expq2 (section 3.11)

Percentage Accurate: 37.9% → 99.4%

Time: 8.3s

Alternatives: 10

Speedup: 68.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 37.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (exp x) (expm1 x)))

C

double code(double x) {
	return exp(x) / expm1(x);
}

Java

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

Python

def code(x):
	return math.exp(x) / math.expm1(x)

Julia

function code(x)
	return Float64(exp(x) / expm1(x))
end

Wolfram

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

Derivation

1. Initial program 40.7%

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

   1. expm1-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \frac{e^{x}}{\mathsf{expm1}\left(x\right)} \]

Alternative 2: 78.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x} + 0.5\\ \mathbf{if}\;x \leq -3.8:\\ \;\;\;\;\frac{t_0 \cdot t_0 - x \cdot \left(x \cdot 0.006944444444444444\right)}{\frac{1}{x} + \left(0.5 - \langle \left( x \cdot 0.08333333333333333 \right)_{\text{binary32}} \rangle_{\text{binary64}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(-0.001388888888888889 \cdot {x}^{3} + \left(\frac{1}{x} + x \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 x) 0.5)))
   (if (<= x -3.8)
     (/
      (- (* t_0 t_0) (* x (* x 0.006944444444444444)))
      (+
       (/ 1.0 x)
       (- 0.5 (cast (! :precision binary32 (* x 0.08333333333333333))))))
     (+
      0.5
      (+
       (* -0.001388888888888889 (pow x 3.0))
       (+ (/ 1.0 x) (* x 0.08333333333333333)))))))

C

double code(double x) {
	double t_0 = (1.0 / x) + 0.5;
	double tmp_1;
	if (x <= -3.8) {
		float tmp_2 = x * 0.08333333333333333f;
		tmp_1 = ((t_0 * t_0) - (x * (x * 0.006944444444444444))) / ((1.0 / x) + (0.5 - ((double) ((double) tmp_2))));
	} else {
		tmp_1 = 0.5 + ((-0.001388888888888889 * pow(x, 3.0)) + ((1.0 / x) + (x * 0.08333333333333333)));
	}
	return tmp_1;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(4) :: tmp
    real(8) :: tmp_1
    real(4) :: tmp_2
    t_0 = (1.0d0 / x) + 0.5d0
    if (x <= (-3.8d0)) then
        tmp_2 = x * 0.08333333333333333e0
        tmp_1 = ((t_0 * t_0) - (x * (x * 0.006944444444444444d0))) / ((1.0d0 / x) + (0.5d0 - real(real(tmp_2, 8), 8)))
    else
        tmp_1 = 0.5d0 + (((-0.001388888888888889d0) * (x ** 3.0d0)) + ((1.0d0 / x) + (x * 0.08333333333333333d0)))
    end if
    code = tmp_1
end function

Julia

function code(x)
	t_0 = Float64(Float64(1.0 / x) + 0.5)
	tmp_1 = 0.0
	if (x <= -3.8)
		tmp_2 = Float32(x * Float32(0.08333333333333333))
		tmp_1 = Float64(Float64(Float64(t_0 * t_0) - Float64(x * Float64(x * 0.006944444444444444))) / Float64(Float64(1.0 / x) + Float64(0.5 - Float64(Float64(tmp_2)))));
	else
		tmp_1 = Float64(0.5 + Float64(Float64(-0.001388888888888889 * (x ^ 3.0)) + Float64(Float64(1.0 / x) + Float64(x * 0.08333333333333333))));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(x)
	t_0 = (1.0 / x) + 0.5;
	tmp_2 = 0.0;
	if (x <= -3.8)
		tmp_3 = single((x * double(single(0.08333333333333333))));
		tmp_2 = ((t_0 * t_0) - (x * (x * 0.006944444444444444))) / ((1.0 / x) + (0.5 - double(double(tmp_3))));
	else
		tmp_2 = 0.5 + ((-0.001388888888888889 * (x ^ 3.0)) + ((1.0 / x) + (x * 0.08333333333333333)));
	end
	tmp_4 = tmp_2;
end

Derivation

1. Split input into 2 regimes

2. if x < -3.7999999999999998

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 2.5%

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

      1. associate-+r+ 2.5%

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

      2. +-commutative 2.5%

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

      3. +-commutative 2.5%

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

      4. *-commutative 2.5%

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

      5. fma-def 2.5%

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

   6. Simplified 2.5%

      \[\leadsto \color{blue}{\frac{1}{x} + \mathsf{fma}\left(x, 0.08333333333333333, 0.5\right)} \]
   7. Step-by-step derivation

      1. +-commutative 2.5%

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

      2. fma-udef 2.5%

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

      3. *-commutative 2.5%

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

      4. associate-+l+ 2.5%

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

      5. +-commutative 2.5%

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

      6. flip-+ 2.3%

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

      7. +-commutative 2.3%

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

      8. +-commutative 2.3%

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

      9. swap-sqr 2.3%

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

      10. *-commutative 2.3%

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

      11. associate-*l* 2.3%

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

      12. metadata-eval 2.3%

          \[\leadsto \frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot \color{blue}{0.006944444444444444}\right)}{\left(0.5 + \frac{1}{x}\right) - 0.08333333333333333 \cdot x} \]

      13. +-commutative 2.3%

          \[\leadsto \frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\color{blue}{\left(\frac{1}{x} + 0.5\right)} - 0.08333333333333333 \cdot x} \]

      14. *-commutative 2.3%

          \[\leadsto \frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\left(\frac{1}{x} + 0.5\right) - \color{blue}{x \cdot 0.08333333333333333}} \]

   8. Applied egg-rr 2.3%

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\left(\frac{1}{x} + 0.5\right) - x \cdot 0.08333333333333333}} \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv 2.3%

         \[\leadsto \frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\color{blue}{\left(\frac{1}{x} + 0.5\right) + \left(-x\right) \cdot 0.08333333333333333}} \]

      2. associate-+l+ 2.3%

         \[\leadsto \frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\color{blue}{\frac{1}{x} + \left(0.5 + \left(-x\right) \cdot 0.08333333333333333\right)}} \]

      3. cancel-sign-sub-inv 2.3%

         \[\leadsto \frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\frac{1}{x} + \color{blue}{\left(0.5 - x \cdot 0.08333333333333333\right)}} \]

   10. Simplified 2.3%

       \[\leadsto \color{blue}{\frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\frac{1}{x} + \left(0.5 - x \cdot 0.08333333333333333\right)}} \]
   11. Step-by-step derivation

       1. rewrite-binary64/binary32 42.2%

          \[\leadsto \color{blue}{\frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\frac{1}{x} + \left(0.5 - \langle \left( x \cdot 0.08333333333333333 \right)_{\text{binary32}} \rangle_{\text{binary64}}\right)}} \]

   12. Applied rewrite-once 42.2%

       \[\leadsto \frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\frac{1}{x} + \left(0.5 - \color{blue}{\langle \color{blue}{\left( \color{blue}{x \cdot 0.08333333333333333} \right)_{\text{binary32}}} \rangle_{\text{binary64}}}\right)} \]

   if -3.7999999999999998 < x

   1. Initial program 7.9%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.5%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8:\\ \;\;\;\;\frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\frac{1}{x} + \left(0.5 - \langle \left( x \cdot 0.08333333333333333 \right)_{\text{binary32}} \rangle_{\text{binary64}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(-0.001388888888888889 \cdot {x}^{3} + \left(\frac{1}{x} + x \cdot 0.08333333333333333\right)\right)\\ \end{array} \]

Alternative 3: 78.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8:\\ \;\;\;\;\frac{x \cdot \left(x \cdot -0.006944444444444444\right)}{0.5 - \langle \left( x \cdot 0.08333333333333333 \right)_{\text{binary32}} \rangle_{\text{binary64}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(-0.001388888888888889 \cdot {x}^{3} + \left(\frac{1}{x} + x \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.8)
   (/
    (* x (* x -0.006944444444444444))
    (- 0.5 (cast (! :precision binary32 (* x 0.08333333333333333)))))
   (+
    0.5
    (+
     (* -0.001388888888888889 (pow x 3.0))
     (+ (/ 1.0 x) (* x 0.08333333333333333))))))

C

double code(double x) {
	double tmp_1;
	if (x <= -3.8) {
		float tmp_2 = x * 0.08333333333333333f;
		tmp_1 = (x * (x * -0.006944444444444444)) / (0.5 - ((double) ((double) tmp_2)));
	} else {
		tmp_1 = 0.5 + ((-0.001388888888888889 * pow(x, 3.0)) + ((1.0 / x) + (x * 0.08333333333333333)));
	}
	return tmp_1;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(4) :: tmp
    real(8) :: tmp_1
    real(4) :: tmp_2
    if (x <= (-3.8d0)) then
        tmp_2 = x * 0.08333333333333333e0
        tmp_1 = (x * (x * (-0.006944444444444444d0))) / (0.5d0 - real(real(tmp_2, 8), 8))
    else
        tmp_1 = 0.5d0 + (((-0.001388888888888889d0) * (x ** 3.0d0)) + ((1.0d0 / x) + (x * 0.08333333333333333d0)))
    end if
    code = tmp_1
end function

Julia

function code(x)
	tmp_1 = 0.0
	if (x <= -3.8)
		tmp_2 = Float32(x * Float32(0.08333333333333333))
		tmp_1 = Float64(Float64(x * Float64(x * -0.006944444444444444)) / Float64(0.5 - Float64(Float64(tmp_2))));
	else
		tmp_1 = Float64(0.5 + Float64(Float64(-0.001388888888888889 * (x ^ 3.0)) + Float64(Float64(1.0 / x) + Float64(x * 0.08333333333333333))));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(x)
	tmp_2 = 0.0;
	if (x <= -3.8)
		tmp_3 = single((x * double(single(0.08333333333333333))));
		tmp_2 = (x * (x * -0.006944444444444444)) / (0.5 - double(double(tmp_3)));
	else
		tmp_2 = 0.5 + ((-0.001388888888888889 * (x ^ 3.0)) + ((1.0 / x) + (x * 0.08333333333333333)));
	end
	tmp_4 = tmp_2;
end

Derivation

1. Split input into 2 regimes

2. if x < -3.7999999999999998

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 2.5%

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

      1. associate-+r+ 2.5%

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

      2. +-commutative 2.5%

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

      3. +-commutative 2.5%

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

      4. *-commutative 2.5%

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

      5. fma-def 2.5%

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

   6. Simplified 2.5%

      \[\leadsto \color{blue}{\frac{1}{x} + \mathsf{fma}\left(x, 0.08333333333333333, 0.5\right)} \]
   7. Step-by-step derivation

      1. +-commutative 2.5%

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

      2. fma-udef 2.5%

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

      3. *-commutative 2.5%

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

      4. associate-+l+ 2.5%

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

      5. +-commutative 2.5%

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

      6. flip-+ 2.3%

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

      7. +-commutative 2.3%

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

      8. +-commutative 2.3%

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

      9. swap-sqr 2.3%

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

      10. *-commutative 2.3%

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

      11. associate-*l* 2.3%

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

      12. metadata-eval 2.3%

          \[\leadsto \frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot \color{blue}{0.006944444444444444}\right)}{\left(0.5 + \frac{1}{x}\right) - 0.08333333333333333 \cdot x} \]

      13. +-commutative 2.3%

          \[\leadsto \frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\color{blue}{\left(\frac{1}{x} + 0.5\right)} - 0.08333333333333333 \cdot x} \]

      14. *-commutative 2.3%

          \[\leadsto \frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\left(\frac{1}{x} + 0.5\right) - \color{blue}{x \cdot 0.08333333333333333}} \]

   8. Applied egg-rr 2.3%

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\left(\frac{1}{x} + 0.5\right) - x \cdot 0.08333333333333333}} \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv 2.3%

         \[\leadsto \frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\color{blue}{\left(\frac{1}{x} + 0.5\right) + \left(-x\right) \cdot 0.08333333333333333}} \]

      2. associate-+l+ 2.3%

         \[\leadsto \frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\color{blue}{\frac{1}{x} + \left(0.5 + \left(-x\right) \cdot 0.08333333333333333\right)}} \]

      3. cancel-sign-sub-inv 2.3%

         \[\leadsto \frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\frac{1}{x} + \color{blue}{\left(0.5 - x \cdot 0.08333333333333333\right)}} \]

   10. Simplified 2.3%

       \[\leadsto \color{blue}{\frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\frac{1}{x} + \left(0.5 - x \cdot 0.08333333333333333\right)}} \]
   11. Step-by-step derivation

       1. rewrite-binary64/binary32 42.2%

          \[\leadsto \color{blue}{\frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\frac{1}{x} + \left(0.5 - \langle \left( x \cdot 0.08333333333333333 \right)_{\text{binary32}} \rangle_{\text{binary64}}\right)}} \]

   12. Applied rewrite-once 42.2%

       \[\leadsto \frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\frac{1}{x} + \left(0.5 - \color{blue}{\langle \color{blue}{\left( \color{blue}{x \cdot 0.08333333333333333} \right)_{\text{binary32}}} \rangle_{\text{binary64}}}\right)} \]

   13. Taylor expanded in x around inf 42.2%

       \[\leadsto \color{blue}{-0.006944444444444444 \cdot \frac{{x}^{2}}{0.5 - \langle \left( 0.08333333333333333 \cdot x \right)_{\text{binary32}} \rangle_{\text{binary64}}}} \]
   14. Step-by-step derivation

       1. associate-*r/ 42.2%

          \[\leadsto \color{blue}{\frac{-0.006944444444444444 \cdot {x}^{2}}{0.5 - \langle \left( 0.08333333333333333 \cdot x \right)_{\text{binary32}} \rangle_{\text{binary64}}}} \]

       2. unpow2 42.2%

          \[\leadsto \frac{-0.006944444444444444 \cdot \color{blue}{\left(x \cdot x\right)}}{0.5 - \langle \left( 0.08333333333333333 \cdot x \right)_{\text{binary32}} \rangle_{\text{binary64}}} \]

       3. *-commutative 42.2%

          \[\leadsto \frac{\color{blue}{\left(x \cdot x\right) \cdot -0.006944444444444444}}{0.5 - \langle \left( 0.08333333333333333 \cdot x \right)_{\text{binary32}} \rangle_{\text{binary64}}} \]

       4. associate-*l* 42.2%

          \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot -0.006944444444444444\right)}}{0.5 - \langle \left( 0.08333333333333333 \cdot x \right)_{\text{binary32}} \rangle_{\text{binary64}}} \]

   15. Simplified 42.2%

       \[\leadsto \color{blue}{\frac{x \cdot \left(x \cdot -0.006944444444444444\right)}{0.5 - \langle \left( 0.08333333333333333 \cdot x \right)_{\text{binary32}} \rangle_{\text{binary64}}}} \]

   if -3.7999999999999998 < x

   1. Initial program 7.9%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.5%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8:\\ \;\;\;\;\frac{x \cdot \left(x \cdot -0.006944444444444444\right)}{0.5 - \langle \left( x \cdot 0.08333333333333333 \right)_{\text{binary32}} \rangle_{\text{binary64}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(-0.001388888888888889 \cdot {x}^{3} + \left(\frac{1}{x} + x \cdot 0.08333333333333333\right)\right)\\ \end{array} \]

Alternative 4: 71.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x} + 0.5\\ \mathbf{if}\;x \leq -3.8:\\ \;\;\;\;\frac{t_0 \cdot t_0 - x \cdot \left(x \cdot 0.006944444444444444\right)}{\frac{1}{x} + \left(0.5 - \sqrt[3]{\left(x \cdot 0.08333333333333333\right) \cdot \left(\left(x \cdot 0.08333333333333333\right) \cdot \left(x \cdot 0.08333333333333333\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(-0.001388888888888889 \cdot {x}^{3} + \left(\frac{1}{x} + x \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 x) 0.5)))
   (if (<= x -3.8)
     (/
      (- (* t_0 t_0) (* x (* x 0.006944444444444444)))
      (+
       (/ 1.0 x)
       (-
        0.5
        (cbrt
         (*
          (* x 0.08333333333333333)
          (* (* x 0.08333333333333333) (* x 0.08333333333333333)))))))
     (+
      0.5
      (+
       (* -0.001388888888888889 (pow x 3.0))
       (+ (/ 1.0 x) (* x 0.08333333333333333)))))))

C

double code(double x) {
	double t_0 = (1.0 / x) + 0.5;
	double tmp;
	if (x <= -3.8) {
		tmp = ((t_0 * t_0) - (x * (x * 0.006944444444444444))) / ((1.0 / x) + (0.5 - cbrt(((x * 0.08333333333333333) * ((x * 0.08333333333333333) * (x * 0.08333333333333333))))));
	} else {
		tmp = 0.5 + ((-0.001388888888888889 * pow(x, 3.0)) + ((1.0 / x) + (x * 0.08333333333333333)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = (1.0 / x) + 0.5;
	double tmp;
	if (x <= -3.8) {
		tmp = ((t_0 * t_0) - (x * (x * 0.006944444444444444))) / ((1.0 / x) + (0.5 - Math.cbrt(((x * 0.08333333333333333) * ((x * 0.08333333333333333) * (x * 0.08333333333333333))))));
	} else {
		tmp = 0.5 + ((-0.001388888888888889 * Math.pow(x, 3.0)) + ((1.0 / x) + (x * 0.08333333333333333)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(1.0 / x) + 0.5)
	tmp = 0.0
	if (x <= -3.8)
		tmp = Float64(Float64(Float64(t_0 * t_0) - Float64(x * Float64(x * 0.006944444444444444))) / Float64(Float64(1.0 / x) + Float64(0.5 - cbrt(Float64(Float64(x * 0.08333333333333333) * Float64(Float64(x * 0.08333333333333333) * Float64(x * 0.08333333333333333)))))));
	else
		tmp = Float64(0.5 + Float64(Float64(-0.001388888888888889 * (x ^ 3.0)) + Float64(Float64(1.0 / x) + Float64(x * 0.08333333333333333))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(1.0 / x), $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[x, -3.8], N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(x * N[(x * 0.006944444444444444), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 - N[Power[N[(N[(x * 0.08333333333333333), $MachinePrecision] * N[(N[(x * 0.08333333333333333), $MachinePrecision] * N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(N[(-0.001388888888888889 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7999999999999998

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 2.5%

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

      1. associate-+r+ 2.5%

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

      2. +-commutative 2.5%

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

      3. +-commutative 2.5%

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

      4. *-commutative 2.5%

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

      5. fma-def 2.5%

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

   6. Simplified 2.5%

      \[\leadsto \color{blue}{\frac{1}{x} + \mathsf{fma}\left(x, 0.08333333333333333, 0.5\right)} \]
   7. Step-by-step derivation

      1. +-commutative 2.5%

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

      2. fma-udef 2.5%

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

      3. *-commutative 2.5%

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

      4. associate-+l+ 2.5%

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

      5. +-commutative 2.5%

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

      6. flip-+ 2.3%

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

      7. +-commutative 2.3%

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

      8. +-commutative 2.3%

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

      9. swap-sqr 2.3%

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

      10. *-commutative 2.3%

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

      11. associate-*l* 2.3%

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

      12. metadata-eval 2.3%

          \[\leadsto \frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot \color{blue}{0.006944444444444444}\right)}{\left(0.5 + \frac{1}{x}\right) - 0.08333333333333333 \cdot x} \]

      13. +-commutative 2.3%

          \[\leadsto \frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\color{blue}{\left(\frac{1}{x} + 0.5\right)} - 0.08333333333333333 \cdot x} \]

      14. *-commutative 2.3%

          \[\leadsto \frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\left(\frac{1}{x} + 0.5\right) - \color{blue}{x \cdot 0.08333333333333333}} \]

   8. Applied egg-rr 2.3%

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\left(\frac{1}{x} + 0.5\right) - x \cdot 0.08333333333333333}} \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv 2.3%

         \[\leadsto \frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\color{blue}{\left(\frac{1}{x} + 0.5\right) + \left(-x\right) \cdot 0.08333333333333333}} \]

      2. associate-+l+ 2.3%

         \[\leadsto \frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\color{blue}{\frac{1}{x} + \left(0.5 + \left(-x\right) \cdot 0.08333333333333333\right)}} \]

      3. cancel-sign-sub-inv 2.3%

         \[\leadsto \frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\frac{1}{x} + \color{blue}{\left(0.5 - x \cdot 0.08333333333333333\right)}} \]

   10. Simplified 2.3%

       \[\leadsto \color{blue}{\frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\frac{1}{x} + \left(0.5 - x \cdot 0.08333333333333333\right)}} \]
   11. Step-by-step derivation

       1. add-cbrt-cube_binary64 18.7%

          \[\leadsto \color{blue}{\frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\frac{1}{x} + \left(0.5 - \sqrt[3]{\left(\left(x \cdot 0.08333333333333333\right) \cdot \left(x \cdot 0.08333333333333333\right)\right) \cdot \left(x \cdot 0.08333333333333333\right)}\right)}} \]

   12. Applied rewrite-once 18.7%

       \[\leadsto \frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\frac{1}{x} + \left(0.5 - \color{blue}{\sqrt[3]{\left(\left(x \cdot 0.08333333333333333\right) \cdot \left(x \cdot 0.08333333333333333\right)\right) \cdot \left(x \cdot 0.08333333333333333\right)}}\right)} \]

   if -3.7999999999999998 < x

   1. Initial program 7.9%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.5%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8:\\ \;\;\;\;\frac{\left(\frac{1}{x} + 0.5\right) \cdot \left(\frac{1}{x} + 0.5\right) - x \cdot \left(x \cdot 0.006944444444444444\right)}{\frac{1}{x} + \left(0.5 - \sqrt[3]{\left(x \cdot 0.08333333333333333\right) \cdot \left(\left(x \cdot 0.08333333333333333\right) \cdot \left(x \cdot 0.08333333333333333\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(-0.001388888888888889 \cdot {x}^{3} + \left(\frac{1}{x} + x \cdot 0.08333333333333333\right)\right)\\ \end{array} \]

Alternative 5: 66.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ 0.5 + \left(-0.001388888888888889 \cdot {x}^{3} + \left(\frac{1}{x} + x \cdot 0.08333333333333333\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  0.5
  (+
   (* -0.001388888888888889 (pow x 3.0))
   (+ (/ 1.0 x) (* x 0.08333333333333333)))))

C

double code(double x) {
	return 0.5 + ((-0.001388888888888889 * pow(x, 3.0)) + ((1.0 / x) + (x * 0.08333333333333333)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 + (((-0.001388888888888889d0) * (x ** 3.0d0)) + ((1.0d0 / x) + (x * 0.08333333333333333d0)))
end function

Java

public static double code(double x) {
	return 0.5 + ((-0.001388888888888889 * Math.pow(x, 3.0)) + ((1.0 / x) + (x * 0.08333333333333333)));
}

Python

def code(x):
	return 0.5 + ((-0.001388888888888889 * math.pow(x, 3.0)) + ((1.0 / x) + (x * 0.08333333333333333)))

Julia

function code(x)
	return Float64(0.5 + Float64(Float64(-0.001388888888888889 * (x ^ 3.0)) + Float64(Float64(1.0 / x) + Float64(x * 0.08333333333333333))))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 + ((-0.001388888888888889 * (x ^ 3.0)) + ((1.0 / x) + (x * 0.08333333333333333)));
end

Wolfram

code[x_] := N[(0.5 + N[(N[(-0.001388888888888889 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] + N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.7%

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

   1. expm1-def 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 64.8%

   \[\leadsto \color{blue}{0.5 + \left(-0.001388888888888889 \cdot {x}^{3} + \left(0.08333333333333333 \cdot x + \frac{1}{x}\right)\right)} \]

5. Final simplification 64.8%

   \[\leadsto 0.5 + \left(-0.001388888888888889 \cdot {x}^{3} + \left(\frac{1}{x} + x \cdot 0.08333333333333333\right)\right) \]

Alternative 6: 66.3% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ 0.5 + \left(\frac{1}{x} + x \cdot 0.08333333333333333\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ 0.5 (+ (/ 1.0 x) (* x 0.08333333333333333))))

C

double code(double x) {
	return 0.5 + ((1.0 / x) + (x * 0.08333333333333333));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 + ((1.0d0 / x) + (x * 0.08333333333333333d0))
end function

Java

public static double code(double x) {
	return 0.5 + ((1.0 / x) + (x * 0.08333333333333333));
}

Python

def code(x):
	return 0.5 + ((1.0 / x) + (x * 0.08333333333333333))

Julia

function code(x)
	return Float64(0.5 + Float64(Float64(1.0 / x) + Float64(x * 0.08333333333333333)))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 + ((1.0 / x) + (x * 0.08333333333333333));
end

Wolfram

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

Derivation

1. Initial program 40.7%

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

   1. expm1-def 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 64.7%

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

5. Final simplification 64.7%

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

Alternative 7: 66.1% accurate, 68.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.7%

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

   1. expm1-def 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 64.2%

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

5. Final simplification 64.2%

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

Alternative 8: 3.2% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -2.0)

C

double code(double x) {
	return -2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -2.0d0
end function

Java

public static double code(double x) {
	return -2.0;
}

Python

def code(x):
	return -2.0

Julia

function code(x)
	return -2.0
end

MATLAB

function tmp = code(x)
	tmp = -2.0;
end

Wolfram

code[x_] := -2.0

Derivation

1. Initial program 40.7%

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

   1. expm1-def 100.0%

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

3. Simplified 100.0%

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

5. Applied egg-rr 3.1%

   \[\leadsto \color{blue}{-2} \]

6. Final simplification 3.1%

   \[\leadsto -2 \]

Alternative 9: 3.2% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 40.7%

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

   1. expm1-def 100.0%

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

3. Simplified 100.0%

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

5. Applied egg-rr 3.1%

   \[\leadsto \color{blue}{-1} \]

6. Final simplification 3.1%

   \[\leadsto -1 \]

Alternative 10: 4.0% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 40.7%

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

   1. expm1-def 100.0%

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

3. Simplified 100.0%

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

5. Applied egg-rr 3.4%

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

6. Final simplification 3.4%

   \[\leadsto 1 \]

Developer target: 38.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023297 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

  :herbie-target
  (/ 1.0 (- 1.0 (exp (- x))))

  (/ (exp x) (- (exp x) 1.0)))