Kahan's exp quotient

Percentage Accurate: 52.7% → 100.0%

Time: 5.4s

Alternatives: 8

Speedup: 15.0×

• 24.2% 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: 52.7% 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 50.5%

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

   1. expm1-define 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. Add Preprocessing
5. Add Preprocessing

Alternative 2: 73.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;\frac{-2 + \frac{-4}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.5)
   (/ (+ -2.0 (/ -4.0 x)) x)
   (/
    (*
     x
     (+
      1.0
      (* x (+ 0.5 (* x (+ 0.16666666666666666 (* x 0.041666666666666664)))))))
    x)))

C

double code(double x) {
	double tmp;
	if (x <= -2.5) {
		tmp = (-2.0 + (-4.0 / x)) / x;
	} else {
		tmp = (x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.5d0)) then
        tmp = ((-2.0d0) + ((-4.0d0) / x)) / x
    else
        tmp = (x * (1.0d0 + (x * (0.5d0 + (x * (0.16666666666666666d0 + (x * 0.041666666666666664d0))))))) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.5) {
		tmp = (-2.0 + (-4.0 / x)) / x;
	} else {
		tmp = (x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))))) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.5:
		tmp = (-2.0 + (-4.0 / x)) / x
	else:
		tmp = (x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))))) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.5)
		tmp = Float64(Float64(-2.0 + Float64(-4.0 / x)) / x);
	else
		tmp = Float64(Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(0.16666666666666666 + Float64(x * 0.041666666666666664))))))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.5)
		tmp = (-2.0 + (-4.0 / x)) / x;
	else
		tmp = (x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.5], N[(N[(-2.0 + N[(-4.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * N[(0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5

   1. Initial program 100.0%

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

      1. expm1-define 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. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around 0 18.8%

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

       1. *-commutative 18.8%

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

   11. Simplified 18.8%

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

   12. Taylor expanded in x around inf 18.8%

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

       1. associate-*r/ 18.8%

          \[\leadsto \color{blue}{\frac{-1 \cdot \left(2 + 4 \cdot \frac{1}{x}\right)}{x}} \]

       2. distribute-lft-in 18.8%

          \[\leadsto \frac{\color{blue}{-1 \cdot 2 + -1 \cdot \left(4 \cdot \frac{1}{x}\right)}}{x} \]

       3. metadata-eval 18.8%

          \[\leadsto \frac{\color{blue}{-2} + -1 \cdot \left(4 \cdot \frac{1}{x}\right)}{x} \]

       4. neg-mul-1 18.8%

          \[\leadsto \frac{-2 + \color{blue}{\left(-4 \cdot \frac{1}{x}\right)}}{x} \]

       5. associate-*r/ 18.8%

          \[\leadsto \frac{-2 + \left(-\color{blue}{\frac{4 \cdot 1}{x}}\right)}{x} \]

       6. metadata-eval 18.8%

          \[\leadsto \frac{-2 + \left(-\frac{\color{blue}{4}}{x}\right)}{x} \]

       7. distribute-neg-frac 18.8%

          \[\leadsto \frac{-2 + \color{blue}{\frac{-4}{x}}}{x} \]

       8. metadata-eval 18.8%

          \[\leadsto \frac{-2 + \frac{\color{blue}{-4}}{x}}{x} \]

   14. Simplified 18.8%

       \[\leadsto \color{blue}{\frac{-2 + \frac{-4}{x}}{x}} \]

   if -2.5 < x

   1. Initial program 32.2%

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

      1. expm1-define 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. Add Preprocessing

   5. Taylor expanded in x around 0 89.2%

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + 0.041666666666666664 \cdot x\right)\right)\right)}}{x} \]
   6. Step-by-step derivation

      1. *-commutative 89.2%

         \[\leadsto \frac{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + \color{blue}{x \cdot 0.041666666666666664}\right)\right)\right)}{x} \]

   7. Simplified 89.2%

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 71.7% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;\frac{-2 + \frac{-4}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.5)
   (/ (+ -2.0 (/ -4.0 x)) x)
   (+
    1.0
    (* x (+ 0.5 (* x (+ 0.16666666666666666 (* x 0.041666666666666664))))))))

C

double code(double x) {
	double tmp;
	if (x <= -2.5) {
		tmp = (-2.0 + (-4.0 / x)) / x;
	} else {
		tmp = 1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.5d0)) then
        tmp = ((-2.0d0) + ((-4.0d0) / x)) / x
    else
        tmp = 1.0d0 + (x * (0.5d0 + (x * (0.16666666666666666d0 + (x * 0.041666666666666664d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.5) {
		tmp = (-2.0 + (-4.0 / x)) / x;
	} else {
		tmp = 1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.5:
		tmp = (-2.0 + (-4.0 / x)) / x
	else:
		tmp = 1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.5)
		tmp = Float64(Float64(-2.0 + Float64(-4.0 / x)) / x);
	else
		tmp = Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(0.16666666666666666 + Float64(x * 0.041666666666666664))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.5)
		tmp = (-2.0 + (-4.0 / x)) / x;
	else
		tmp = 1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.5], N[(N[(-2.0 + N[(-4.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 + N[(x * N[(0.5 + N[(x * N[(0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5

   1. Initial program 100.0%

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

      1. expm1-define 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. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around 0 18.8%

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

       1. *-commutative 18.8%

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

   11. Simplified 18.8%

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

   12. Taylor expanded in x around inf 18.8%

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

       1. associate-*r/ 18.8%

          \[\leadsto \color{blue}{\frac{-1 \cdot \left(2 + 4 \cdot \frac{1}{x}\right)}{x}} \]

       2. distribute-lft-in 18.8%

          \[\leadsto \frac{\color{blue}{-1 \cdot 2 + -1 \cdot \left(4 \cdot \frac{1}{x}\right)}}{x} \]

       3. metadata-eval 18.8%

          \[\leadsto \frac{\color{blue}{-2} + -1 \cdot \left(4 \cdot \frac{1}{x}\right)}{x} \]

       4. neg-mul-1 18.8%

          \[\leadsto \frac{-2 + \color{blue}{\left(-4 \cdot \frac{1}{x}\right)}}{x} \]

       5. associate-*r/ 18.8%

          \[\leadsto \frac{-2 + \left(-\color{blue}{\frac{4 \cdot 1}{x}}\right)}{x} \]

       6. metadata-eval 18.8%

          \[\leadsto \frac{-2 + \left(-\frac{\color{blue}{4}}{x}\right)}{x} \]

       7. distribute-neg-frac 18.8%

          \[\leadsto \frac{-2 + \color{blue}{\frac{-4}{x}}}{x} \]

       8. metadata-eval 18.8%

          \[\leadsto \frac{-2 + \frac{\color{blue}{-4}}{x}}{x} \]

   14. Simplified 18.8%

       \[\leadsto \color{blue}{\frac{-2 + \frac{-4}{x}}{x}} \]

   if -2.5 < x

   1. Initial program 32.2%

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

      1. expm1-define 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. Add Preprocessing

   5. Taylor expanded in x around 0 87.7%

      \[\leadsto \color{blue}{1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + 0.041666666666666664 \cdot x\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 87.7%

         \[\leadsto 1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + \color{blue}{x \cdot 0.041666666666666664}\right)\right) \]

   7. Simplified 87.7%

      \[\leadsto \color{blue}{1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 67.7% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.55:\\ \;\;\;\;\frac{-2 + \frac{-4}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.55)
   (/ (+ -2.0 (/ -4.0 x)) x)
   (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666))))))

C

double code(double x) {
	double tmp;
	if (x <= -2.55) {
		tmp = (-2.0 + (-4.0 / x)) / x;
	} 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 <= (-2.55d0)) then
        tmp = ((-2.0d0) + ((-4.0d0) / x)) / x
    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 <= -2.55) {
		tmp = (-2.0 + (-4.0 / x)) / x;
	} else {
		tmp = 1.0 + (x * (0.5 + (x * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.55:
		tmp = (-2.0 + (-4.0 / x)) / x
	else:
		tmp = 1.0 + (x * (0.5 + (x * 0.16666666666666666)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.55)
		tmp = Float64(Float64(-2.0 + Float64(-4.0 / x)) / x);
	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 <= -2.55)
		tmp = (-2.0 + (-4.0 / x)) / x;
	else
		tmp = 1.0 + (x * (0.5 + (x * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.55], N[(N[(-2.0 + N[(-4.0 / x), $MachinePrecision]), $MachinePrecision] / x), $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 < -2.5499999999999998

   1. Initial program 100.0%

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

      1. expm1-define 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. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around 0 18.8%

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

       1. *-commutative 18.8%

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

   11. Simplified 18.8%

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

   12. Taylor expanded in x around inf 18.8%

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

       1. associate-*r/ 18.8%

          \[\leadsto \color{blue}{\frac{-1 \cdot \left(2 + 4 \cdot \frac{1}{x}\right)}{x}} \]

       2. distribute-lft-in 18.8%

          \[\leadsto \frac{\color{blue}{-1 \cdot 2 + -1 \cdot \left(4 \cdot \frac{1}{x}\right)}}{x} \]

       3. metadata-eval 18.8%

          \[\leadsto \frac{\color{blue}{-2} + -1 \cdot \left(4 \cdot \frac{1}{x}\right)}{x} \]

       4. neg-mul-1 18.8%

          \[\leadsto \frac{-2 + \color{blue}{\left(-4 \cdot \frac{1}{x}\right)}}{x} \]

       5. associate-*r/ 18.8%

          \[\leadsto \frac{-2 + \left(-\color{blue}{\frac{4 \cdot 1}{x}}\right)}{x} \]

       6. metadata-eval 18.8%

          \[\leadsto \frac{-2 + \left(-\frac{\color{blue}{4}}{x}\right)}{x} \]

       7. distribute-neg-frac 18.8%

          \[\leadsto \frac{-2 + \color{blue}{\frac{-4}{x}}}{x} \]

       8. metadata-eval 18.8%

          \[\leadsto \frac{-2 + \frac{\color{blue}{-4}}{x}}{x} \]

   14. Simplified 18.8%

       \[\leadsto \color{blue}{\frac{-2 + \frac{-4}{x}}{x}} \]

   if -2.5499999999999998 < x

   1. Initial program 32.2%

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

      1. expm1-define 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. Add Preprocessing

   5. Taylor expanded in x around 0 84.5%

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

      1. *-commutative 84.5%

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

   7. Simplified 84.5%

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

Alternative 5: 55.9% accurate, 10.5× 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-define 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. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around 0 18.8%

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

       1. *-commutative 18.8%

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

   11. Simplified 18.8%

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

   12. Taylor expanded in x around inf 18.8%

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

   if -1.55000000000000004 < x

   1. Initial program 32.2%

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

      1. expm1-define 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. Add Preprocessing

   5. Taylor expanded in x around 0 73.2%

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

4. Final simplification 58.5%

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

Alternative 6: 54.8% accurate, 13.1× 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-define 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. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around 0 18.8%

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

       1. *-commutative 18.8%

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

   11. Simplified 18.8%

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

   12. Taylor expanded in x around inf 18.8%

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

   if -2 < x

   1. Initial program 32.2%

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

      1. expm1-define 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. Add Preprocessing

   5. Taylor expanded in x around 0 71.9%

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

Alternative 7: 54.9% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + x \cdot -0.5} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.5%

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

   1. expm1-define 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. Add Preprocessing
5. Step-by-step derivation

   1. clear-num 100.0%

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

   2. inv-pow 100.0%

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

6. Applied egg-rr 100.0%

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

   1. unpow-1 100.0%

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

8. Applied egg-rr 100.0%

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

9. Taylor expanded in x around 0 57.7%

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

    1. *-commutative 57.7%

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

11. Simplified 57.7%

    \[\leadsto \frac{1}{\color{blue}{1 + x \cdot -0.5}} \]
12. Add Preprocessing

Alternative 8: 51.5% 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 50.5%

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

   1. expm1-define 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. Add Preprocessing

5. Taylor expanded in x around 0 53.9%

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

Developer Target 1: 52.2% 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 2024118 
(FPCore (x)
  :name "Kahan's exp quotient"
  :precision binary64

  :alt
  (! :herbie-platform default (if (and (< x 1) (> x -1)) (/ (- (exp x) 1) (log (exp x))) (/ (- (exp x) 1) x)))

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