Kahan's exp quotient

Percentage Accurate: 53.1% → 100.0%

Time: 11.2s

Alternatives: 15

Speedup: 8.8×

• 24.6% 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 50.6%

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

   1. lift--.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lower-expm1.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 70.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;\frac{e^{x} + -1}{x} \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.25 \cdot \left(x \cdot x\right)}{\mathsf{fma}\left(0.125, t\_0, -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.25, 0.5\right), 1\right), \frac{-1}{\mathsf{fma}\left(x \cdot x, 0.25, -1\right)} \cdot \mathsf{fma}\left(x, 0.5, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0 \cdot \mathsf{fma}\left(x, x \cdot 0.001736111111111111, -0.027777777777777776\right)}{\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= (/ (+ (exp x) -1.0) x) 5.0)
     (fma
      (/ (* 0.25 (* x x)) (fma 0.125 t_0 -1.0))
      (fma x (fma x 0.25 0.5) 1.0)
      (* (/ -1.0 (fma (* x x) 0.25 -1.0)) (fma x 0.5 1.0)))
     (/
      (/
       (* t_0 (fma x (* x 0.001736111111111111) -0.027777777777777776))
       (fma x 0.041666666666666664 -0.16666666666666666))
      x))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double tmp;
	if (((exp(x) + -1.0) / x) <= 5.0) {
		tmp = fma(((0.25 * (x * x)) / fma(0.125, t_0, -1.0)), fma(x, fma(x, 0.25, 0.5), 1.0), ((-1.0 / fma((x * x), 0.25, -1.0)) * fma(x, 0.5, 1.0)));
	} else {
		tmp = ((t_0 * fma(x, (x * 0.001736111111111111), -0.027777777777777776)) / fma(x, 0.041666666666666664, -0.16666666666666666)) / x;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (Float64(Float64(exp(x) + -1.0) / x) <= 5.0)
		tmp = fma(Float64(Float64(0.25 * Float64(x * x)) / fma(0.125, t_0, -1.0)), fma(x, fma(x, 0.25, 0.5), 1.0), Float64(Float64(-1.0 / fma(Float64(x * x), 0.25, -1.0)) * fma(x, 0.5, 1.0)));
	else
		tmp = Float64(Float64(Float64(t_0 * fma(x, Float64(x * 0.001736111111111111), -0.027777777777777776)) / fma(x, 0.041666666666666664, -0.16666666666666666)) / x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision], 5.0], N[(N[(N[(0.25 * N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(0.125 * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * 0.25 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(-1.0 / N[(N[(x * x), $MachinePrecision] * 0.25 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(x * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * N[(x * N[(x * 0.001736111111111111), $MachinePrecision] + -0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(x * 0.041666666666666664 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (exp.f64 x) #s(literal 1 binary64)) x) < 5

   1. Initial program 37.7%

      \[\frac{e^{x} - 1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 66.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)} \]

   5. Applied rewrites 66.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 67.0%

      \[\leadsto \mathsf{fma}\left(\frac{0.25 \cdot \left(x \cdot x\right)}{\mathsf{fma}\left(0.125, x \cdot \left(x \cdot x\right), -1\right)}, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.25, 0.5\right), 1\right)}, -\frac{1}{\mathsf{fma}\left(x, 0.5, -1\right)}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 67.0%

      \[\leadsto \mathsf{fma}\left(\frac{0.25 \cdot \left(x \cdot x\right)}{\mathsf{fma}\left(0.125, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.25, 0.5\right), \color{blue}{1}\right), \frac{-1}{\mathsf{fma}\left(x \cdot x, 0.25, -1\right)} \cdot \mathsf{fma}\left(x, 0.5, 1\right)\right) \]

   if 5 < (/.f64 (-.f64 (exp.f64 x) #s(literal 1 binary64)) x)

   1. Initial program 100.0%

      \[\frac{e^{x} - 1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) + 1\right)}}{x} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) + x \cdot 1}}{x} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) + \color{blue}{x}}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right), x\right)}}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)}, x\right)}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2}\right)}, x\right)}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{6} + \frac{1}{24} \cdot x, \frac{1}{2}\right)}, x\right)}{x} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{\frac{1}{24} \cdot x + \frac{1}{6}}, \frac{1}{2}\right), x\right)}{x} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24}} + \frac{1}{6}, \frac{1}{2}\right), x\right)}{x} \]

      10. lower-fma.f64 76.9

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)}, 0.5\right), x\right)}{x} \]

   5. Applied rewrites 76.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}}{x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{{x}^{4} \cdot \color{blue}{\left(\frac{1}{24} + \frac{1}{6} \cdot \frac{1}{x}\right)}}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 76.9%

      \[\leadsto \frac{\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 81.1%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(x, x \cdot 0.001736111111111111, -0.027777777777777776\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)}{\mathsf{fma}\left(x, \color{blue}{0.041666666666666664}, -0.16666666666666666\right)}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x} + -1}{x} \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.25 \cdot \left(x \cdot x\right)}{\mathsf{fma}\left(0.125, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.25, 0.5\right), 1\right), \frac{-1}{\mathsf{fma}\left(x \cdot x, 0.25, -1\right)} \cdot \mathsf{fma}\left(x, 0.5, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x, x \cdot 0.001736111111111111, -0.027777777777777776\right)}{\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 52.5% accurate, 0.4× 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 2024222 
(FPCore (x)
  :name "Kahan's exp quotient"
  :precision binary64

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

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