expq2 (section 3.11)

Percentage Accurate: 38.4% → 99.2%

Time: 3.4s

Alternatives: 4

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: 38.4% 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: 99.2% 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 35.0%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. Final simplification 100.0%

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

Alternative 2: 96.2% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.0%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 69.4%

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

   1. +-commutative 69.4%

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

6. Simplified 69.4%

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

   1. add-sqr-sqrt 32.5%

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

   2. pow2 32.5%

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

8. Applied egg-rr 32.5%

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

9. Taylor expanded in x around 0 31.3%

   \[\leadsto {\left(\sqrt{\color{blue}{\frac{1}{x}}}\right)}^{2} \]
10. Step-by-step derivation

    1. unpow2 31.3%

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

    2. add-sqr-sqrt 69.5%

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

    3. expm1-log1p-u 30.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)} \]

    4. expm1-udef 59.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{x}\right)} - 1} \]

    5. log1p-udef 59.3%

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

    6. add-exp-log 98.3%

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

11. Applied egg-rr 98.3%

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

    1. associate--l+ 98.6%

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

13. Simplified 98.6%

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

14. Final simplification 98.6%

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

Alternative 3: 65.4% 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 35.0%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 69.5%

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

5. Final simplification 69.5%

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

Alternative 4: 3.4% 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 35.0%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 69.4%

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

   1. +-commutative 69.4%

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

6. Simplified 69.4%

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

7. Taylor expanded in x around inf 3.1%

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

8. Final simplification 3.1%

   \[\leadsto 0.5 \]

Developer target: 39.1% 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 2023182 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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