expq2 (section 3.11)

Percentage Accurate: 36.8% → 99.4%

Time: 5.6s

Alternatives: 8

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: 36.8% 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: 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 40.5%

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

   1. expm1-def 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 2: 98.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return math.exp((x * 0.5)) * ((x * -0.041666666666666664) + (1.0 / x))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.5%

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

   1. expm1-def 99.2%

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

3. Simplified 99.2%

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

   1. add-sqr-sqrt 99.2%

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

   2. *-un-lft-identity 99.2%

      \[\leadsto \frac{\sqrt{e^{x}} \cdot \sqrt{e^{x}}}{\color{blue}{1 \cdot \mathsf{expm1}\left(x\right)}} \]

   3. times-frac 99.2%

      \[\leadsto \color{blue}{\frac{\sqrt{e^{x}}}{1} \cdot \frac{\sqrt{e^{x}}}{\mathsf{expm1}\left(x\right)}} \]

5. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{\sqrt{e^{x}}}{1} \cdot \frac{\sqrt{e^{x}}}{\mathsf{expm1}\left(x\right)}} \]
6. Step-by-step derivation

   1. pow1/2 99.2%

      \[\leadsto \frac{\sqrt{e^{x}}}{1} \cdot \frac{\color{blue}{{\left(e^{x}\right)}^{0.5}}}{\mathsf{expm1}\left(x\right)} \]

   2. pow-exp 99.2%

      \[\leadsto \frac{\sqrt{e^{x}}}{1} \cdot \frac{\color{blue}{e^{x \cdot 0.5}}}{\mathsf{expm1}\left(x\right)} \]

7. Applied egg-rr 99.2%

   \[\leadsto \frac{\sqrt{e^{x}}}{1} \cdot \frac{\color{blue}{e^{x \cdot 0.5}}}{\mathsf{expm1}\left(x\right)} \]
8. Step-by-step derivation

   1. pow1/2 99.2%

      \[\leadsto \frac{\sqrt{e^{x}}}{1} \cdot \frac{\color{blue}{{\left(e^{x}\right)}^{0.5}}}{\mathsf{expm1}\left(x\right)} \]

   2. pow-exp 99.2%

      \[\leadsto \frac{\sqrt{e^{x}}}{1} \cdot \frac{\color{blue}{e^{x \cdot 0.5}}}{\mathsf{expm1}\left(x\right)} \]

9. Applied egg-rr 99.2%

   \[\leadsto \frac{\color{blue}{e^{x \cdot 0.5}}}{1} \cdot \frac{e^{x \cdot 0.5}}{\mathsf{expm1}\left(x\right)} \]

10. Taylor expanded in x around 0 98.3%

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

11. Final simplification 98.3%

    \[\leadsto e^{x \cdot 0.5} \cdot \left(x \cdot -0.041666666666666664 + \frac{1}{x}\right) \]

Alternative 3: 83.9% accurate, 18.5× speedup

Math

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

FPCore

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

C

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

Python

def code(x):
	tmp = 0
	if x <= -4.5:
		tmp = -2.0 / (x * x)
	else:
		tmp = 0.5 + ((1.0 / x) + (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(0.5 + Float64(Float64(1.0 / x) + 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 = 0.5 + ((1.0 / x) + (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[(0.5 + N[(N[(1.0 / x), $MachinePrecision] + 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. 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 3.1%

      \[\leadsto \color{blue}{0.5 + \frac{1}{x}} \]
   5. Step-by-step derivation

      1. +-commutative 3.1%

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

   6. Simplified 3.1%

      \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]
   7. Step-by-step derivation

      1. flip-+ 3.1%

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

      2. clear-num 3.1%

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

      3. sub-neg 3.1%

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

      4. metadata-eval 3.1%

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

      5. sub-neg 3.1%

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

      6. inv-pow 3.1%

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

      7. inv-pow 3.1%

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

      8. pow-prod-up 3.1%

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

      9. metadata-eval 3.1%

         \[\leadsto \frac{1}{\frac{\frac{1}{x} + -0.5}{{x}^{\color{blue}{-2}} + \left(-0.5 \cdot 0.5\right)}} \]

      10. metadata-eval 3.1%

          \[\leadsto \frac{1}{\frac{\frac{1}{x} + -0.5}{{x}^{-2} + \left(-\color{blue}{0.25}\right)}} \]

      11. metadata-eval 3.1%

          \[\leadsto \frac{1}{\frac{\frac{1}{x} + -0.5}{{x}^{-2} + \color{blue}{-0.25}}} \]

   8. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{x} + -0.5}{{x}^{-2} + -0.25}}} \]

   9. Taylor expanded in x around 0 47.5%

      \[\leadsto \frac{1}{\color{blue}{x + -0.5 \cdot {x}^{2}}} \]
   10. Step-by-step derivation

       1. *-commutative 47.5%

          \[\leadsto \frac{1}{x + \color{blue}{{x}^{2} \cdot -0.5}} \]

       2. unpow2 47.5%

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

   11. Simplified 47.5%

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

   12. Taylor expanded in x around inf 47.5%

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

       1. unpow2 47.5%

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

   14. Simplified 47.5%

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

   if -4.5 < x

   1. Initial program 6.5%

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

      1. expm1-def 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in x around 0 98.7%

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

4. Final simplification 80.1%

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

Alternative 4: 83.4% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.5%

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

   1. expm1-def 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in x around 0 63.7%

   \[\leadsto \color{blue}{0.5 + \frac{1}{x}} \]
5. Step-by-step derivation

   1. +-commutative 63.7%

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

6. Simplified 63.7%

   \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]
7. Step-by-step derivation

   1. flip-+ 32.8%

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

   2. clear-num 32.7%

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

   3. sub-neg 32.7%

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

   4. metadata-eval 32.7%

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

   5. sub-neg 32.7%

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

   6. inv-pow 32.7%

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

   7. inv-pow 32.7%

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

   8. pow-prod-up 32.7%

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

   9. metadata-eval 32.7%

      \[\leadsto \frac{1}{\frac{\frac{1}{x} + -0.5}{{x}^{\color{blue}{-2}} + \left(-0.5 \cdot 0.5\right)}} \]

   10. metadata-eval 32.7%

       \[\leadsto \frac{1}{\frac{\frac{1}{x} + -0.5}{{x}^{-2} + \left(-\color{blue}{0.25}\right)}} \]

   11. metadata-eval 32.7%

       \[\leadsto \frac{1}{\frac{\frac{1}{x} + -0.5}{{x}^{-2} + \color{blue}{-0.25}}} \]

8. Applied egg-rr 32.7%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{x} + -0.5}{{x}^{-2} + -0.25}}} \]

9. Taylor expanded in x around 0 79.6%

   \[\leadsto \frac{1}{\color{blue}{x + -0.5 \cdot {x}^{2}}} \]
10. Step-by-step derivation

    1. *-commutative 79.6%

       \[\leadsto \frac{1}{x + \color{blue}{{x}^{2} \cdot -0.5}} \]

    2. unpow2 79.6%

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

11. Simplified 79.6%

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

12. Final simplification 79.6%

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

Alternative 5: 83.6% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.75) (/ -2.0 (* x x)) (+ 0.5 (/ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.75) {
		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 <= (-1.75d0)) 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 <= -1.75) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = 0.5 + (1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.75:
		tmp = -2.0 / (x * x)
	else:
		tmp = 0.5 + (1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.75)
		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 <= -1.75)
		tmp = -2.0 / (x * x);
	else
		tmp = 0.5 + (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.75], 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 < -1.75

   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 3.1%

      \[\leadsto \color{blue}{0.5 + \frac{1}{x}} \]
   5. Step-by-step derivation

      1. +-commutative 3.1%

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

   6. Simplified 3.1%

      \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]
   7. Step-by-step derivation

      1. flip-+ 3.1%

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

      2. clear-num 3.1%

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

      3. sub-neg 3.1%

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

      4. metadata-eval 3.1%

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

      5. sub-neg 3.1%

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

      6. inv-pow 3.1%

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

      7. inv-pow 3.1%

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

      8. pow-prod-up 3.1%

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

      9. metadata-eval 3.1%

         \[\leadsto \frac{1}{\frac{\frac{1}{x} + -0.5}{{x}^{\color{blue}{-2}} + \left(-0.5 \cdot 0.5\right)}} \]

      10. metadata-eval 3.1%

          \[\leadsto \frac{1}{\frac{\frac{1}{x} + -0.5}{{x}^{-2} + \left(-\color{blue}{0.25}\right)}} \]

      11. metadata-eval 3.1%

          \[\leadsto \frac{1}{\frac{\frac{1}{x} + -0.5}{{x}^{-2} + \color{blue}{-0.25}}} \]

   8. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{x} + -0.5}{{x}^{-2} + -0.25}}} \]

   9. Taylor expanded in x around 0 47.5%

      \[\leadsto \frac{1}{\color{blue}{x + -0.5 \cdot {x}^{2}}} \]
   10. Step-by-step derivation

       1. *-commutative 47.5%

          \[\leadsto \frac{1}{x + \color{blue}{{x}^{2} \cdot -0.5}} \]

       2. unpow2 47.5%

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

   11. Simplified 47.5%

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

   12. Taylor expanded in x around inf 47.5%

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

       1. unpow2 47.5%

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

   14. Simplified 47.5%

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

   if -1.75 < x

   1. Initial program 6.5%

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

      1. expm1-def 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in x around 0 98.2%

      \[\leadsto \color{blue}{0.5 + \frac{1}{x}} \]
   5. Step-by-step derivation

      1. +-commutative 98.2%

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

   6. Simplified 98.2%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{1}{x}\\ \end{array} \]

Alternative 6: 67.0% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.5%

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

   1. expm1-def 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in x around 0 63.7%

   \[\leadsto \color{blue}{0.5 + \frac{1}{x}} \]
5. Step-by-step derivation

   1. +-commutative 63.7%

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

6. Simplified 63.7%

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

7. Final simplification 63.7%

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

Alternative 7: 67.1% accurate, 68.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.5%

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

   1. expm1-def 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in x around 0 63.5%

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

5. Final simplification 63.5%

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

Alternative 8: 3.3% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

function tmp = code(x)
	tmp = 0.5;
end

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 40.5%

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

   1. expm1-def 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in x around 0 63.7%

   \[\leadsto \color{blue}{0.5 + \frac{1}{x}} \]
5. Step-by-step derivation

   1. +-commutative 63.7%

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

6. Simplified 63.7%

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

7. Taylor expanded in x around inf 3.3%

   \[\leadsto \color{blue}{0.5} \]

8. Final simplification 3.3%

   \[\leadsto 0.5 \]

Developer target: 37.4% 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 2023283 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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