expq2 (section 3.11)

Percentage Accurate: 38.1% → 100.0%

Time: 5.5s

Alternatives: 6

Speedup: 68.3×

Specification

\[710 > x\]

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: 38.1% 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: 100.0% 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 39.0%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. Add Preprocessing

5. Final simplification 99.9%

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

Alternative 2: 99.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0023:\\ \;\;\;\;\frac{1}{1 - e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.0023)
   (/ 1.0 (- 1.0 (exp (- x))))
   (+ 0.5 (+ (* x 0.08333333333333333) (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -0.0023) {
		tmp = 1.0 / (1.0 - exp(-x));
	} else {
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.0023d0)) then
        tmp = 1.0d0 / (1.0d0 - exp(-x))
    else
        tmp = 0.5d0 + ((x * 0.08333333333333333d0) + (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.0023) {
		tmp = 1.0 / (1.0 - Math.exp(-x));
	} else {
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.0023:
		tmp = 1.0 / (1.0 - math.exp(-x))
	else:
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.0023)
		tmp = Float64(1.0 / Float64(1.0 - exp(Float64(-x))));
	else
		tmp = Float64(0.5 + Float64(Float64(x * 0.08333333333333333) + Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.0023)
		tmp = 1.0 / (1.0 - exp(-x));
	else
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.0023], N[(1.0 / N[(1.0 - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(N[(x * 0.08333333333333333), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0023

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 96.6%

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

      2. pow1/2 96.6%

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

      3. clear-num 96.6%

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

      4. inv-pow 96.6%

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

      5. metadata-eval 96.6%

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

      6. pow-pow 96.6%

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

      7. expm1-udef 96.6%

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

      8. div-sub 0.0%

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

      9. pow1 0.0%

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

      10. pow1 0.0%

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

      11. pow-div 96.6%

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

      12. metadata-eval 96.6%

          \[\leadsto {\left({\left(e^{x}\right)}^{\color{blue}{0}} - \frac{1}{e^{x}}\right)}^{\left(\left(-1\right) \cdot 0.5\right)} \cdot \sqrt{\frac{e^{x}}{\mathsf{expm1}\left(x\right)}} \]

      13. metadata-eval 96.6%

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

      14. rec-exp 96.6%

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

      15. metadata-eval 96.6%

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

      16. metadata-eval 96.6%

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

      17. pow1/2 96.6%

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

      18. clear-num 96.6%

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

      19. inv-pow 96.6%

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

      20. metadata-eval 96.6%

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

   6. Applied egg-rr 96.6%

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

      1. pow-sqr 99.9%

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

      2. metadata-eval 99.9%

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

      3. unpow-1 99.9%

         \[\leadsto \color{blue}{\frac{1}{1 - e^{-x}}} \]

   8. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1}{1 - e^{-x}}} \]

   if -0.0023 < x

   1. Initial program 7.7%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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. Add Preprocessing

   5. Taylor expanded in x around 0 98.5%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0023:\\ \;\;\;\;\frac{1}{1 - e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.0%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. Add Preprocessing

5. Taylor expanded in x around 0 96.7%

   \[\leadsto \frac{e^{x}}{\color{blue}{x}} \]

6. Final simplification 96.7%

   \[\leadsto \frac{e^{x}}{x} \]
7. Add Preprocessing

Alternative 4: 66.5% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.0%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. Add Preprocessing

5. Taylor expanded in x around 0 66.1%

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

6. Final simplification 66.1%

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

Alternative 5: 66.4% 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 39.0%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. Add Preprocessing

5. Taylor expanded in x around 0 66.1%

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

   1. +-commutative 66.1%

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

7. Simplified 66.1%

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

8. Final simplification 66.1%

   \[\leadsto 0.5 + \frac{1}{x} \]
9. Add Preprocessing

Alternative 6: 66.4% 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 39.0%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. Add Preprocessing

5. Taylor expanded in x around 0 65.6%

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

6. Final simplification 65.6%

   \[\leadsto \frac{1}{x} \]
7. Add Preprocessing

Developer target: 100.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{expm1}\left(-x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0) (expm1 (- x))))

C

double code(double x) {
	return -1.0 / expm1(-x);
}

Java

public static double code(double x) {
	return -1.0 / Math.expm1(-x);
}

Python

def code(x):
	return -1.0 / math.expm1(-x)

Julia

function code(x)
	return Float64(Float64(-1.0) / expm1(Float64(-x)))
end

Wolfram

code[x_] := N[((-1.0) / N[(Exp[(-x)] - 1), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024018 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64
  :pre (> 710.0 x)

  :herbie-target
  (/ (- 1.0) (expm1 (- x)))

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