expq2 (section 3.11)

Percentage Accurate: 37.8% → 100.0%

Time: 3.4s

Alternatives: 10

Speedup: 17.9×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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: 37.8% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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 43.4%

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

   1. lift--.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lower-expm1.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 99.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right)\\ \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;\frac{e^{x}}{\left(x - -1\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, x, 1\right)}{t\_0 \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fma 0.16666666666666666 x 0.5) x 1.0)))
   (if (<= x -1.35)
     (/ (exp x) (- (- x -1.0) 1.0))
     (/ (fma t_0 x 1.0) (* t_0 x)))))

C

double code(double x) {
	double t_0 = fma(fma(0.16666666666666666, x, 0.5), x, 1.0);
	double tmp;
	if (x <= -1.35) {
		tmp = exp(x) / ((x - -1.0) - 1.0);
	} else {
		tmp = fma(t_0, x, 1.0) / (t_0 * x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(fma(0.16666666666666666, x, 0.5), x, 1.0)
	tmp = 0.0
	if (x <= -1.35)
		tmp = Float64(exp(x) / Float64(Float64(x - -1.0) - 1.0));
	else
		tmp = Float64(fma(t_0, x, 1.0) / Float64(t_0 * x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.35], N[(N[Exp[x], $MachinePrecision] / N[(N[(x - -1.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * x + 1.0), $MachinePrecision] / N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3500000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. rgt-mult-inverse N/A

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

      3. remove-double-neg N/A

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

      4. fp-cancel-sub-sign N/A

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

      5. distribute-lft-neg-in N/A

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

      6. associate-/l* N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-frac-neg N/A

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

      9. lower--.f64 N/A

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

      10. distribute-frac-neg N/A

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

      11. *-inverses N/A

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

      12. metadata-eval 99.2

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

   5. Applied rewrites 99.2%

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

   if -1.3500000000000001 < x

   1. Initial program 6.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. rgt-mult-inverse N/A

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

      3. remove-double-neg N/A

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

      4. fp-cancel-sub-sign N/A

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

      5. distribute-lft-neg-in N/A

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

      6. associate-/l* N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-frac-neg N/A

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

      9. lower--.f64 N/A

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

      10. distribute-frac-neg N/A

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

      11. *-inverses N/A

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

      12. metadata-eval 5.4

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 5.6

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

   8. Applied rewrites 5.6%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{x - -1}{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{x - -1}{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) \cdot x}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{x - -1}{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) \cdot x}} \]

   11. Applied rewrites 99.0%

       \[\leadsto \frac{x - -1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x}} \]

   12. Taylor expanded in x around 0

       \[\leadsto \frac{\color{blue}{1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x, 1\right) \cdot x} \]
   13. Step-by-step derivation

       1. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{\color{blue}{1 - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x, 1\right) \cdot x} \]

       2. fp-cancel-sub-sign-inv N/A

          \[\leadsto \frac{\color{blue}{1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x, 1\right) \cdot x} \]

       3. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + 1}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x, 1\right) \cdot x} \]

       4. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{x} \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) + 1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x, 1\right) \cdot x} \]

       5. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) \cdot x} + 1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x, 1\right) \cdot x} \]

       6. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right), x, 1\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x, 1\right) \cdot x} \]

       7. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1}, x, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x, 1\right) \cdot x} \]

       8. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot x} + 1, x, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x, 1\right) \cdot x} \]

       9. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot x, x, 1\right)}, x, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x, 1\right) \cdot x} \]

       10. +-commutative N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, x, 1\right), x, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x, 1\right) \cdot x} \]

       11. lower-fma.f64 99.7

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, x, 0.5\right)}, x, 1\right), x, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x} \]

   14. Applied rewrites 99.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 90.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{x - -1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (- x -1.0)
  (*
   (fma (fma (fma 0.041666666666666664 x 0.16666666666666666) x 0.5) x 1.0)
   x)))

C

double code(double x) {
	return (x - -1.0) / (fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x);
}

Julia

function code(x)
	return Float64(Float64(x - -1.0) / Float64(fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x))
end

Wolfram

code[x_] := N[(N[(x - -1.0), $MachinePrecision] / N[(N[(N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.4%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. rgt-mult-inverse N/A

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

   3. remove-double-neg N/A

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

   4. fp-cancel-sub-sign N/A

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

   5. distribute-lft-neg-in N/A

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

   6. associate-/l* N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-frac-neg N/A

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

   9. lower--.f64 N/A

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

   10. distribute-frac-neg N/A

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

   11. *-inverses N/A

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

   12. metadata-eval 42.4

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

5. Applied rewrites 42.4%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

   3. fp-cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lower--.f64 4.6

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

8. Applied rewrites 4.6%

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

9. Taylor expanded in x around 0

   \[\leadsto \frac{x - -1}{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{x - -1}{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) \cdot x}} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{x - -1}{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) \cdot x}} \]

11. Applied rewrites 89.0%

    \[\leadsto \frac{x - -1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x}} \]
12. Add Preprocessing

Alternative 4: 90.2% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (*
   (fma (fma (fma 0.041666666666666664 x 0.16666666666666666) x 0.5) x 1.0)
   x)))

C

double code(double x) {
	return 1.0 / (fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x);
}

Julia

function code(x)
	return Float64(1.0 / Float64(fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x))
end

Wolfram

code[x_] := N[(1.0 / N[(N[(N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.4%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. rgt-mult-inverse N/A

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

   3. remove-double-neg N/A

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

   4. fp-cancel-sub-sign N/A

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

   5. distribute-lft-neg-in N/A

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

   6. associate-/l* N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-frac-neg N/A

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

   9. lower--.f64 N/A

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

   10. distribute-frac-neg N/A

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

   11. *-inverses N/A

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

   12. metadata-eval 42.4

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

5. Applied rewrites 42.4%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

   3. fp-cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lower--.f64 4.6

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

8. Applied rewrites 4.6%

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

9. Taylor expanded in x around 0

   \[\leadsto \frac{x - -1}{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{x - -1}{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) \cdot x}} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{x - -1}{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) \cdot x}} \]

11. Applied rewrites 89.0%

    \[\leadsto \frac{x - -1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x}} \]

12. Taylor expanded in x around 0

    \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x, \frac{1}{6}\right), x, \frac{1}{2}\right), x, 1\right) \cdot x} \]
13. Step-by-step derivation

14. Applied rewrites 88.3%

    \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x} \]
15. Add Preprocessing

Alternative 5: 88.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \frac{x - -1}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (- x -1.0) (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x)))

C

double code(double x) {
	return (x - -1.0) / (fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x);
}

Julia

function code(x)
	return Float64(Float64(x - -1.0) / Float64(fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x))
end

Wolfram

code[x_] := N[(N[(x - -1.0), $MachinePrecision] / N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.4%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. rgt-mult-inverse N/A

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

   3. remove-double-neg N/A

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

   4. fp-cancel-sub-sign N/A

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

   5. distribute-lft-neg-in N/A

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

   6. associate-/l* N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-frac-neg N/A

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

   9. lower--.f64 N/A

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

   10. distribute-frac-neg N/A

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

   11. *-inverses N/A

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

   12. metadata-eval 42.4

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

5. Applied rewrites 42.4%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

   3. fp-cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lower--.f64 4.6

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

8. Applied rewrites 4.6%

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

9. Taylor expanded in x around 0

   \[\leadsto \frac{x - -1}{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)}} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{x - -1}{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) \cdot x}} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{x - -1}{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) \cdot x}} \]

11. Applied rewrites 86.6%

    \[\leadsto \frac{x - -1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x}} \]
12. Add Preprocessing

Alternative 6: 87.4% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x)))

C

double code(double x) {
	return 1.0 / (fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x);
}

Julia

function code(x)
	return Float64(1.0 / Float64(fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x))
end

Wolfram

code[x_] := N[(1.0 / N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.4%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. rgt-mult-inverse N/A

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

   3. remove-double-neg N/A

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

   4. fp-cancel-sub-sign N/A

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

   5. distribute-lft-neg-in N/A

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

   6. associate-/l* N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-frac-neg N/A

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

   9. lower--.f64 N/A

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

   10. distribute-frac-neg N/A

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

   11. *-inverses N/A

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

   12. metadata-eval 42.4

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

5. Applied rewrites 42.4%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

   3. fp-cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lower--.f64 4.6

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

8. Applied rewrites 4.6%

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

9. Taylor expanded in x around 0

   \[\leadsto \frac{x - -1}{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)}} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{x - -1}{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) \cdot x}} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{x - -1}{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) \cdot x}} \]

11. Applied rewrites 86.6%

    \[\leadsto \frac{x - -1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x}} \]

12. Taylor expanded in x around 0

    \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x, 1\right) \cdot x} \]
13. Step-by-step derivation

14. Applied rewrites 86.1%

    \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x} \]
15. Add Preprocessing

Alternative 7: 82.8% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.4%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. rgt-mult-inverse N/A

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

   3. remove-double-neg N/A

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

   4. fp-cancel-sub-sign N/A

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

   5. distribute-lft-neg-in N/A

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

   6. associate-/l* N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-frac-neg N/A

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

   9. lower--.f64 N/A

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

   10. distribute-frac-neg N/A

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

   11. *-inverses N/A

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

   12. metadata-eval 42.4

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

5. Applied rewrites 42.4%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

   3. fp-cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lower--.f64 4.6

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

8. Applied rewrites 4.6%

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

9. Taylor expanded in x around 0

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

    1. *-commutative N/A

       \[\leadsto \frac{x - -1}{\color{blue}{\left(1 + \frac{1}{2} \cdot x\right) \cdot x}} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{x - -1}{\color{blue}{\left(1 + \frac{1}{2} \cdot x\right) \cdot x}} \]

    3. +-commutative N/A

       \[\leadsto \frac{x - -1}{\color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} \cdot x} \]

    4. lower-fma.f64 82.1

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

11. Applied rewrites 82.1%

    \[\leadsto \frac{x - -1}{\color{blue}{\mathsf{fma}\left(0.5, x, 1\right) \cdot x}} \]
12. Add Preprocessing

Alternative 8: 82.3% accurate, 9.3× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (* (fma 0.5 x 1.0) x)))

C

double code(double x) {
	return 1.0 / (fma(0.5, x, 1.0) * x);
}

Julia

function code(x)
	return Float64(1.0 / Float64(fma(0.5, x, 1.0) * x))
end

Wolfram

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

Derivation

1. Initial program 43.4%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. rgt-mult-inverse N/A

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

   3. remove-double-neg N/A

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

   4. fp-cancel-sub-sign N/A

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

   5. distribute-lft-neg-in N/A

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

   6. associate-/l* N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-frac-neg N/A

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

   9. lower--.f64 N/A

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

   10. distribute-frac-neg N/A

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

   11. *-inverses N/A

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

   12. metadata-eval 42.4

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

5. Applied rewrites 42.4%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

   3. fp-cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lower--.f64 4.6

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

8. Applied rewrites 4.6%

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

9. Taylor expanded in x around 0

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

    1. *-commutative N/A

       \[\leadsto \frac{x - -1}{\color{blue}{\left(1 + \frac{1}{2} \cdot x\right) \cdot x}} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{x - -1}{\color{blue}{\left(1 + \frac{1}{2} \cdot x\right) \cdot x}} \]

    3. +-commutative N/A

       \[\leadsto \frac{x - -1}{\color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} \cdot x} \]

    4. lower-fma.f64 82.1

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

11. Applied rewrites 82.1%

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

12. Taylor expanded in x around 0

    \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\frac{1}{2}, x, 1\right) \cdot x} \]
13. Step-by-step derivation

14. Applied rewrites 81.6%

    \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(0.5, x, 1\right) \cdot x} \]
15. Add Preprocessing

Alternative 9: 66.7% accurate, 17.9× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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 43.4%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 61.5

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

5. Applied rewrites 61.5%

   \[\leadsto \color{blue}{\frac{1}{x}} \]
6. Add Preprocessing

Alternative 10: 3.2% accurate, 215.0× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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 43.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot x}{x}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{1 + \color{blue}{x \cdot \frac{1}{2}}}{x} \]

   2. fp-cancel-sign-sub-inv N/A

      \[\leadsto \frac{\color{blue}{1 - \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{1}{2}}}{x} \]

   3. div-sub N/A

      \[\leadsto \color{blue}{\frac{1}{x} - \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{1}{2}}{x}} \]

   4. distribute-lft-neg-in N/A

      \[\leadsto \frac{1}{x} - \frac{\color{blue}{\mathsf{neg}\left(x \cdot \frac{1}{2}\right)}}{x} \]

   5. *-commutative N/A

      \[\leadsto \frac{1}{x} - \frac{\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot x}\right)}{x} \]

   6. distribute-frac-neg N/A

      \[\leadsto \frac{1}{x} - \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot x}{x}\right)\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{x} - \left(\mathsf{neg}\left(\frac{\color{blue}{x \cdot \frac{1}{2}}}{x}\right)\right) \]

   8. *-rgt-identity N/A

      \[\leadsto \frac{1}{x} - \left(\mathsf{neg}\left(\frac{x \cdot \frac{1}{2}}{\color{blue}{x \cdot 1}}\right)\right) \]

   9. times-frac N/A

      \[\leadsto \frac{1}{x} - \left(\mathsf{neg}\left(\color{blue}{\frac{x}{x} \cdot \frac{\frac{1}{2}}{1}}\right)\right) \]

   10. *-inverses N/A

       \[\leadsto \frac{1}{x} - \left(\mathsf{neg}\left(\color{blue}{1} \cdot \frac{\frac{1}{2}}{1}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \frac{1}{x} - \left(\mathsf{neg}\left(1 \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \frac{1}{x} - \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right) \]

   13. lower--.f64 N/A

       \[\leadsto \color{blue}{\frac{1}{x} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

   14. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{1}{x}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   15. metadata-eval 61.3

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

5. Applied rewrites 61.3%

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

6. Taylor expanded in x around inf

   \[\leadsto \frac{1}{2} \]
7. Step-by-step derivation

8. Applied rewrites 3.3%

   \[\leadsto 0.5 \]
9. Add Preprocessing

Developer Target 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(Float64(-1.0) / expm1(Float64(-x)))
end

Wolfram

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

Reproduce

herbie shell --seed 2025017 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64
  :pre (> 710.0 x)

  :alt
  (! :herbie-platform default (/ (- 1) (expm1 (- x))))

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