expq2 (section 3.11)

Percentage Accurate: 36.4% → 100.0%

Time: 3.4s

Alternatives: 2

Speedup: 41.8×

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: 36.4% 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: 100.0% accurate, 1.9× 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(-1.0 / expm1(Float64(-x)))
end

Wolfram

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

Derivation

1. Initial program 38.2%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. Taylor expanded in x around inf 38.2%

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

   1. expm1-def 100.0%

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

   2. *-lft-identity 100.0%

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

   3. metadata-eval 100.0%

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

   4. times-frac 100.0%

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

   5. neg-mul-1 100.0%

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

   6. associate-/l* 100.0%

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

   7. neg-sub0 100.0%

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

   8. expm1-def 38.2%

      \[\leadsto \frac{-1}{\frac{0 - \color{blue}{\left(e^{x} - 1\right)}}{e^{x}}} \]

   9. associate--r- 38.2%

      \[\leadsto \frac{-1}{\frac{\color{blue}{\left(0 - e^{x}\right) + 1}}{e^{x}}} \]

   10. neg-sub0 38.2%

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

   11. +-commutative 38.2%

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

   12. sub-neg 38.2%

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

   13. div-sub 3.5%

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

   14. exp-neg 3.5%

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

   15. *-inverses 38.2%

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

   16. expm1-def 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

   \[\leadsto \frac{-1}{\mathsf{expm1}\left(-x\right)} \]
9. Add Preprocessing

Alternative 2: 68.0% accurate, 41.8× 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 38.2%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. Taylor expanded in x around 0 66.0%

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

6. Final simplification 66.0%

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

Developer target: 100.0% accurate, 1.9× 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 2024011 
(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)))