expq2 (section 3.11)

Percentage Accurate: 36.9% → 99.4%

Time: 3.0s

Alternatives: 6

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: 36.9% 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 41.8%

   \[\frac{e^{x}}{e^{x} - 1} \]
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: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} \leq 10^{-56}:\\ \;\;\;\;e^{x} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(\frac{1}{x} + x \cdot 0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp x) 1e-56)
   (* (exp x) -0.5)
   (+ 0.5 (+ (/ 1.0 x) (* x 0.08333333333333333)))))

C

double code(double x) {
	double tmp;
	if (exp(x) <= 1e-56) {
		tmp = exp(x) * -0.5;
	} 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 (exp(x) <= 1d-56) then
        tmp = exp(x) * (-0.5d0)
    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 (Math.exp(x) <= 1e-56) {
		tmp = Math.exp(x) * -0.5;
	} else {
		tmp = 0.5 + ((1.0 / x) + (x * 0.08333333333333333));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.exp(x) <= 1e-56:
		tmp = math.exp(x) * -0.5
	else:
		tmp = 0.5 + ((1.0 / x) + (x * 0.08333333333333333))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (exp(x) <= 1e-56)
		tmp = Float64(exp(x) * -0.5);
	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 (exp(x) <= 1e-56)
		tmp = exp(x) * -0.5;
	else
		tmp = 0.5 + ((1.0 / x) + (x * 0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Exp[x], $MachinePrecision], 1e-56], N[(N[Exp[x], $MachinePrecision] * -0.5), $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 (exp.f64 x) < 1e-56

   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. Step-by-step derivation

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 99.1%

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

   7. Taylor expanded in x around inf 99.1%

      \[\leadsto \color{blue}{-0.5 \cdot e^{x}} \]

   if 1e-56 < (exp.f64 x)

   1. Initial program 9.1%

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

      1. expm1-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \leq 10^{-56}:\\ \;\;\;\;e^{x} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(\frac{1}{x} + x \cdot 0.08333333333333333\right)\\ \end{array} \]

Alternative 3: 98.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.8%

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

   1. clear-num 100.0%

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

   2. associate-/r/ 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around 0 98.2%

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

7. Final simplification 98.2%

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

Alternative 4: 67.2% 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 41.8%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 64.3%

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

5. Final simplification 64.3%

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

Alternative 5: 67.1% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.8%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 63.8%

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

   1. +-commutative 63.8%

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

6. Simplified 63.8%

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

7. Final simplification 63.8%

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

Alternative 6: 67.0% 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 41.8%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 63.3%

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

5. Final simplification 63.3%

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

Developer target: 37.5% 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 2023192 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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