Kahan's exp quotient

Percentage Accurate: 53.7% → 100.0%
Time: 1.6s
Alternatives: 3
Speedup: 1.0×

Specification

?
\[\begin{array}{l} \\ \frac{e^{x} - 1}{x} \end{array} \]
(FPCore (x) :precision binary64 (/ (- (exp x) 1.0) x))
double code(double x) {
	return (exp(x) - 1.0) / x;
}

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

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

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

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

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

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

\begin{array}{l}

\\
\frac{e^{x} - 1}{x}
\end{array}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 3 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 53.7% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \frac{e^{x} - 1}{x} \end{array} \]
(FPCore (x) :precision binary64 (/ (- (exp x) 1.0) x))
double code(double x) {
	return (exp(x) - 1.0) / x;
}

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

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

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

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

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

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

\begin{array}{l}

\\
\frac{e^{x} - 1}{x}
\end{array}

Alternative 1: 100.0% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \frac{\mathsf{expm1}\left(x\right)}{x} \end{array} \]
(FPCore (x) :precision binary64 (/ (expm1 x) x))
double code(double x) {
	return expm1(x) / x;
}

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

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

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

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

\begin{array}{l}

\\
\frac{\mathsf{expm1}\left(x\right)}{x}
\end{array}
Derivation

  1. Initial program 53.7%

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

    1. expm1-def100.0%

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

  3. Simplified100.0%

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

  4. Final simplification100.0%

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

Alternative 2: 50.7% accurate, 21.0× speedup?

\[\begin{array}{l} \\ x \cdot 0.5 + 1 \end{array} \]
(FPCore (x) :precision binary64 (+ (* x 0.5) 1.0))
double code(double x) {
	return (x * 0.5) + 1.0;
}

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

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

def code(x):
	return (x * 0.5) + 1.0

function code(x)
	return Float64(Float64(x * 0.5) + 1.0)
end

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

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

\begin{array}{l}

\\
x \cdot 0.5 + 1
\end{array}
Derivation

  1. Initial program 53.7%

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

    1. expm1-def100.0%

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

  3. Simplified100.0%

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

  4. Taylor expanded in x around 0 51.3%

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

  5. Final simplification51.3%

    \[\leadsto x \cdot 0.5 + 1 \]

Alternative 3: 50.5% accurate, 105.0× speedup?

\[\begin{array}{l} \\ 1 \end{array} \]
(FPCore (x) :precision binary64 1.0)
double code(double x) {
	return 1.0;
}

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

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

def code(x):
	return 1.0

function code(x)
	return 1.0
end

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

code[x_] := 1.0

\begin{array}{l}

\\
1
\end{array}
Derivation

  1. Initial program 53.7%

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

    1. expm1-def100.0%

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

  3. Simplified100.0%

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

  4. Taylor expanded in x around 0 50.7%

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

  5. Final simplification50.7%

    \[\leadsto 1 \]

Developer target: 53.3% accurate, 0.3× speedup?

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

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

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;
}

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

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

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

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]]]

\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}

Reproduce

?
herbie shell --seed 2023178 
(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))