expq2 (section 3.11)

Percentage Accurate: 37.5% → 99.3%

Time: 4.4s

Alternatives: 5

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: 37.5% 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: 99.3% 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 36.1%

   \[\frac{e^{x}}{e^{x} - 1} \]
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: 66.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 36.1%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 68.0%

   \[\leadsto \color{blue}{0.5 + \left(-0.001388888888888889 \cdot {x}^{3} + \left(0.08333333333333333 \cdot x + \frac{1}{x}\right)\right)} \]
5. Step-by-step derivation

   1. +-commutative 68.0%

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

   2. associate-+r+ 68.0%

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

   3. +-commutative 68.0%

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

   4. associate-+l+ 68.0%

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

   5. unpow3 68.0%

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

   6. associate-*r* 68.0%

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

   7. distribute-rgt-out 68.0%

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

   8. fma-def 68.0%

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

   9. fma-def 68.0%

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

6. Simplified 68.0%

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

   1. fma-udef 68.0%

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

8. Applied egg-rr 68.0%

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

9. Final simplification 68.0%

   \[\leadsto \frac{1}{x} + \mathsf{fma}\left(x, -0.001388888888888889 \cdot \left(x \cdot x\right) + 0.08333333333333333, 0.5\right) \]

Alternative 3: 66.4% 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 36.1%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 68.0%

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

5. Final simplification 68.0%

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

Alternative 4: 66.3% 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 36.1%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 67.8%

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

   1. +-commutative 67.8%

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

6. Simplified 67.8%

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

7. Final simplification 67.8%

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

Alternative 5: 66.3% accurate, 68.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.1%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 67.5%

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

5. Final simplification 67.5%

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

Developer target: 38.2% 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 2023196 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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