Kahan's exp quotient

Percentage Accurate: 53.1% → 100.0%

Time: 9.0s

Alternatives: 10

Speedup: 13.1×

• 23.9% 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: 53.1% 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 52.2%

   \[\frac{e^{x} - 1}{x} \]
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. Final simplification 100.0%

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

Alternative 2: 73.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{1}{x} - 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \left(x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e-28)
   (/ (/ 1.0 x) (- (/ 1.0 x) 0.5))
   (*
    (/ 1.0 x)
    (*
     x
     (+
      1.0
      (*
       x
       (+ 0.5 (* x (+ 0.16666666666666666 (* x 0.041666666666666664))))))))))

C

double code(double x) {
	double tmp;
	if (x <= 5e-28) {
		tmp = (1.0 / x) / ((1.0 / x) - 0.5);
	} else {
		tmp = (1.0 / x) * (x * (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 <= 5d-28) then
        tmp = (1.0d0 / x) / ((1.0d0 / x) - 0.5d0)
    else
        tmp = (1.0d0 / x) * (x * (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 <= 5e-28) {
		tmp = (1.0 / x) / ((1.0 / x) - 0.5);
	} else {
		tmp = (1.0 / x) * (x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5e-28:
		tmp = (1.0 / x) / ((1.0 / x) - 0.5)
	else:
		tmp = (1.0 / x) * (x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e-28)
		tmp = Float64(Float64(1.0 / x) / Float64(Float64(1.0 / x) - 0.5));
	else
		tmp = Float64(Float64(1.0 / x) * Float64(x * 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 <= 5e-28)
		tmp = (1.0 / x) / ((1.0 / x) - 0.5);
	else
		tmp = (1.0 / x) * (x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5e-28], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(1.0 / x), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] * 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]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000002e-28

   1. Initial program 38.4%

      \[\frac{e^{x} - 1}{x} \]
   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%

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

      3. div-inv 100.0%

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

      4. unpow-prod-down 99.8%

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

      5. inv-pow 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. un-div-inv 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around 0 71.2%

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

       1. *-commutative 71.2%

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

   11. Simplified 71.2%

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

   12. Taylor expanded in x around inf 71.2%

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

   if 5.0000000000000002e-28 < x

   1. Initial program 94.4%

      \[\frac{e^{x} - 1}{x} \]
   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%

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

      3. div-inv 100.0%

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

      4. unpow-prod-down 100.0%

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

      5. inv-pow 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 72.3%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{1}{x} - 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \left(x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.0% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4:\\ \;\;\;\;\frac{-2 + \frac{-4}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.4)
   (/ (+ -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 <= -2.4) {
		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 <= (-2.4d0)) 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 <= -2.4) {
		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 <= -2.4:
		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 <= -2.4)
		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 <= -2.4)
		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, -2.4], 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 < -2.39999999999999991

   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%

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

      3. div-inv 100.0%

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

      4. unpow-prod-down 100.0%

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

      5. inv-pow 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

      2. un-div-inv 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around 0 18.8%

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

       1. *-commutative 18.8%

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

   11. Simplified 18.8%

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

   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. associate-*r/ 18.8%

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

       5. metadata-eval 18.8%

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

       6. associate-*r/ 18.8%

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

       7. 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 -2.39999999999999991 < x

   1. Initial program 34.9%

      \[\frac{e^{x} - 1}{x} \]
   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 88.9%

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

      1. *-commutative 88.9%

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

   7. Simplified 88.9%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4:\\ \;\;\;\;\frac{-2 + \frac{-4}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.8% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\frac{-2 + \frac{-4}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot \left(0.25 + x \cdot 0.125\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.0)
   (/ (+ -2.0 (/ -4.0 x)) x)
   (+ 1.0 (* x (+ 0.5 (* x (+ 0.25 (* x 0.125))))))))

C

double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = (-2.0 + (-4.0 / x)) / x;
	} else {
		tmp = 1.0 + (x * (0.5 + (x * (0.25 + (x * 0.125)))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.0d0)) then
        tmp = ((-2.0d0) + ((-4.0d0) / x)) / x
    else
        tmp = 1.0d0 + (x * (0.5d0 + (x * (0.25d0 + (x * 0.125d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = (-2.0 + (-4.0 / x)) / x;
	} else {
		tmp = 1.0 + (x * (0.5 + (x * (0.25 + (x * 0.125)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.0:
		tmp = (-2.0 + (-4.0 / x)) / x
	else:
		tmp = 1.0 + (x * (0.5 + (x * (0.25 + (x * 0.125)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.0)
		tmp = Float64(Float64(-2.0 + Float64(-4.0 / x)) / x);
	else
		tmp = Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(0.25 + Float64(x * 0.125))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.0)
		tmp = (-2.0 + (-4.0 / x)) / x;
	else
		tmp = 1.0 + (x * (0.5 + (x * (0.25 + (x * 0.125)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.0], N[(N[(-2.0 + N[(-4.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 + N[(x * N[(0.5 + N[(x * N[(0.25 + N[(x * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

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%

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

      3. div-inv 100.0%

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

      4. unpow-prod-down 100.0%

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

      5. inv-pow 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

      2. un-div-inv 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around 0 18.8%

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

       1. *-commutative 18.8%

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

   11. Simplified 18.8%

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

   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. associate-*r/ 18.8%

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

       5. metadata-eval 18.8%

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

       6. associate-*r/ 18.8%

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

       7. 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 < x

   1. Initial program 34.9%

      \[\frac{e^{x} - 1}{x} \]
   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%

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

      3. div-inv 100.0%

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

      4. unpow-prod-down 99.8%

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

      5. inv-pow 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. un-div-inv 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around 0 69.6%

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

       1. *-commutative 69.6%

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

   11. Simplified 69.6%

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

   12. Taylor expanded in x around 0 89.0%

       \[\leadsto \color{blue}{1 + x \cdot \left(0.5 + x \cdot \left(0.25 + 0.125 \cdot x\right)\right)} \]
   13. Step-by-step derivation

       1. *-commutative 89.0%

          \[\leadsto 1 + x \cdot \left(0.5 + x \cdot \left(0.25 + \color{blue}{x \cdot 0.125}\right)\right) \]

   14. Simplified 89.0%

       \[\leadsto \color{blue}{1 + x \cdot \left(0.5 + x \cdot \left(0.25 + x \cdot 0.125\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\frac{-2 + \frac{-4}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot \left(0.25 + x \cdot 0.125\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.65:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{1}{x} - 0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.65)
   (/ (/ 1.0 x) (- (/ 1.0 x) 0.5))
   (+ 1.0 (* x (+ 0.5 (* x (* x 0.041666666666666664)))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.65) {
		tmp = (1.0 / x) / ((1.0 / x) - 0.5);
	} else {
		tmp = 1.0 + (x * (0.5 + (x * (x * 0.041666666666666664))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.65d0) then
        tmp = (1.0d0 / x) / ((1.0d0 / x) - 0.5d0)
    else
        tmp = 1.0d0 + (x * (0.5d0 + (x * (x * 0.041666666666666664d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.65) {
		tmp = (1.0 / x) / ((1.0 / x) - 0.5);
	} else {
		tmp = 1.0 + (x * (0.5 + (x * (x * 0.041666666666666664))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.65:
		tmp = (1.0 / x) / ((1.0 / x) - 0.5)
	else:
		tmp = 1.0 + (x * (0.5 + (x * (x * 0.041666666666666664))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.65)
		tmp = Float64(Float64(1.0 / x) / Float64(Float64(1.0 / x) - 0.5));
	else
		tmp = Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(x * 0.041666666666666664)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.65)
		tmp = (1.0 / x) / ((1.0 / x) - 0.5);
	else
		tmp = 1.0 + (x * (0.5 + (x * (x * 0.041666666666666664))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.65], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(1.0 / x), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(0.5 + N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.650000000000000022

   1. Initial program 38.5%

      \[\frac{e^{x} - 1}{x} \]
   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%

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

      3. div-inv 100.0%

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

      4. unpow-prod-down 99.8%

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

      5. inv-pow 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. un-div-inv 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around 0 71.7%

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

       1. *-commutative 71.7%

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

   11. Simplified 71.7%

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

   12. Taylor expanded in x around inf 71.7%

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

   if 0.650000000000000022 < x

   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. Taylor expanded in x around 0 64.1%

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

      1. *-commutative 64.1%

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

   7. Simplified 64.1%

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

   8. Taylor expanded in x around inf 64.1%

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

   9. Taylor expanded in x around inf 64.0%

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

       1. *-commutative 64.0%

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

   11. Simplified 64.0%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.65:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{1}{x} - 0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.4% accurate, 7.5× speedup

Math

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

FPCore

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

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)));
	}
	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)))
    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)));
	}
	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)))
	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 * 0.16666666666666666))));
	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)));
	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 * 0.16666666666666666), $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%

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

      3. div-inv 100.0%

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

      4. unpow-prod-down 100.0%

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

      5. inv-pow 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

      2. un-div-inv 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around 0 18.8%

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

       1. *-commutative 18.8%

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

   11. Simplified 18.8%

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

   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. associate-*r/ 18.8%

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

       5. metadata-eval 18.8%

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

       6. associate-*r/ 18.8%

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

       7. 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 34.9%

      \[\frac{e^{x} - 1}{x} \]
   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 84.1%

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

      1. *-commutative 84.1%

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

   7. Simplified 84.1%

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

4. Final simplification 66.8%

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

Alternative 7: 55.7% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

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%

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

      3. div-inv 100.0%

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

      4. unpow-prod-down 100.0%

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

      5. inv-pow 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

      2. un-div-inv 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around 0 18.8%

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

       1. *-commutative 18.8%

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

   11. Simplified 18.8%

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

   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. associate-*r/ 18.8%

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

       5. metadata-eval 18.8%

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

       6. associate-*r/ 18.8%

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

       7. 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 < x

   1. Initial program 34.9%

      \[\frac{e^{x} - 1}{x} \]
   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 70.7%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\frac{-2 + \frac{-4}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.7% 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%

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

      3. div-inv 100.0%

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

      4. unpow-prod-down 100.0%

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

      5. inv-pow 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

      2. un-div-inv 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around 0 18.8%

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

       1. *-commutative 18.8%

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

   11. Simplified 18.8%

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

   12. Taylor expanded in x around inf 18.8%

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

   if -1.55000000000000004 < x

   1. Initial program 34.9%

      \[\frac{e^{x} - 1}{x} \]
   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 70.7%

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

4. Final simplification 56.9%

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

Alternative 9: 54.6% 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%

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

      3. div-inv 100.0%

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

      4. unpow-prod-down 100.0%

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

      5. inv-pow 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

      2. un-div-inv 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around 0 18.8%

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

       1. *-commutative 18.8%

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

   11. Simplified 18.8%

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

   12. Taylor expanded in x around inf 18.8%

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

   if -2 < x

   1. Initial program 34.9%

      \[\frac{e^{x} - 1}{x} \]
   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 69.2%

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

4. Final simplification 55.8%

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

Alternative 10: 51.2% 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 52.2%

   \[\frac{e^{x} - 1}{x} \]
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 52.3%

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

6. Final simplification 52.3%

   \[\leadsto 1 \]
7. Add Preprocessing

Developer target: 52.6% 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 2024067 
(FPCore (x)
  :name "Kahan's exp quotient"
  :precision binary64

  :alt
  (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))