\right)\right) \]

       15. cube-mult N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), x\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \mathsf{*.f64}\left(\left(x \cdot {x}^{2}\right), x\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{2}\right)\right), x\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), x\right)\right) \]

       19. *-lowering-*.f64 67.1%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), x\right)\right) \]

   12. Simplified 67.1%

       \[\leadsto \color{blue}{\frac{-24 + \frac{-96}{x}}{\left(x \cdot \left(x \cdot x\right)\right) \cdot x}} \]

   if -3.9500000000000002 < x

   1. Initial program 5.6%

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

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

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{1 + \frac{1}{2} \cdot x}{x}\right), \color{blue}{\left(\frac{1}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)}\right) \]

   7. Simplified 99.6%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.95:\\ \;\;\;\;\frac{-24 + \frac{-96}{x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot -0.001388888888888889\right) + \left(0.5 + \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.9% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;\frac{-24 + \frac{-96}{x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(0.5 + x \cdot 0.08333333333333333\right) + 1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.2)
   (/ (+ -24.0 (/ -96.0 x)) (* x (* x (* x x))))
   (/ (+ (* x (+ 0.5 (* x 0.08333333333333333))) 1.0) x)))

C

double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = (-24.0 + (-96.0 / x)) / (x * (x * (x * x)));
	} else {
		tmp = ((x * (0.5 + (x * 0.08333333333333333))) + 1.0) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4.2d0)) then
        tmp = ((-24.0d0) + ((-96.0d0) / x)) / (x * (x * (x * x)))
    else
        tmp = ((x * (0.5d0 + (x * 0.08333333333333333d0))) + 1.0d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = (-24.0 + (-96.0 / x)) / (x * (x * (x * x)));
	} else {
		tmp = ((x * (0.5 + (x * 0.08333333333333333))) + 1.0) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.2:
		tmp = (-24.0 + (-96.0 / x)) / (x * (x * (x * x)))
	else:
		tmp = ((x * (0.5 + (x * 0.08333333333333333))) + 1.0) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.2)
		tmp = Float64(Float64(-24.0 + Float64(-96.0 / x)) / Float64(x * Float64(x * Float64(x * x))));
	else
		tmp = Float64(Float64(Float64(x * Float64(0.5 + Float64(x * 0.08333333333333333))) + 1.0) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.2)
		tmp = (-24.0 + (-96.0 / x)) / (x * (x * (x * x)));
	else
		tmp = ((x * (0.5 + (x * 0.08333333333333333))) + 1.0) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.2], N[(N[(-24.0 + N[(-96.0 / x), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(0.5 + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.20000000000000018

   1. Initial program 100.0%

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. 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%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

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

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 67.1%

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

   9. Simplified 67.1%

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

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{-1 \cdot \frac{24 + 96 \cdot \frac{1}{x}}{{x}^{4}}} \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{-1 \cdot \left(24 + 96 \cdot \frac{1}{x}\right)}{\color{blue}{{x}^{4}}} \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(24 + 96 \cdot \frac{1}{x}\right)\right), \color{blue}{\left({x}^{4}\right)}\right) \]

       3. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot 24 + -1 \cdot \left(96 \cdot \frac{1}{x}\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-24 + -1 \cdot \left(96 \cdot \frac{1}{x}\right)\right), \left({x}^{4}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \left(-1 \cdot \left(96 \cdot \frac{1}{x}\right)\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

       6. neg-mul-1 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \left(\mathsf{neg}\left(96 \cdot \frac{1}{x}\right)\right)\right), \left({x}^{4}\right)\right) \]

       7. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \left(\mathsf{neg}\left(\frac{96 \cdot 1}{x}\right)\right)\right), \left({x}^{4}\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \left(\mathsf{neg}\left(\frac{96}{x}\right)\right)\right), \left({x}^{4}\right)\right) \]

       9. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \left(\frac{\mathsf{neg}\left(96\right)}{x}\right)\right), \left({x}^{4}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(96\right)\right), x\right)\right), \left({x}^{4}\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \left({x}^{4}\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right) \]

       13. pow-plus N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \left({x}^{3} \cdot \color{blue}{x}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \mathsf{*.f64}\left(\left({x}^{3}\right), \color{blue}{x}\right)\right) \]

       15. cube-mult N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), x\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \mathsf{*.f64}\left(\left(x \cdot {x}^{2}\right), x\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{2}\right)\right), x\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), x\right)\right) \]

       19. *-lowering-*.f64 67.1%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(-24, \mathsf{/.f64}\left(-96, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), x\right)\right) \]

   12. Simplified 67.1%

       \[\leadsto \color{blue}{\frac{-24 + \frac{-96}{x}}{\left(x \cdot \left(x \cdot x\right)\right) \cdot x}} \]

   if -4.20000000000000018 < x

   1. Initial program 5.6%

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. 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%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

   9. 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. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;\frac{-24 + \frac{-96}{x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(0.5 + x \cdot 0.08333333333333333\right) + 1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 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. Step-by-step derivation

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

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

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

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

   3. accelerator-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. 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%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

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

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

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

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

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

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

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

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. +-commutative N/A

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

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

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

   13. *-commutative N/A

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

   14. *-lowering-*.f64 88.8%

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

9. Simplified 88.8%

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

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

11. Applied egg-rr 88.8%

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

Alternative 11: 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. Step-by-step derivation

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

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

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

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

   3. accelerator-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. 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%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

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

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

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

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

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

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

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

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. +-commutative N/A

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

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

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

   13. *-commutative N/A

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

   14. *-lowering-*.f64 88.8%

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

9. Simplified 88.8%

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

Alternative 12: 91.9% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Python

def code(x):
	tmp = 0
	if x <= -4.0:
		tmp = -24.0 / (x * (x * (x * x)))
	else:
		tmp = ((x * (0.5 + (x * 0.08333333333333333))) + 1.0) / 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(Float64(x * Float64(0.5 + Float64(x * 0.08333333333333333))) + 1.0) / 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 = ((x * (0.5 + (x * 0.08333333333333333))) + 1.0) / 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[(N[(x * N[(0.5 + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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. Step-by-step derivation

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

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

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

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

      3. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. 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%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

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

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 67.1%

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

   9. Simplified 67.1%

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

   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. *-lowering-*.f64 N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 67.1%

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

   12. Simplified 67.1%

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

   if -4 < x

   1. Initial program 5.6%

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. 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%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

   9. 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. Final simplification 88.8%

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

Alternative 13: 91.3% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.6%

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

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

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

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

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

   3. accelerator-lowering-expm1.f64 100.0%

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

3. Simplified 100.0%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. 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%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

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

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

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

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

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

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

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

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. +-commutative N/A

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

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

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

   13. *-commutative N/A

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

   14. *-lowering-*.f64 88.8%

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

9. Simplified 88.8%

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

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

11. Applied egg-rr 88.8%

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

12. Taylor expanded in x around inf

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

    1. unpow2 N/A

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

    2. associate-*r* N/A

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

    3. *-commutative N/A

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

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

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

    5. *-commutative N/A

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

    6. *-lowering-*.f64 88.5%

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

14. Simplified 88.5%

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

Alternative 14: 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. Step-by-step derivation

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

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

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

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

      3. accelerator-lowering-expm1.f64 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\color{blue}{\frac{e^{x} - 1}{e^{x}}}\right)\right)\right) \]

      5. 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%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(-x\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + -1\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \left(-1 + \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot x - \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot x + \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{-1}{6} + \color{blue}{\frac{1}{24} \cdot x}\right)\right)\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{24} \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \left(x \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 67.1%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 67.1%

      \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)}} \]

   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. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-24, \mathsf{*.f64}\left(\left({x}^{3}\right), \color{blue}{x}\right)\right) \]

       5. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(-24, \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), x\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(-24, \mathsf{*.f64}\left(\left(x \cdot {x}^{2}\right), x\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-24, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{2}\right)\right), x\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(-24, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), x\right)\right) \]

       9. *-lowering-*.f64 67.1%

          \[\leadsto \mathsf{/.f64}\left(-24, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), x\right)\right) \]

   12. Simplified 67.1%

       \[\leadsto \color{blue}{\frac{-24}{\left(x \cdot \left(x \cdot x\right)\right) \cdot x}} \]

   if -4 < x

   1. Initial program 5.6%

      \[\frac{e^{x}}{e^{x} - 1} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

      3. accelerator-lowering-expm1.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{x}}{\mathsf{expm1}\left(x\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto 1 \cdot \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{1 \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right)}{\color{blue}{x}} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{1}{x} \cdot 1 + \color{blue}{\frac{1}{x} \cdot \left(x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right)} \]

      5. *-rgt-identity N/A

         \[\leadsto \frac{1}{x} + \color{blue}{\frac{1}{x}} \cdot \left(x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right) \]

      6. +-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%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{12}\right), \frac{1}{2}\right)\right) \]

   7. 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 15: 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. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

   2. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

   3. accelerator-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{e^{x}}{\mathsf{expm1}\left(x\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{x}} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 67.8%

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{x}\right) \]

7. Simplified 67.8%

   \[\leadsto \color{blue}{\frac{1}{x}} \]
8. Add Preprocessing

Alternative 16: 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. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(e^{x} - 1\right)}\right) \]

   2. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{e^{x}} - 1\right)\right) \]

   3. accelerator-lowering-expm1.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{expm1.f64}\left(x\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{e^{x}}{\mathsf{expm1}\left(x\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot x}{x}} \]
6. Step-by-step derivation

   1. *-rgt-identity N/A

      \[\leadsto \frac{\left(1 + \frac{1}{2} \cdot x\right) \cdot 1}{x} \]

   2. associate-/l* N/A

      \[\leadsto \left(1 + \frac{1}{2} \cdot x\right) \cdot \color{blue}{\frac{1}{x}} \]

   3. +-commutative N/A

      \[\leadsto \left(\frac{1}{2} \cdot x + 1\right) \cdot \frac{\color{blue}{1}}{x} \]

   4. distribute-rgt1-in 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%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \frac{1}{2}\right) \]

7. Simplified 67.5%

   \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]

8. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{2}} \]
9. Step-by-step derivation

10. Simplified 3.2%

    \[\leadsto \color{blue}{0.5} \]
11. Add Preprocessing

Developer Target 1: 100.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{expm1}\left(-x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0) (expm1 (- x))))

C

double code(double x) {
	return -1.0 / expm1(-x);
}

Java

public static double code(double x) {
	return -1.0 / Math.expm1(-x);
}

Python

def code(x):
	return -1.0 / math.expm1(-x)

Julia

function code(x)
	return Float64(Float64(-1.0) / expm1(Float64(-x)))
end

Wolfram

code[x_] := N[((-1.0) / N[(Exp[(-x)] - 1), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 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)))