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

Percentage Accurate: 53.9% → 100.0%
Time: 4.8s
Alternatives: 9
Speedup: 8.8×

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 9 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: 53.9% 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 51.8%

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

    1. lift--.f64N/A

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

    2. lift-exp.f64N/A

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

    3. lower-expm1.f64100.0

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

  4. Applied rewrites100.0%

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

Alternative 2: 67.1% accurate, 0.8× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{x} - 1}{x} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right) \cdot x\right) \cdot x\\ \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (if (<= (/ (- (exp x) 1.0) x) 2.0)
   1.0
   (* (* (fma 0.041666666666666664 x 0.16666666666666666) x) x)))
double code(double x) {
	double tmp;
	if (((exp(x) - 1.0) / x) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = (fma(0.041666666666666664, x, 0.16666666666666666) * x) * x;
	}
	return tmp;
}

function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) - 1.0) / x) <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(fma(0.041666666666666664, x, 0.16666666666666666) * x) * x);
	end
	return tmp
end

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

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;\frac{e^{x} - 1}{x} \leq 2:\\
\;\;\;\;1\\

\mathbf{else}:\\
\;\;\;\;\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right) \cdot x\right) \cdot x\\


\end{array}
\end{array}
Derivation
  1. Split input into 2 regimes

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

    1. Initial program 36.7%

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

    3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
    4. Step-by-step derivation

      1. Applied rewrites67.6%

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

      if 2 < (/.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

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

        1. +-commutativeN/A

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

        2. *-commutativeN/A

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

        3. lower-fma.f64N/A

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

        4. +-commutativeN/A

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

        5. *-commutativeN/A

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

        6. lower-fma.f64N/A

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

        7. +-commutativeN/A

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

        8. lower-fma.f6465.8

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right)}, x, 0.5\right), x, 1\right) \]

      5. Applied rewrites65.8%

        \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right)} \]

      6. Taylor expanded in x around inf

        \[\leadsto {x}^{3} \cdot \color{blue}{\left(\frac{1}{24} + \frac{1}{6} \cdot \frac{1}{x}\right)} \]
      7. Step-by-step derivation

        1. Applied rewrites65.8%

          \[\leadsto \left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right) \cdot x\right) \cdot \color{blue}{x} \]
      8. Recombined 2 regimes into one program.
      9. Add Preprocessing

      Alternative 3: 63.1% accurate, 0.9× speedup?

      \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{x} - 1}{x} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, x, 0.5\right) \cdot x\\ \end{array} \end{array} \]
      (FPCore (x)
 :precision binary64
 (if (<= (/ (- (exp x) 1.0) x) 2.0) 1.0 (* (fma 0.16666666666666666 x 0.5) x)))
      double code(double x) {
	double tmp;
	if (((exp(x) - 1.0) / x) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = fma(0.16666666666666666, x, 0.5) * x;
	}
	return tmp;
}

      function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) - 1.0) / x) <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(fma(0.16666666666666666, x, 0.5) * x);
	end
	return tmp
end

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

      \begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;\frac{e^{x} - 1}{x} \leq 2:\\
\;\;\;\;1\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(0.16666666666666666, x, 0.5\right) \cdot x\\


\end{array}
\end{array}
      Derivation
      1. Split input into 2 regimes

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

        1. Initial program 36.7%

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

        3. Taylor expanded in x around 0

          \[\leadsto \color{blue}{1} \]
        4. Step-by-step derivation

          1. Applied rewrites67.6%

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

          if 2 < (/.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

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

            1. +-commutativeN/A

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

            2. *-commutativeN/A

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

            3. lower-fma.f64N/A

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

            4. +-commutativeN/A

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

            5. lower-fma.f6445.7

              \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, x, 0.5\right)}, x, 1\right) \]

          5. Applied rewrites45.7%

            \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right)} \]

          6. Taylor expanded in x around inf

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

          8. Applied rewrites45.7%

            \[\leadsto \mathsf{fma}\left(0.16666666666666666, x, 0.5\right) \cdot \color{blue}{x} \]
        5. Recombined 2 regimes into one program.
        6. Add Preprocessing

        Alternative 4: 63.1% accurate, 0.9× speedup?

        \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{x} - 1}{x} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]
        (FPCore (x)
 :precision binary64
 (if (<= (/ (- (exp x) 1.0) x) 2.0) 1.0 (* (* x x) 0.16666666666666666)))
        double code(double x) {
	double tmp;
	if (((exp(x) - 1.0) / x) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = (x * x) * 0.16666666666666666;
	}
	return tmp;
}

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

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

        def code(x):
	tmp = 0
	if ((math.exp(x) - 1.0) / x) <= 2.0:
		tmp = 1.0
	else:
		tmp = (x * x) * 0.16666666666666666
	return tmp

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

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

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

        \begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;\frac{e^{x} - 1}{x} \leq 2:\\
\;\;\;\;1\\

\mathbf{else}:\\
\;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\


\end{array}
\end{array}
        Derivation
        1. Split input into 2 regimes

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

          1. Initial program 36.7%

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

          3. Taylor expanded in x around 0

            \[\leadsto \color{blue}{1} \]
          4. Step-by-step derivation

            1. Applied rewrites67.6%

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

            if 2 < (/.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

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

              1. +-commutativeN/A

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

              2. *-commutativeN/A

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

              3. lower-fma.f64N/A

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

              4. +-commutativeN/A

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

              5. lower-fma.f6445.7

                \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, x, 0.5\right)}, x, 1\right) \]

            5. Applied rewrites45.7%

              \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right)} \]

            6. Taylor expanded in x around inf

              \[\leadsto \frac{1}{6} \cdot \color{blue}{{x}^{2}} \]
            7. Step-by-step derivation

              1. Applied rewrites45.7%

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

            Alternative 5: 68.9% accurate, 3.3× speedup?

            \[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x}{x} \end{array} \]
            (FPCore (x)
 :precision binary64
 (/
  (*
   (fma (fma (fma 0.041666666666666664 x 0.16666666666666666) x 0.5) x 1.0)
   x)
  x))
            double code(double x) {
	return (fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x) / x;
}

            function code(x)
	return Float64(Float64(fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x) / x)
end

            code[x_] := N[(N[(N[(N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] / x), $MachinePrecision]

            \begin{array}{l}

\\
\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x}{x}
\end{array}
            Derivation

            1. Initial program 51.8%

              \[\frac{e^{x} - 1}{x} \]
            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. *-commutativeN/A

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

              2. lower-*.f64N/A

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

              3. +-commutativeN/A

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

              4. *-commutativeN/A

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

              5. lower-fma.f64N/A

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

              6. +-commutativeN/A

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

              7. *-commutativeN/A

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

              8. lower-fma.f64N/A

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

              9. +-commutativeN/A

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

              10. lower-fma.f6468.2

                \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right)}, x, 0.5\right), x, 1\right) \cdot x}{x} \]

            5. Applied rewrites68.2%

              \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x}}{x} \]
            6. Add Preprocessing

            Alternative 6: 66.9% accurate, 6.1× speedup?

            \[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \end{array} \]
            (FPCore (x)
 :precision binary64
 (fma (fma (fma 0.041666666666666664 x 0.16666666666666666) x 0.5) x 1.0))
            double code(double x) {
	return fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0);
}

            function code(x)
	return fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0)
end

            code[x_] := N[(N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision]

            \begin{array}{l}

\\
\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right)
\end{array}
            Derivation

            1. Initial program 51.8%

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

            3. Taylor expanded in x around 0

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

              1. +-commutativeN/A

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

              2. *-commutativeN/A

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

              3. lower-fma.f64N/A

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

              4. +-commutativeN/A

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

              5. *-commutativeN/A

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

              6. lower-fma.f64N/A

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

              7. +-commutativeN/A

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

              8. lower-fma.f6466.4

                \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right)}, x, 0.5\right), x, 1\right) \]

            5. Applied rewrites66.4%

              \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right)} \]
            6. Add Preprocessing

            Alternative 7: 66.0% accurate, 6.8× speedup?

            \[\begin{array}{l} \\ \mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x, 1\right) \end{array} \]
            (FPCore (x) :precision binary64 (fma (* (* x x) 0.041666666666666664) x 1.0))
            double code(double x) {
	return fma(((x * x) * 0.041666666666666664), x, 1.0);
}

            function code(x)
	return fma(Float64(Float64(x * x) * 0.041666666666666664), x, 1.0)
end

            code[x_] := N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * x + 1.0), $MachinePrecision]

            \begin{array}{l}

\\
\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x, 1\right)
\end{array}
            Derivation

            1. Initial program 51.8%

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

            3. Taylor expanded in x around 0

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

              1. +-commutativeN/A

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

              2. *-commutativeN/A

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

              3. lower-fma.f64N/A

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

              4. +-commutativeN/A

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

              5. *-commutativeN/A

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

              6. lower-fma.f64N/A

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

              7. +-commutativeN/A

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

              8. lower-fma.f6466.4

                \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right)}, x, 0.5\right), x, 1\right) \]

            5. Applied rewrites66.4%

              \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right)} \]

            6. Taylor expanded in x around inf

              \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2}, x, 1\right) \]
            7. Step-by-step derivation

              1. Applied rewrites66.1%

                \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x, 1\right) \]
              2. Add Preprocessing

              Alternative 8: 63.3% accurate, 8.8× speedup?

              \[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \end{array} \]
              (FPCore (x) :precision binary64 (fma (fma 0.16666666666666666 x 0.5) x 1.0))
              double code(double x) {
	return fma(fma(0.16666666666666666, x, 0.5), x, 1.0);
}

              function code(x)
	return fma(fma(0.16666666666666666, x, 0.5), x, 1.0)
end

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

              \begin{array}{l}

\\
\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right)
\end{array}
              Derivation

              1. Initial program 51.8%

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

              3. Taylor expanded in x around 0

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

                1. +-commutativeN/A

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

                2. *-commutativeN/A

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

                3. lower-fma.f64N/A

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

                4. +-commutativeN/A

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

                5. lower-fma.f6462.1

                  \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, x, 0.5\right)}, x, 1\right) \]

              5. Applied rewrites62.1%

                \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right)} \]
              6. Add Preprocessing

              Alternative 9: 50.3% accurate, 115.0× 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 51.8%

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

              3. Taylor expanded in x around 0

                \[\leadsto \color{blue}{1} \]
              4. Step-by-step derivation

                1. Applied rewrites52.2%

                  \[\leadsto \color{blue}{1} \]
                2. Add Preprocessing

                Reproduce

                ?
                herbie shell --seed 1 
(FPCore (x)
  :name "(/ (- (exp x) 1.0) x)"
  :precision binary64
  (/ (- (exp x) 1.0) x))