Kahan's exp quotient

Percentage Accurate: 52.8% → 100.0%

Time: 8.7s

Alternatives: 10

Speedup: 13.1×

• 24.5% 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: 52.8% 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 49.3%

   \[\frac{e^{x} - 1}{x} \]
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: 74.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.25:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot \left(0.041666666666666664 + x \cdot \left(0.008333333333333333 + x \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.25)
   (/ -2.0 x)
   (+
    1.0
    (*
     x
     (+
      0.5
      (*
       x
       (+
        0.16666666666666666
        (*
         x
         (+
          0.041666666666666664
          (* x (+ 0.008333333333333333 (* x 0.001388888888888889))))))))))))

C

double code(double x) {
	double tmp;
	if (x <= -3.25) {
		tmp = -2.0 / x;
	} else {
		tmp = 1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * (0.041666666666666664 + (x * (0.008333333333333333 + (x * 0.001388888888888889)))))))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.25d0)) then
        tmp = (-2.0d0) / x
    else
        tmp = 1.0d0 + (x * (0.5d0 + (x * (0.16666666666666666d0 + (x * (0.041666666666666664d0 + (x * (0.008333333333333333d0 + (x * 0.001388888888888889d0)))))))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.25) {
		tmp = -2.0 / x;
	} else {
		tmp = 1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * (0.041666666666666664 + (x * (0.008333333333333333 + (x * 0.001388888888888889)))))))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.25:
		tmp = -2.0 / x
	else:
		tmp = 1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * (0.041666666666666664 + (x * (0.008333333333333333 + (x * 0.001388888888888889)))))))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.25)
		tmp = Float64(-2.0 / x);
	else
		tmp = Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(0.16666666666666666 + Float64(x * Float64(0.041666666666666664 + Float64(x * Float64(0.008333333333333333 + Float64(x * 0.001388888888888889))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.25)
		tmp = -2.0 / x;
	else
		tmp = 1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * (0.041666666666666664 + (x * (0.008333333333333333 + (x * 0.001388888888888889)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.25], N[(-2.0 / x), $MachinePrecision], N[(1.0 + N[(x * N[(0.5 + N[(x * N[(0.16666666666666666 + N[(x * N[(0.041666666666666664 + N[(x * N[(0.008333333333333333 + N[(x * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.25

   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. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)\right)} \]

      2. expm1-undefine 5.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)} - 1} \]

      3. log1p-undefine 5.4%

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

      4. rem-exp-log 5.4%

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

   6. Applied egg-rr 5.4%

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

      1. add-exp-log 5.4%

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

      2. expm1-define 5.4%

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

      3. log1p-define 100.0%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)}\right) \]

      4. clear-num 100.0%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{x}{\mathsf{expm1}\left(x\right)}}}\right)\right) \]

      5. expm1-log1p-u 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. Taylor expanded in x around inf 18.8%

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

   if -3.25 < x

   1. Initial program 33.5%

      \[\frac{e^{x} - 1}{x} \]
   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 94.4%

      \[\leadsto \color{blue}{1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot \left(0.041666666666666664 + x \cdot \left(0.008333333333333333 + 0.001388888888888889 \cdot x\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 94.4%

         \[\leadsto 1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot \left(0.041666666666666664 + x \cdot \left(0.008333333333333333 + \color{blue}{x \cdot 0.001388888888888889}\right)\right)\right)\right) \]

   7. Simplified 94.4%

      \[\leadsto \color{blue}{1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot \left(0.041666666666666664 + x \cdot \left(0.008333333333333333 + x \cdot 0.001388888888888889\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.25:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot \left(0.041666666666666664 + x \cdot \left(0.008333333333333333 + x \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.25:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot \left(0.041666666666666664 + x \cdot 0.008333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.25)
   (/ -2.0 x)
   (+
    1.0
    (*
     x
     (+
      0.5
      (*
       x
       (+
        0.16666666666666666
        (* x (+ 0.041666666666666664 (* x 0.008333333333333333))))))))))

C

double code(double x) {
	double tmp;
	if (x <= -3.25) {
		tmp = -2.0 / x;
	} else {
		tmp = 1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * (0.041666666666666664 + (x * 0.008333333333333333)))))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.25d0)) then
        tmp = (-2.0d0) / x
    else
        tmp = 1.0d0 + (x * (0.5d0 + (x * (0.16666666666666666d0 + (x * (0.041666666666666664d0 + (x * 0.008333333333333333d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.25) {
		tmp = -2.0 / x;
	} else {
		tmp = 1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * (0.041666666666666664 + (x * 0.008333333333333333)))))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.25:
		tmp = -2.0 / x
	else:
		tmp = 1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * (0.041666666666666664 + (x * 0.008333333333333333)))))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.25)
		tmp = Float64(-2.0 / x);
	else
		tmp = Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(0.16666666666666666 + Float64(x * Float64(0.041666666666666664 + Float64(x * 0.008333333333333333))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.25)
		tmp = -2.0 / x;
	else
		tmp = 1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * (0.041666666666666664 + (x * 0.008333333333333333)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.25], N[(-2.0 / x), $MachinePrecision], N[(1.0 + N[(x * N[(0.5 + N[(x * N[(0.16666666666666666 + N[(x * N[(0.041666666666666664 + N[(x * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.25

   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. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)\right)} \]

      2. expm1-undefine 5.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)} - 1} \]

      3. log1p-undefine 5.4%

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

      4. rem-exp-log 5.4%

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

   6. Applied egg-rr 5.4%

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

      1. add-exp-log 5.4%

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

      2. expm1-define 5.4%

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

      3. log1p-define 100.0%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)}\right) \]

      4. clear-num 100.0%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{x}{\mathsf{expm1}\left(x\right)}}}\right)\right) \]

      5. expm1-log1p-u 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. Taylor expanded in x around inf 18.8%

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

   if -3.25 < x

   1. Initial program 33.5%

      \[\frac{e^{x} - 1}{x} \]
   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 93.3%

      \[\leadsto \color{blue}{1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot \left(0.041666666666666664 + 0.008333333333333333 \cdot x\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 93.3%

         \[\leadsto 1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot \left(0.041666666666666664 + \color{blue}{x \cdot 0.008333333333333333}\right)\right)\right) \]

   7. Simplified 93.3%

      \[\leadsto \color{blue}{1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot \left(0.041666666666666664 + x \cdot 0.008333333333333333\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.25:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot \left(0.041666666666666664 + x \cdot 0.008333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.4% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.45:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.45)
   (/ -2.0 x)
   (+
    1.0
    (* x (+ 0.5 (* x (+ 0.16666666666666666 (* x 0.041666666666666664))))))))

C

double code(double x) {
	double tmp;
	if (x <= -2.45) {
		tmp = -2.0 / 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 <= (-2.45d0)) then
        tmp = (-2.0d0) / 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 <= -2.45) {
		tmp = -2.0 / x;
	} else {
		tmp = 1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.45:
		tmp = -2.0 / 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 <= -2.45)
		tmp = Float64(-2.0 / 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 <= -2.45)
		tmp = -2.0 / 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, -2.45], N[(-2.0 / 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 < -2.4500000000000002

   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. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)\right)} \]

      2. expm1-undefine 5.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)} - 1} \]

      3. log1p-undefine 5.4%

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

      4. rem-exp-log 5.4%

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

   6. Applied egg-rr 5.4%

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

      1. add-exp-log 5.4%

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

      2. expm1-define 5.4%

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

      3. log1p-define 100.0%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)}\right) \]

      4. clear-num 100.0%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{x}{\mathsf{expm1}\left(x\right)}}}\right)\right) \]

      5. expm1-log1p-u 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. Taylor expanded in x around inf 18.8%

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

   if -2.4500000000000002 < x

   1. Initial program 33.5%

      \[\frac{e^{x} - 1}{x} \]
   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 89.9%

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

      1. *-commutative 89.9%

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

   7. Simplified 89.9%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.45:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.3% accurate, 7.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -2.5) {
		tmp = -2.0 / 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) / 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 / 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 / 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(-2.0 / 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 / 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[(-2.0 / 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. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)\right)} \]

      2. expm1-undefine 5.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)} - 1} \]

      3. log1p-undefine 5.4%

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

      4. rem-exp-log 5.4%

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

   6. Applied egg-rr 5.4%

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

      1. add-exp-log 5.4%

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

      2. expm1-define 5.4%

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

      3. log1p-define 100.0%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)}\right) \]

      4. clear-num 100.0%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{x}{\mathsf{expm1}\left(x\right)}}}\right)\right) \]

      5. expm1-log1p-u 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. Taylor expanded in x around inf 18.8%

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

   if -2.5 < x

   1. Initial program 33.5%

      \[\frac{e^{x} - 1}{x} \]
   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 85.4%

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

      1. *-commutative 85.4%

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

   7. Simplified 85.4%

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

4. Final simplification 69.5%

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

Alternative 6: 66.6% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.5) (/ -2.0 x) (+ 1.0 (* x (* x 0.16666666666666666)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.5) {
		tmp = -2.0 / x;
	} else {
		tmp = 1.0 + (x * (x * 0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.5d0)) then
        tmp = (-2.0d0) / x
    else
        tmp = 1.0d0 + (x * (x * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.5) {
		tmp = -2.0 / x;
	} else {
		tmp = 1.0 + (x * (x * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.5:
		tmp = -2.0 / x
	else:
		tmp = 1.0 + (x * (x * 0.16666666666666666))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.5)
		tmp = Float64(-2.0 / x);
	else
		tmp = Float64(1.0 + Float64(x * Float64(x * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.5)
		tmp = -2.0 / x;
	else
		tmp = 1.0 + (x * (x * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.5], N[(-2.0 / x), $MachinePrecision], N[(1.0 + N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5

   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. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)\right)} \]

      2. expm1-undefine 5.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)} - 1} \]

      3. log1p-undefine 5.4%

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

      4. rem-exp-log 5.4%

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

   6. Applied egg-rr 5.4%

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

      1. add-exp-log 5.4%

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

      2. expm1-define 5.4%

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

      3. log1p-define 100.0%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)}\right) \]

      4. clear-num 100.0%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{x}{\mathsf{expm1}\left(x\right)}}}\right)\right) \]

      5. expm1-log1p-u 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. Taylor expanded in x around inf 18.8%

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

   if -1.5 < x

   1. Initial program 33.5%

      \[\frac{e^{x} - 1}{x} \]
   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 85.4%

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

      1. *-commutative 85.4%

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

   7. Simplified 85.4%

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

   8. Taylor expanded in x around inf 85.4%

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

      1. associate-*r/ 85.4%

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

      2. metadata-eval 85.4%

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

   10. Simplified 85.4%

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

   11. Taylor expanded in x around inf 84.2%

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

       1. *-commutative 84.2%

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

   13. Simplified 84.2%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.1% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.45)
   (/ 1.0 (+ 1.0 (* x -0.5)))
   (+ 1.0 (* x (* x 0.16666666666666666)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.45) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else {
		tmp = 1.0 + (x * (x * 0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.45d0) then
        tmp = 1.0d0 / (1.0d0 + (x * (-0.5d0)))
    else
        tmp = 1.0d0 + (x * (x * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.45) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else {
		tmp = 1.0 + (x * (x * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.45:
		tmp = 1.0 / (1.0 + (x * -0.5))
	else:
		tmp = 1.0 + (x * (x * 0.16666666666666666))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.45)
		tmp = Float64(1.0 / Float64(1.0 + Float64(x * -0.5)));
	else
		tmp = Float64(1.0 + Float64(x * Float64(x * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.45)
		tmp = 1.0 / (1.0 + (x * -0.5));
	else
		tmp = 1.0 + (x * (x * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.45], N[(1.0 / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   1. Initial program 35.1%

      \[\frac{e^{x} - 1}{x} \]
   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. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)\right)} \]

      2. expm1-undefine 71.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)} - 1} \]

      3. log1p-undefine 71.1%

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

      4. rem-exp-log 71.1%

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

   6. Applied egg-rr 71.1%

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

      1. add-exp-log 71.1%

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

      2. expm1-define 71.1%

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

      3. log1p-define 100.0%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)}\right) \]

      4. clear-num 100.0%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{x}{\mathsf{expm1}\left(x\right)}}}\right)\right) \]

      5. expm1-log1p-u 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 74.4%

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

   if 1.44999999999999996 < 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 51.2%

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

      1. *-commutative 51.2%

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

   7. Simplified 51.2%

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

   8. Taylor expanded in x around inf 51.2%

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

      1. associate-*r/ 51.2%

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

      2. metadata-eval 51.2%

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

   10. Simplified 51.2%

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

   11. Taylor expanded in x around inf 51.1%

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

       1. *-commutative 51.1%

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

   13. Simplified 51.1%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.8% 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. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)\right)} \]

      2. expm1-undefine 5.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)} - 1} \]

      3. log1p-undefine 5.4%

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

      4. rem-exp-log 5.4%

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

   6. Applied egg-rr 5.4%

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

      1. add-exp-log 5.4%

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

      2. expm1-define 5.4%

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

      3. log1p-define 100.0%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)}\right) \]

      4. clear-num 100.0%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{x}{\mathsf{expm1}\left(x\right)}}}\right)\right) \]

      5. expm1-log1p-u 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. Taylor expanded in x around inf 18.8%

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

   if -1.55000000000000004 < x

   1. Initial program 33.5%

      \[\frac{e^{x} - 1}{x} \]
   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 72.1%

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

4. Final simplification 59.4%

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

Alternative 9: 54.7% 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. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)\right)} \]

      2. expm1-undefine 5.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)} - 1} \]

      3. log1p-undefine 5.4%

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

      4. rem-exp-log 5.4%

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

   6. Applied egg-rr 5.4%

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

      1. add-exp-log 5.4%

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

      2. expm1-define 5.4%

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

      3. log1p-define 100.0%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)}\right) \]

      4. clear-num 100.0%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{x}{\mathsf{expm1}\left(x\right)}}}\right)\right) \]

      5. expm1-log1p-u 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. Taylor expanded in x around inf 18.8%

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

   if -2 < x

   1. Initial program 33.5%

      \[\frac{e^{x} - 1}{x} \]
   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 70.5%

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

4. Final simplification 58.2%

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

Alternative 10: 51.4% 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 49.3%

   \[\frac{e^{x} - 1}{x} \]
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 54.9%

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

6. Final simplification 54.9%

   \[\leadsto 1 \]
7. Add Preprocessing

Developer target: 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 2024053 
(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))