Kahan's exp quotient

Percentage Accurate: 52.9% → 100.0%

Time: 4.4s

Alternatives: 6

Speedup: 20.7×

• 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: 52.9% 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 47.9%

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

   1. expm1-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 68.2% accurate, 9.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. 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%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot {\left(\frac{1}{\mathsf{expm1}\left(x\right)}\right)}^{-1}} \]
   6. 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%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around 0 18.8%

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

      1. add-log-exp 6.1%

         \[\leadsto \color{blue}{\log \left(e^{\frac{\frac{1}{x}}{\frac{1}{x} - 0.5}}\right)} \]

      2. *-un-lft-identity 6.1%

         \[\leadsto \log \color{blue}{\left(1 \cdot e^{\frac{\frac{1}{x}}{\frac{1}{x} - 0.5}}\right)} \]

      3. log-prod 6.1%

         \[\leadsto \color{blue}{\log 1 + \log \left(e^{\frac{\frac{1}{x}}{\frac{1}{x} - 0.5}}\right)} \]

      4. metadata-eval 6.1%

         \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\frac{1}{x}}{\frac{1}{x} - 0.5}}\right) \]

      5. clear-num 6.1%

         \[\leadsto 0 + \log \left(e^{\color{blue}{\frac{1}{\frac{\frac{1}{x} - 0.5}{\frac{1}{x}}}}}\right) \]

      6. add-log-exp 18.8%

         \[\leadsto 0 + \color{blue}{\frac{1}{\frac{\frac{1}{x} - 0.5}{\frac{1}{x}}}} \]

      7. div-sub 18.8%

         \[\leadsto 0 + \frac{1}{\color{blue}{\frac{\frac{1}{x}}{\frac{1}{x}} - \frac{0.5}{\frac{1}{x}}}} \]

      8. inv-pow 18.8%

         \[\leadsto 0 + \frac{1}{\frac{\color{blue}{{x}^{-1}}}{\frac{1}{x}} - \frac{0.5}{\frac{1}{x}}} \]

      9. inv-pow 18.8%

         \[\leadsto 0 + \frac{1}{\frac{{x}^{-1}}{\color{blue}{{x}^{-1}}} - \frac{0.5}{\frac{1}{x}}} \]

      10. pow-div 18.8%

          \[\leadsto 0 + \frac{1}{\color{blue}{{x}^{\left(-1 - -1\right)}} - \frac{0.5}{\frac{1}{x}}} \]

      11. metadata-eval 18.8%

          \[\leadsto 0 + \frac{1}{{x}^{\color{blue}{0}} - \frac{0.5}{\frac{1}{x}}} \]

      12. metadata-eval 18.8%

          \[\leadsto 0 + \frac{1}{\color{blue}{1} - \frac{0.5}{\frac{1}{x}}} \]

   10. Applied egg-rr 18.8%

       \[\leadsto \color{blue}{0 + \frac{1}{1 - \frac{0.5}{\frac{1}{x}}}} \]
   11. Step-by-step derivation

       1. +-lft-identity 18.8%

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

       2. associate-/r/ 18.8%

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

       3. metadata-eval 18.8%

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

   12. Simplified 18.8%

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

   if -1 < x

   1. Initial program 35.9%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 84.1%

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

      1. +-commutative 84.1%

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

      2. *-commutative 84.1%

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

      3. *-commutative 84.1%

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

      4. unpow2 84.1%

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

      5. associate-*l* 84.1%

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

      6. distribute-lft-out 84.1%

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

   6. 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 71.8%

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

Alternative 3: 55.8% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.56:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -1.56) (/ -2.0 x) (+ 1.0 (* x 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= -1.56) {
		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 <= (-1.56d0)) 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 <= -1.56) {
		tmp = -2.0 / x;
	} else {
		tmp = 1.0 + (x * 0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.56:
		tmp = -2.0 / x
	else:
		tmp = 1.0 + (x * 0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.56)
		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 <= -1.56)
		tmp = -2.0 / x;
	else
		tmp = 1.0 + (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.56], N[(-2.0 / x), $MachinePrecision], N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5600000000000001

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. 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%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot {\left(\frac{1}{\mathsf{expm1}\left(x\right)}\right)}^{-1}} \]
   6. 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%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around 0 18.8%

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

   9. Taylor expanded in x around inf 18.8%

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

   if -1.5600000000000001 < x

   1. Initial program 35.9%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 68.9%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.56:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot 0.5\\ \end{array} \]

Alternative 4: 54.7% accurate, 15.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.9%

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

   1. expm1-def 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{x}} \]
4. 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 99.9%

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

   5. inv-pow 99.9%

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

5. Applied egg-rr 99.9%

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

   1. unpow-1 99.9%

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

   2. un-div-inv 100.0%

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

7. Applied egg-rr 100.0%

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

8. Taylor expanded in x around 0 58.4%

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

   1. add-log-exp 56.1%

      \[\leadsto \color{blue}{\log \left(e^{\frac{\frac{1}{x}}{\frac{1}{x} - 0.5}}\right)} \]

   2. *-un-lft-identity 56.1%

      \[\leadsto \log \color{blue}{\left(1 \cdot e^{\frac{\frac{1}{x}}{\frac{1}{x} - 0.5}}\right)} \]

   3. log-prod 56.1%

      \[\leadsto \color{blue}{\log 1 + \log \left(e^{\frac{\frac{1}{x}}{\frac{1}{x} - 0.5}}\right)} \]

   4. metadata-eval 56.1%

      \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\frac{1}{x}}{\frac{1}{x} - 0.5}}\right) \]

   5. clear-num 56.1%

      \[\leadsto 0 + \log \left(e^{\color{blue}{\frac{1}{\frac{\frac{1}{x} - 0.5}{\frac{1}{x}}}}}\right) \]

   6. add-log-exp 58.4%

      \[\leadsto 0 + \color{blue}{\frac{1}{\frac{\frac{1}{x} - 0.5}{\frac{1}{x}}}} \]

   7. div-sub 58.4%

      \[\leadsto 0 + \frac{1}{\color{blue}{\frac{\frac{1}{x}}{\frac{1}{x}} - \frac{0.5}{\frac{1}{x}}}} \]

   8. inv-pow 58.4%

      \[\leadsto 0 + \frac{1}{\frac{\color{blue}{{x}^{-1}}}{\frac{1}{x}} - \frac{0.5}{\frac{1}{x}}} \]

   9. inv-pow 58.4%

      \[\leadsto 0 + \frac{1}{\frac{{x}^{-1}}{\color{blue}{{x}^{-1}}} - \frac{0.5}{\frac{1}{x}}} \]

   10. pow-div 58.4%

       \[\leadsto 0 + \frac{1}{\color{blue}{{x}^{\left(-1 - -1\right)}} - \frac{0.5}{\frac{1}{x}}} \]

   11. metadata-eval 58.4%

       \[\leadsto 0 + \frac{1}{{x}^{\color{blue}{0}} - \frac{0.5}{\frac{1}{x}}} \]

   12. metadata-eval 58.4%

       \[\leadsto 0 + \frac{1}{\color{blue}{1} - \frac{0.5}{\frac{1}{x}}} \]

10. Applied egg-rr 58.4%

    \[\leadsto \color{blue}{0 + \frac{1}{1 - \frac{0.5}{\frac{1}{x}}}} \]
11. Step-by-step derivation

    1. +-lft-identity 58.4%

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

    2. associate-/r/ 58.4%

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

    3. metadata-eval 58.4%

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

12. Simplified 58.4%

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

13. Final simplification 58.4%

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

Alternative 5: 54.7% accurate, 20.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -2

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. 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%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot {\left(\frac{1}{\mathsf{expm1}\left(x\right)}\right)}^{-1}} \]
   6. 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%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around 0 18.8%

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

   9. Taylor expanded in x around inf 18.8%

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

   if -2 < x

   1. Initial program 35.9%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 67.7%

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

4. Final simplification 58.6%

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

Alternative 6: 51.3% 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 47.9%

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

   1. expm1-def 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 55.9%

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

5. Final simplification 55.9%

   \[\leadsto 1 \]

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 2023313 
(FPCore (x)
  :name "Kahan's exp quotient"
  :precision binary64

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

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