expq2 (section 3.11)

Percentage Accurate: 37.8% → 100.0%

Time: 10.7s

Alternatives: 13

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: 37.8% 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 44.4%

   \[\frac{e^{x}}{e^{x} - 1} \]
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: 98.1% 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 44.4%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 97.4%

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

Alternative 3: 95.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot -0.041666666666666664 + 0.16666666666666666\right) + -0.5\\ t_1 := 0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\\ t_2 := \left(x \cdot x\right) \cdot \left(t\_1 \cdot t\_0\right)\\ \mathbf{if}\;x \leq -6.8 \cdot 10^{+51}:\\ \;\;\;\;\frac{-96}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\frac{x \cdot \left(1 - t\_1 \cdot \left(\left(\left(x \cdot x\right) \cdot t\_0\right) \cdot t\_2\right)\right)}{1 - t\_2}}{-1 + x \cdot t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (+ (* x (+ (* x -0.041666666666666664) 0.16666666666666666)) -0.5))
        (t_1 (+ 0.5 (* x (+ -0.16666666666666666 (* x 0.041666666666666664)))))
        (t_2 (* (* x x) (* t_1 t_0))))
   (if (<= x -6.8e+51)
     (/ -96.0 (* x (* x (* x (* x x)))))
     (/
      -1.0
      (/
       (/ (* x (- 1.0 (* t_1 (* (* (* x x) t_0) t_2)))) (- 1.0 t_2))
       (+ -1.0 (* x t_0)))))))

C

double code(double x) {
	double t_0 = (x * ((x * -0.041666666666666664) + 0.16666666666666666)) + -0.5;
	double t_1 = 0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664)));
	double t_2 = (x * x) * (t_1 * t_0);
	double tmp;
	if (x <= -6.8e+51) {
		tmp = -96.0 / (x * (x * (x * (x * x))));
	} else {
		tmp = -1.0 / (((x * (1.0 - (t_1 * (((x * x) * t_0) * t_2)))) / (1.0 - t_2)) / (-1.0 + (x * t_0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (x * ((x * (-0.041666666666666664d0)) + 0.16666666666666666d0)) + (-0.5d0)
    t_1 = 0.5d0 + (x * ((-0.16666666666666666d0) + (x * 0.041666666666666664d0)))
    t_2 = (x * x) * (t_1 * t_0)
    if (x <= (-6.8d+51)) then
        tmp = (-96.0d0) / (x * (x * (x * (x * x))))
    else
        tmp = (-1.0d0) / (((x * (1.0d0 - (t_1 * (((x * x) * t_0) * t_2)))) / (1.0d0 - t_2)) / ((-1.0d0) + (x * t_0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * ((x * -0.041666666666666664) + 0.16666666666666666)) + -0.5;
	double t_1 = 0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664)));
	double t_2 = (x * x) * (t_1 * t_0);
	double tmp;
	if (x <= -6.8e+51) {
		tmp = -96.0 / (x * (x * (x * (x * x))));
	} else {
		tmp = -1.0 / (((x * (1.0 - (t_1 * (((x * x) * t_0) * t_2)))) / (1.0 - t_2)) / (-1.0 + (x * t_0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * ((x * -0.041666666666666664) + 0.16666666666666666)) + -0.5
	t_1 = 0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664)))
	t_2 = (x * x) * (t_1 * t_0)
	tmp = 0
	if x <= -6.8e+51:
		tmp = -96.0 / (x * (x * (x * (x * x))))
	else:
		tmp = -1.0 / (((x * (1.0 - (t_1 * (((x * x) * t_0) * t_2)))) / (1.0 - t_2)) / (-1.0 + (x * t_0)))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * Float64(Float64(x * -0.041666666666666664) + 0.16666666666666666)) + -0.5)
	t_1 = Float64(0.5 + Float64(x * Float64(-0.16666666666666666 + Float64(x * 0.041666666666666664))))
	t_2 = Float64(Float64(x * x) * Float64(t_1 * t_0))
	tmp = 0.0
	if (x <= -6.8e+51)
		tmp = Float64(-96.0 / Float64(x * Float64(x * Float64(x * Float64(x * x)))));
	else
		tmp = Float64(-1.0 / Float64(Float64(Float64(x * Float64(1.0 - Float64(t_1 * Float64(Float64(Float64(x * x) * t_0) * t_2)))) / Float64(1.0 - t_2)) / Float64(-1.0 + Float64(x * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * ((x * -0.041666666666666664) + 0.16666666666666666)) + -0.5;
	t_1 = 0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664)));
	t_2 = (x * x) * (t_1 * t_0);
	tmp = 0.0;
	if (x <= -6.8e+51)
		tmp = -96.0 / (x * (x * (x * (x * x))));
	else
		tmp = -1.0 / (((x * (1.0 - (t_1 * (((x * x) * t_0) * t_2)))) / (1.0 - t_2)) / (-1.0 + (x * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * N[(N[(x * -0.041666666666666664), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 + N[(x * N[(-0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * x), $MachinePrecision] * N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.8e+51], N[(-96.0 / N[(x * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(N[(x * N[(1.0 - N[(t$95$1 * N[(N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$2), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(x * t$95$0), $MachinePrecision]), $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 86.1%

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

   9. Simplified 86.1%

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

   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

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

       13. pow-sqr N/A

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

       14. unpow2 N/A

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

       15. associate-*l* N/A

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

       16. unpow2 N/A

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

       17. cube-mult N/A

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

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

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

       19. cube-mult N/A

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

       20. unpow2 N/A

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

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

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

       22. unpow2 N/A

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

       23. *-lowering-*.f64 86.1%

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

   12. Simplified 86.1%

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

   13. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{-96}{{x}^{5}}} \]
   14. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(-96, \color{blue}{\left({x}^{5}\right)}\right) \]

       2. metadata-eval N/A

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

       3. pow-plus N/A

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

       4. *-commutative N/A

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

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

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

       6. metadata-eval N/A

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

       7. pow-plus N/A

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

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

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

       9. cube-mult N/A

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 95.8%

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

   15. Simplified 95.8%

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

   if -6.79999999999999969e51 < x

   1. Initial program 16.3%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 89.1%

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

   9. Simplified 89.1%

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

   11. Applied egg-rr 90.7%

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

   13. Applied egg-rr 95.2%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+51}:\\ \;\;\;\;\frac{-96}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\frac{x \cdot \left(1 - \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot -0.041666666666666664 + 0.16666666666666666\right) + -0.5\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(x \cdot \left(x \cdot -0.041666666666666664 + 0.16666666666666666\right) + -0.5\right)\right)\right)\right)\right)}{1 - \left(x \cdot x\right) \cdot \left(\left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right) \cdot \left(x \cdot \left(x \cdot -0.041666666666666664 + 0.16666666666666666\right) + -0.5\right)\right)}}{-1 + x \cdot \left(x \cdot \left(x \cdot -0.041666666666666664 + 0.16666666666666666\right) + -0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.4% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot -0.041666666666666664 + 0.16666666666666666\right) + -0.5\\ \mathbf{if}\;x \leq -1 \cdot 10^{+105}:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot \left(1 + t\_0 \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right)}{-1 + x \cdot t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (+ (* x (+ (* x -0.041666666666666664) 0.16666666666666666)) -0.5)))
   (if (<= x -1e+105)
     (/ -24.0 (* x (* x (* x x))))
     (/
      -1.0
      (/
       (*
        x
        (+
         1.0
         (*
          t_0
          (*
           (* x x)
           (+
            0.5
            (* x (+ -0.16666666666666666 (* x 0.041666666666666664))))))))
       (+ -1.0 (* x t_0)))))))

C

double code(double x) {
	double t_0 = (x * ((x * -0.041666666666666664) + 0.16666666666666666)) + -0.5;
	double tmp;
	if (x <= -1e+105) {
		tmp = -24.0 / (x * (x * (x * x)));
	} else {
		tmp = -1.0 / ((x * (1.0 + (t_0 * ((x * x) * (0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664)))))))) / (-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 = (x * ((x * (-0.041666666666666664d0)) + 0.16666666666666666d0)) + (-0.5d0)
    if (x <= (-1d+105)) then
        tmp = (-24.0d0) / (x * (x * (x * x)))
    else
        tmp = (-1.0d0) / ((x * (1.0d0 + (t_0 * ((x * x) * (0.5d0 + (x * ((-0.16666666666666666d0) + (x * 0.041666666666666664d0)))))))) / ((-1.0d0) + (x * t_0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * ((x * -0.041666666666666664) + 0.16666666666666666)) + -0.5;
	double tmp;
	if (x <= -1e+105) {
		tmp = -24.0 / (x * (x * (x * x)));
	} else {
		tmp = -1.0 / ((x * (1.0 + (t_0 * ((x * x) * (0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664)))))))) / (-1.0 + (x * t_0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * ((x * -0.041666666666666664) + 0.16666666666666666)) + -0.5
	tmp = 0
	if x <= -1e+105:
		tmp = -24.0 / (x * (x * (x * x)))
	else:
		tmp = -1.0 / ((x * (1.0 + (t_0 * ((x * x) * (0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664)))))))) / (-1.0 + (x * t_0)))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * Float64(Float64(x * -0.041666666666666664) + 0.16666666666666666)) + -0.5)
	tmp = 0.0
	if (x <= -1e+105)
		tmp = Float64(-24.0 / Float64(x * Float64(x * Float64(x * x))));
	else
		tmp = Float64(-1.0 / Float64(Float64(x * Float64(1.0 + Float64(t_0 * Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(-0.16666666666666666 + Float64(x * 0.041666666666666664)))))))) / Float64(-1.0 + Float64(x * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * ((x * -0.041666666666666664) + 0.16666666666666666)) + -0.5;
	tmp = 0.0;
	if (x <= -1e+105)
		tmp = -24.0 / (x * (x * (x * x)));
	else
		tmp = -1.0 / ((x * (1.0 + (t_0 * ((x * x) * (0.5 + (x * (-0.16666666666666666 + (x * 0.041666666666666664)))))))) / (-1.0 + (x * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * N[(N[(x * -0.041666666666666664), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]}, If[LessEqual[x, -1e+105], N[(-24.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(x * N[(1.0 + N[(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]), $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 < -9.9999999999999994e104

   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

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

       3. pow-sqr N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. unpow2 N/A

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

       7. cube-mult N/A

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

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

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

       9. cube-mult N/A

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 100.0%

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

   12. Simplified 100.0%

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

   if -9.9999999999999994e104 < x

   1. Initial program 24.7%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 83.8%

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

   9. Simplified 83.8%

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

   11. Applied egg-rr 91.6%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+105}:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot \left(1 + \left(x \cdot \left(x \cdot -0.041666666666666664 + 0.16666666666666666\right) + -0.5\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(-0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right)}{-1 + x \cdot \left(x \cdot \left(x \cdot -0.041666666666666664 + 0.16666666666666666\right) + -0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot -0.041666666666666664 + 0.16666666666666666\right) + -0.5\\ \mathbf{if}\;x \leq -1 \cdot 10^{+105}:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot \left(1 + t\_0 \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)}{-1 + x \cdot t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (+ (* x (+ (* x -0.041666666666666664) 0.16666666666666666)) -0.5)))
   (if (<= x -1e+105)
     (/ -24.0 (* x (* x (* x x))))
     (/
      -1.0
      (/
       (*
        x
        (+ 1.0 (* t_0 (* (* x x) (+ 0.5 (* (* x x) 0.041666666666666664))))))
       (+ -1.0 (* x t_0)))))))

C

double code(double x) {
	double t_0 = (x * ((x * -0.041666666666666664) + 0.16666666666666666)) + -0.5;
	double tmp;
	if (x <= -1e+105) {
		tmp = -24.0 / (x * (x * (x * x)));
	} else {
		tmp = -1.0 / ((x * (1.0 + (t_0 * ((x * x) * (0.5 + ((x * x) * 0.041666666666666664)))))) / (-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 = (x * ((x * (-0.041666666666666664d0)) + 0.16666666666666666d0)) + (-0.5d0)
    if (x <= (-1d+105)) then
        tmp = (-24.0d0) / (x * (x * (x * x)))
    else
        tmp = (-1.0d0) / ((x * (1.0d0 + (t_0 * ((x * x) * (0.5d0 + ((x * x) * 0.041666666666666664d0)))))) / ((-1.0d0) + (x * t_0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * ((x * -0.041666666666666664) + 0.16666666666666666)) + -0.5;
	double tmp;
	if (x <= -1e+105) {
		tmp = -24.0 / (x * (x * (x * x)));
	} else {
		tmp = -1.0 / ((x * (1.0 + (t_0 * ((x * x) * (0.5 + ((x * x) * 0.041666666666666664)))))) / (-1.0 + (x * t_0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * ((x * -0.041666666666666664) + 0.16666666666666666)) + -0.5
	tmp = 0
	if x <= -1e+105:
		tmp = -24.0 / (x * (x * (x * x)))
	else:
		tmp = -1.0 / ((x * (1.0 + (t_0 * ((x * x) * (0.5 + ((x * x) * 0.041666666666666664)))))) / (-1.0 + (x * t_0)))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * Float64(Float64(x * -0.041666666666666664) + 0.16666666666666666)) + -0.5)
	tmp = 0.0
	if (x <= -1e+105)
		tmp = Float64(-24.0 / Float64(x * Float64(x * Float64(x * x))));
	else
		tmp = Float64(-1.0 / Float64(Float64(x * Float64(1.0 + Float64(t_0 * Float64(Float64(x * x) * Float64(0.5 + Float64(Float64(x * x) * 0.041666666666666664)))))) / Float64(-1.0 + Float64(x * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * ((x * -0.041666666666666664) + 0.16666666666666666)) + -0.5;
	tmp = 0.0;
	if (x <= -1e+105)
		tmp = -24.0 / (x * (x * (x * x)));
	else
		tmp = -1.0 / ((x * (1.0 + (t_0 * ((x * x) * (0.5 + ((x * x) * 0.041666666666666664)))))) / (-1.0 + (x * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * N[(N[(x * -0.041666666666666664), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]}, If[LessEqual[x, -1e+105], N[(-24.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(x * N[(1.0 + N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $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 < -9.9999999999999994e104

   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

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

       3. pow-sqr N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. unpow2 N/A

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

       7. cube-mult N/A

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

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

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

       9. cube-mult N/A

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 100.0%

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

   12. Simplified 100.0%

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

   if -9.9999999999999994e104 < x

   1. Initial program 24.7%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 83.8%

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

   9. Simplified 83.8%

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

   11. Applied egg-rr 91.6%

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

   12. Taylor expanded in x around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 91.5%

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

   14. Simplified 91.5%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+105}:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot \left(1 + \left(x \cdot \left(x \cdot -0.041666666666666664 + 0.16666666666666666\right) + -0.5\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)}{-1 + x \cdot \left(x \cdot \left(x \cdot -0.041666666666666664 + 0.16666666666666666\right) + -0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot -0.041666666666666664 + 0.16666666666666666\right) + -0.5\\ \mathbf{if}\;x \leq -1 \cdot 10^{+105}:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot \left(1 + t\_0 \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)\right)}{-1 + x \cdot t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (+ (* x (+ (* x -0.041666666666666664) 0.16666666666666666)) -0.5)))
   (if (<= x -1e+105)
     (/ -24.0 (* x (* x (* x x))))
     (/
      -1.0
      (/
       (* x (+ 1.0 (* t_0 (* (* x x) (+ 0.5 (* x -0.16666666666666666))))))
       (+ -1.0 (* x t_0)))))))

C

double code(double x) {
	double t_0 = (x * ((x * -0.041666666666666664) + 0.16666666666666666)) + -0.5;
	double tmp;
	if (x <= -1e+105) {
		tmp = -24.0 / (x * (x * (x * x)));
	} else {
		tmp = -1.0 / ((x * (1.0 + (t_0 * ((x * x) * (0.5 + (x * -0.16666666666666666)))))) / (-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 = (x * ((x * (-0.041666666666666664d0)) + 0.16666666666666666d0)) + (-0.5d0)
    if (x <= (-1d+105)) then
        tmp = (-24.0d0) / (x * (x * (x * x)))
    else
        tmp = (-1.0d0) / ((x * (1.0d0 + (t_0 * ((x * x) * (0.5d0 + (x * (-0.16666666666666666d0))))))) / ((-1.0d0) + (x * t_0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * ((x * -0.041666666666666664) + 0.16666666666666666)) + -0.5;
	double tmp;
	if (x <= -1e+105) {
		tmp = -24.0 / (x * (x * (x * x)));
	} else {
		tmp = -1.0 / ((x * (1.0 + (t_0 * ((x * x) * (0.5 + (x * -0.16666666666666666)))))) / (-1.0 + (x * t_0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * ((x * -0.041666666666666664) + 0.16666666666666666)) + -0.5
	tmp = 0
	if x <= -1e+105:
		tmp = -24.0 / (x * (x * (x * x)))
	else:
		tmp = -1.0 / ((x * (1.0 + (t_0 * ((x * x) * (0.5 + (x * -0.16666666666666666)))))) / (-1.0 + (x * t_0)))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * Float64(Float64(x * -0.041666666666666664) + 0.16666666666666666)) + -0.5)
	tmp = 0.0
	if (x <= -1e+105)
		tmp = Float64(-24.0 / Float64(x * Float64(x * Float64(x * x))));
	else
		tmp = Float64(-1.0 / Float64(Float64(x * Float64(1.0 + Float64(t_0 * Float64(Float64(x * x) * Float64(0.5 + Float64(x * -0.16666666666666666)))))) / Float64(-1.0 + Float64(x * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * ((x * -0.041666666666666664) + 0.16666666666666666)) + -0.5;
	tmp = 0.0;
	if (x <= -1e+105)
		tmp = -24.0 / (x * (x * (x * x)));
	else
		tmp = -1.0 / ((x * (1.0 + (t_0 * ((x * x) * (0.5 + (x * -0.16666666666666666)))))) / (-1.0 + (x * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * N[(N[(x * -0.041666666666666664), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]}, If[LessEqual[x, -1e+105], N[(-24.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(x * N[(1.0 + N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $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 < -9.9999999999999994e104

   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

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

       3. pow-sqr N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. unpow2 N/A

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

       7. cube-mult N/A

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

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

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

       9. cube-mult N/A

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 100.0%

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

   12. Simplified 100.0%

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

   if -9.9999999999999994e104 < x

   1. Initial program 24.7%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 83.8%

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

   9. Simplified 83.8%

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

   11. Applied egg-rr 91.6%

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

   12. Taylor expanded in x around 0

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

       1. *-lowering-*.f64 90.3%

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

   14. Simplified 90.3%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+105}:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot \left(1 + \left(x \cdot \left(x \cdot -0.041666666666666664 + 0.16666666666666666\right) + -0.5\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)\right)}{-1 + x \cdot \left(x \cdot \left(x \cdot -0.041666666666666664 + 0.16666666666666666\right) + -0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.0% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -680:\\ \;\;\;\;\frac{-96}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -680.0)
   (/ -96.0 (* x (* x (* x (* x x)))))
   (+ (/ 1.0 x) (+ 0.5 (* x 0.08333333333333333)))))

C

double code(double x) {
	double tmp;
	if (x <= -680.0) {
		tmp = -96.0 / (x * (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 <= (-680.0d0)) then
        tmp = (-96.0d0) / (x * (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 <= -680.0) {
		tmp = -96.0 / (x * (x * (x * (x * x))));
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -680.0:
		tmp = -96.0 / (x * (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 <= -680.0)
		tmp = Float64(-96.0 / Float64(x * 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 <= -680.0)
		tmp = -96.0 / (x * (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, -680.0], N[(-96.0 / N[(x * N[(x * N[(x * N[(x * x), $MachinePrecision]), $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 < -680

   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 74.0%

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

   9. Simplified 74.0%

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

   10. Taylor expanded in x around inf

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

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

       13. pow-sqr N/A

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

       14. unpow2 N/A

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

       15. associate-*l* N/A

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

       16. unpow2 N/A

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

       17. cube-mult N/A

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

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

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

       19. cube-mult N/A

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

       20. unpow2 N/A

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

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

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

       22. unpow2 N/A

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

       23. *-lowering-*.f64 74.0%

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

   12. Simplified 74.0%

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

   13. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{-96}{{x}^{5}}} \]
   14. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(-96, \color{blue}{\left({x}^{5}\right)}\right) \]

       2. metadata-eval N/A

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

       3. pow-plus N/A

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

       4. *-commutative N/A

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

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

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

       6. metadata-eval N/A

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

       7. pow-plus N/A

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

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

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

       9. cube-mult N/A

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 82.3%

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

   15. Simplified 82.3%

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

   if -680 < x

   1. Initial program 8.2%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]
   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 97.7%

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

   7. Simplified 97.7%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -680:\\ \;\;\;\;\frac{-96}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.3% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.1)
   (/ -24.0 (* x (* x (* x x))))
   (+ (/ 1.0 x) (+ 0.5 (* x 0.08333333333333333)))))

C

double code(double x) {
	double tmp;
	if (x <= -4.1) {
		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 <= (-4.1d0)) 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 <= -4.1) {
		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 <= -4.1:
		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 <= -4.1)
		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 <= -4.1)
		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, -4.1], 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 < -4.0999999999999996

   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 73.4%

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

   9. Simplified 73.4%

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

   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

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

       3. pow-sqr N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. unpow2 N/A

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

       7. cube-mult N/A

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

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

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

       9. cube-mult N/A

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 73.4%

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

   12. Simplified 73.4%

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

   if -4.0999999999999996 < x

   1. Initial program 7.6%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 98.3%

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

   7. Simplified 98.3%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.6% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.5)
   (/ -2.0 (* x x))
   (+ (/ 1.0 x) (+ 0.5 (* x 0.08333333333333333)))))

C

double code(double x) {
	double tmp;
	if (x <= -4.5) {
		tmp = -2.0 / (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 <= (-4.5d0)) then
        tmp = (-2.0d0) / (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 <= -4.5) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.5:
		tmp = -2.0 / (x * x)
	else:
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.5)
		tmp = Float64(-2.0 / 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 <= -4.5)
		tmp = -2.0 / (x * x);
	else
		tmp = (1.0 / x) + (0.5 + (x * 0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.5], N[(-2.0 / N[(x * x), $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 < -4.5

   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

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 53.6%

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

   9. Simplified 53.6%

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

   10. Taylor expanded in x around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 53.6%

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

   12. Simplified 53.6%

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

   if -4.5 < x

   1. Initial program 7.6%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 98.3%

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

   7. Simplified 98.3%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(0.5 + x \cdot 0.08333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.2% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.76:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.76) (/ -2.0 (* x x)) (+ 0.5 (/ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.76) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = 0.5 + (1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.76d0)) then
        tmp = (-2.0d0) / (x * x)
    else
        tmp = 0.5d0 + (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.76) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = 0.5 + (1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.76:
		tmp = -2.0 / (x * x)
	else:
		tmp = 0.5 + (1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.76)
		tmp = Float64(-2.0 / Float64(x * x));
	else
		tmp = Float64(0.5 + Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.76)
		tmp = -2.0 / (x * x);
	else
		tmp = 0.5 + (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.76], N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.76000000000000001

   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

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 53.6%

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

   9. Simplified 53.6%

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

   10. Taylor expanded in x around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 53.6%

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

   12. Simplified 53.6%

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

   if -1.76000000000000001 < x

   1. Initial program 7.6%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 97.6%

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

   7. Simplified 97.6%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.76:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.7% accurate, 68.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.4%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 60.1%

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

7. Simplified 60.1%

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

Alternative 12: 3.4% accurate, 68.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * 0.08333333333333333

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.4%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 60.0%

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

7. Simplified 60.0%

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

8. Taylor expanded in x around inf

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

   1. metadata-eval N/A

      \[\leadsto \left(\frac{1}{12} \cdot 1\right) \cdot x \]

   2. lft-mult-inverse N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

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

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

   6. associate-*l* N/A

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

   7. lft-mult-inverse N/A

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

   8. metadata-eval 3.5%

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

10. Simplified 3.5%

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

Alternative 13: 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 44.4%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 59.9%

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

7. Simplified 59.9%

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

8. Taylor expanded in x around inf

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

10. Simplified 3.4%

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

Developer Target 1: 100.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024192 
(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)))