Kahan's exp quotient

Percentage Accurate: 53.3% → 100.0%

Time: 4.9s

Alternatives: 5

Speedup: 14.9×

• 24.8% 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: 53.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.7%

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

   1. expm1-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 67.2% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0 / (1.0 + (x * -0.5))
	else:
		tmp = (x * 0.5) + (1.0 + (0.16666666666666666 * (x * x)))
	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(Float64(x * 0.5) + Float64(1.0 + Float64(0.16666666666666666 * Float64(x * x))));
	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 = (x * 0.5) + (1.0 + (0.16666666666666666 * (x * x)));
	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[(N[(x * 0.5), $MachinePrecision] + N[(1.0 + N[(0.16666666666666666 * N[(x * x), $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%

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

   5. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around 0 18.8%

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

   if -1 < x

   1. Initial program 35.2%

      \[\frac{e^{x} - 1}{x} \]
   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 87.9%

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

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

         \[\leadsto 0.5 \cdot x + \left(\left(\color{blue}{0} + \log \left(e^{0.16666666666666666 \cdot {x}^{2}}\right)\right) + 1\right) \]

      5. add-log-exp 87.9%

         \[\leadsto 0.5 \cdot x + \left(\left(0 + \color{blue}{0.16666666666666666 \cdot {x}^{2}}\right) + 1\right) \]

      6. unpow2 87.9%

         \[\leadsto 0.5 \cdot x + \left(\left(0 + 0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}\right) + 1\right) \]

   6. Applied egg-rr 87.9%

      \[\leadsto 0.5 \cdot x + \left(\color{blue}{\left(0 + 0.16666666666666666 \cdot \left(x \cdot x\right)\right)} + 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.4%

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

Alternative 3: 67.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.86:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.86) (/ 1.0 (+ 1.0 (* x -0.5))) (* 0.16666666666666666 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.86) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else {
		tmp = 0.16666666666666666 * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.86d0) then
        tmp = 1.0d0 / (1.0d0 + (x * (-0.5d0)))
    else
        tmp = 0.16666666666666666d0 * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.86) {
		tmp = 1.0 / (1.0 + (x * -0.5));
	} else {
		tmp = 0.16666666666666666 * (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.86:
		tmp = 1.0 / (1.0 + (x * -0.5))
	else:
		tmp = 0.16666666666666666 * (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.86)
		tmp = Float64(1.0 / Float64(1.0 + Float64(x * -0.5)));
	else
		tmp = Float64(0.16666666666666666 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.86)
		tmp = 1.0 / (1.0 + (x * -0.5));
	else
		tmp = 0.16666666666666666 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.86], N[(1.0 / N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8600000000000001

   1. Initial program 35.2%

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

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

   5. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around 0 74.3%

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

   if 1.8600000000000001 < x

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 61.5%

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

   5. Taylor expanded in x around inf 61.5%

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

      1. unpow2 61.5%

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

   7. Simplified 61.5%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.86:\\ \;\;\;\;\frac{1}{1 + x \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 4: 63.1% accurate, 14.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.5) 1.0 (* 0.16666666666666666 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 2.5) {
		tmp = 1.0;
	} else {
		tmp = 0.16666666666666666 * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.5d0) then
        tmp = 1.0d0
    else
        tmp = 0.16666666666666666d0 * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.5) {
		tmp = 1.0;
	} else {
		tmp = 0.16666666666666666 * (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.5:
		tmp = 1.0
	else:
		tmp = 0.16666666666666666 * (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.5)
		tmp = 1.0;
	else
		tmp = Float64(0.16666666666666666 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.5)
		tmp = 1.0;
	else
		tmp = 0.16666666666666666 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.5], 1.0, N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5

   1. Initial program 35.2%

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

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

   if 2.5 < x

   1. Initial program 100.0%

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

      1. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 61.5%

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

   5. Taylor expanded in x around inf 61.5%

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

      1. unpow2 61.5%

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

   7. Simplified 61.5%

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

4. Final simplification 67.4%

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

Alternative 5: 50.9% accurate, 105.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 50.7%

   \[\frac{e^{x} - 1}{x} \]
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 53.5%

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

5. Final simplification 53.5%

   \[\leadsto 1 \]

Developer target: 52.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023274 
(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))