Kahan's exp quotient

Percentage Accurate: 53.2% → 100.0%

Time: 4.2s

Alternatives: 7

Speedup: 20.7×

• 24.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

   1. expm1-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 67.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;x \leq 2.4:\\ \;\;\;\;1 + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.55)
   (/ -2.0 x)
   (if (<= x 2.4) (+ 1.0 (* x 0.5)) (* x (+ 0.5 (* x 0.16666666666666666))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = -2.0 / x;
	} else if (x <= 2.4) {
		tmp = 1.0 + (x * 0.5);
	} else {
		tmp = x * (0.5 + (x * 0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.55d0)) then
        tmp = (-2.0d0) / x
    else if (x <= 2.4d0) then
        tmp = 1.0d0 + (x * 0.5d0)
    else
        tmp = x * (0.5d0 + (x * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = -2.0 / x;
	} else if (x <= 2.4) {
		tmp = 1.0 + (x * 0.5);
	} else {
		tmp = x * (0.5 + (x * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.55:
		tmp = -2.0 / x
	elif x <= 2.4:
		tmp = 1.0 + (x * 0.5)
	else:
		tmp = x * (0.5 + (x * 0.16666666666666666))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.55)
		tmp = Float64(-2.0 / x);
	elseif (x <= 2.4)
		tmp = Float64(1.0 + Float64(x * 0.5));
	else
		tmp = Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.55)
		tmp = -2.0 / x;
	elseif (x <= 2.4)
		tmp = 1.0 + (x * 0.5);
	else
		tmp = x * (0.5 + (x * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.55], N[(-2.0 / x), $MachinePrecision], If[LessEqual[x, 2.4], N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.55000000000000004

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{x}} \]
   4. Step-by-step derivation

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \mathsf{expm1}\left(x\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ 100.0%

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

      2. associate-/l* 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around 0 18.8%

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

      1. *-commutative 18.8%

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

   10. Simplified 18.8%

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

   11. Taylor expanded in x around inf 18.7%

       \[\leadsto \color{blue}{\frac{-2}{x}} \]

   if -1.55000000000000004 < x < 2.39999999999999991

   1. Initial program 7.9%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 98.7%

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

   if 2.39999999999999991 < x

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 51.8%

      \[\leadsto \color{blue}{1 + \left(0.16666666666666666 \cdot {x}^{2} + 0.5 \cdot x\right)} \]

   5. Taylor expanded in x around inf 51.8%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot {x}^{2} + 0.5 \cdot x} \]
   6. Step-by-step derivation

      1. unpow2 51.8%

         \[\leadsto 0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)} + 0.5 \cdot x \]

      2. associate-*r* 51.8%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot x\right) \cdot x} + 0.5 \cdot x \]

      3. distribute-rgt-out 51.8%

         \[\leadsto \color{blue}{x \cdot \left(0.16666666666666666 \cdot x + 0.5\right)} \]

      4. *-commutative 51.8%

         \[\leadsto x \cdot \left(\color{blue}{x \cdot 0.16666666666666666} + 0.5\right) \]

   7. Simplified 51.8%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot 0.16666666666666666 + 0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;x \leq 2.4:\\ \;\;\;\;1 + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\\ \end{array} \]

Alternative 3: 67.7% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (/ 1.0 (+ 1.0 (* x -0.5)))
   (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else {
		tmp = 1.0 + (x * (0.5 + (x * 0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 1.0d0 / (1.0d0 + (x * (-0.5d0)))
    else
        tmp = 1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else {
		tmp = 1.0 + (x * (0.5 + (x * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0 / (1.0 + (x * -0.5))
	else:
		tmp = 1.0 + (x * (0.5 + (x * 0.16666666666666666)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(1.0 / Float64(1.0 + Float64(x * -0.5)));
	else
		tmp = Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 1.0 / (1.0 + (x * -0.5));
	else
		tmp = 1.0 + (x * (0.5 + (x * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{x}} \]
   4. Step-by-step derivation

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \mathsf{expm1}\left(x\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ 100.0%

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

      2. associate-/l* 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around 0 18.8%

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

      1. *-commutative 18.8%

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

   10. Simplified 18.8%

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

   if -1 < x

   1. Initial program 39.8%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{x}} \]
   4. Step-by-step derivation

      1. clear-num 100.0%

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

      2. associate-/r/ 99.8%

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

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \mathsf{expm1}\left(x\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ 100.0%

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

      2. associate-/l* 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around 0 65.6%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(-0.5 \cdot x + 0.08333333333333333 \cdot {x}^{2}\right)}} \]
   9. Step-by-step derivation

      1. unpow2 65.6%

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

      2. associate-*r* 65.6%

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

      3. *-commutative 65.6%

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

      4. distribute-rgt-out 65.6%

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

   10. Simplified 65.6%

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

   11. Taylor expanded in x around 0 82.9%

       \[\leadsto \color{blue}{1 + \left(0.16666666666666666 \cdot {x}^{2} + 0.5 \cdot x\right)} \]
   12. Step-by-step derivation

       1. +-commutative 82.9%

          \[\leadsto 1 + \color{blue}{\left(0.5 \cdot x + 0.16666666666666666 \cdot {x}^{2}\right)} \]

       2. *-commutative 82.9%

          \[\leadsto 1 + \left(\color{blue}{x \cdot 0.5} + 0.16666666666666666 \cdot {x}^{2}\right) \]

       3. *-commutative 82.9%

          \[\leadsto 1 + \left(x \cdot 0.5 + \color{blue}{{x}^{2} \cdot 0.16666666666666666}\right) \]

       4. unpow2 82.9%

          \[\leadsto 1 + \left(x \cdot 0.5 + \color{blue}{\left(x \cdot x\right)} \cdot 0.16666666666666666\right) \]

       5. associate-*l* 82.9%

          \[\leadsto 1 + \left(x \cdot 0.5 + \color{blue}{x \cdot \left(x \cdot 0.16666666666666666\right)}\right) \]

       6. distribute-lft-out 82.9%

          \[\leadsto 1 + \color{blue}{x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)} \]

   13. Simplified 82.9%

       \[\leadsto \color{blue}{1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\\ \end{array} \]

Alternative 4: 67.4% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.6)
   (/ 1.0 (+ 1.0 (* x -0.5)))
   (* x (+ 0.5 (* x 0.16666666666666666)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else {
		tmp = x * (0.5 + (x * 0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.6d0) then
        tmp = 1.0d0 / (1.0d0 + (x * (-0.5d0)))
    else
        tmp = x * (0.5d0 + (x * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else {
		tmp = x * (0.5 + (x * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.6:
		tmp = 1.0 / (1.0 + (x * -0.5))
	else:
		tmp = x * (0.5 + (x * 0.16666666666666666))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.6)
		tmp = Float64(1.0 / Float64(1.0 + Float64(x * -0.5)));
	else
		tmp = Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.6)
		tmp = 1.0 / (1.0 + (x * -0.5));
	else
		tmp = x * (0.5 + (x * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.6], N[(1.0 / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001

   1. Initial program 38.6%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{x}} \]
   4. Step-by-step derivation

      1. clear-num 100.0%

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

      2. associate-/r/ 99.8%

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

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \mathsf{expm1}\left(x\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ 100.0%

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

      2. associate-/l* 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around 0 72.1%

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

      1. *-commutative 72.1%

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

   10. Simplified 72.1%

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

   if 1.6000000000000001 < x

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 51.8%

      \[\leadsto \color{blue}{1 + \left(0.16666666666666666 \cdot {x}^{2} + 0.5 \cdot x\right)} \]

   5. Taylor expanded in x around inf 51.8%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot {x}^{2} + 0.5 \cdot x} \]
   6. Step-by-step derivation

      1. unpow2 51.8%

         \[\leadsto 0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)} + 0.5 \cdot x \]

      2. associate-*r* 51.8%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot x\right) \cdot x} + 0.5 \cdot x \]

      3. distribute-rgt-out 51.8%

         \[\leadsto \color{blue}{x \cdot \left(0.16666666666666666 \cdot x + 0.5\right)} \]

      4. *-commutative 51.8%

         \[\leadsto x \cdot \left(\color{blue}{x \cdot 0.16666666666666666} + 0.5\right) \]

   7. Simplified 51.8%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot 0.16666666666666666 + 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\\ \end{array} \]

Alternative 5: 55.5% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -1.55) (/ -2.0 x) (+ 1.0 (* x 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = -2.0 / x;
	} else {
		tmp = 1.0 + (x * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.55d0)) then
        tmp = (-2.0d0) / x
    else
        tmp = 1.0d0 + (x * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = -2.0 / x;
	} else {
		tmp = 1.0 + (x * 0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.55:
		tmp = -2.0 / x
	else:
		tmp = 1.0 + (x * 0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.55)
		tmp = Float64(-2.0 / x);
	else
		tmp = Float64(1.0 + Float64(x * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.55)
		tmp = -2.0 / x;
	else
		tmp = 1.0 + (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.55], N[(-2.0 / x), $MachinePrecision], N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{x}} \]
   4. Step-by-step derivation

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \mathsf{expm1}\left(x\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ 100.0%

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

      2. associate-/l* 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around 0 18.8%

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

      1. *-commutative 18.8%

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

   10. Simplified 18.8%

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

   11. Taylor expanded in x around inf 18.7%

       \[\leadsto \color{blue}{\frac{-2}{x}} \]

   if -1.55000000000000004 < x

   1. Initial program 39.8%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 66.2%

      \[\leadsto \color{blue}{1 + 0.5 \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot 0.5\\ \end{array} \]

Alternative 6: 54.4% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -2.0) (/ -2.0 x) 1.0))

C

double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = -2.0 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.0d0)) then
        tmp = (-2.0d0) / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = -2.0 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.0:
		tmp = -2.0 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.0)
		tmp = Float64(-2.0 / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.0)
		tmp = -2.0 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.0], N[(-2.0 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -2

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{x}} \]
   4. Step-by-step derivation

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \mathsf{expm1}\left(x\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ 100.0%

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

      2. associate-/l* 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around 0 18.8%

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

      1. *-commutative 18.8%

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

   10. Simplified 18.8%

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

   11. Taylor expanded in x around inf 18.7%

       \[\leadsto \color{blue}{\frac{-2}{x}} \]

   if -2 < x

   1. Initial program 39.8%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 64.3%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 51.0% accurate, 105.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 54.6%

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

   1. expm1-def 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 49.9%

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

5. Final simplification 49.9%

   \[\leadsto 1 \]

Developer target: 52.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x} - 1\\ \mathbf{if}\;x < 1 \land x > -1:\\ \;\;\;\;\frac{t_0}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (exp x) 1.0)))
   (if (and (< x 1.0) (> x -1.0)) (/ t_0 (log (exp x))) (/ t_0 x))))

C

double code(double x) {
	double t_0 = exp(x) - 1.0;
	double tmp;
	if ((x < 1.0) && (x > -1.0)) {
		tmp = t_0 / log(exp(x));
	} else {
		tmp = t_0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(x) - 1.0d0
    if ((x < 1.0d0) .and. (x > (-1.0d0))) then
        tmp = t_0 / log(exp(x))
    else
        tmp = t_0 / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.exp(x) - 1.0;
	double tmp;
	if ((x < 1.0) && (x > -1.0)) {
		tmp = t_0 / Math.log(Math.exp(x));
	} else {
		tmp = t_0 / x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.exp(x) - 1.0
	tmp = 0
	if (x < 1.0) and (x > -1.0):
		tmp = t_0 / math.log(math.exp(x))
	else:
		tmp = t_0 / x
	return tmp

Julia

function code(x)
	t_0 = Float64(exp(x) - 1.0)
	tmp = 0.0
	if ((x < 1.0) && (x > -1.0))
		tmp = Float64(t_0 / log(exp(x)));
	else
		tmp = Float64(t_0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = exp(x) - 1.0;
	tmp = 0.0;
	if ((x < 1.0) && (x > -1.0))
		tmp = t_0 / log(exp(x));
	else
		tmp = t_0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision]}, If[And[Less[x, 1.0], Greater[x, -1.0]], N[(t$95$0 / N[Log[N[Exp[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 / x), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023318 
(FPCore (x)
  :name "Kahan's exp quotient"
  :precision binary64

  :herbie-target
  (if (and (< x 1.0) (> x -1.0)) (/ (- (exp x) 1.0) (log (exp x))) (/ (- (exp x) 1.0) x))

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