Kahan's exp quotient

Percentage Accurate: 53.5% → 100.0%

Time: 9.9s

Alternatives: 19

Speedup: 8.8×

• 25.1% 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: 53.5% 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 49.3%

   \[\frac{e^{x} - 1}{x} \]
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: 75.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)\\ \mathbf{if}\;\frac{-1 + e^{x}}{x} \leq 2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(x \cdot x\right), t\_0 \cdot t\_0, 1\right)}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (fma x 0.16666666666666666 0.5))))
   (if (<= (/ (+ -1.0 (exp x)) x) 2.0)
     (/ 1.0 (fma x -0.5 1.0))
     (/ (fma (* 0.16666666666666666 (* x x)) (* t_0 t_0) 1.0) 1.0))))

C

double code(double x) {
	double t_0 = x * fma(x, 0.16666666666666666, 0.5);
	double tmp;
	if (((-1.0 + exp(x)) / x) <= 2.0) {
		tmp = 1.0 / fma(x, -0.5, 1.0);
	} else {
		tmp = fma((0.16666666666666666 * (x * x)), (t_0 * t_0), 1.0) / 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * fma(x, 0.16666666666666666, 0.5))
	tmp = 0.0
	if (Float64(Float64(-1.0 + exp(x)) / x) <= 2.0)
		tmp = Float64(1.0 / fma(x, -0.5, 1.0));
	else
		tmp = Float64(fma(Float64(0.16666666666666666 * Float64(x * x)), Float64(t_0 * t_0), 1.0) / 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(-1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 2.0], N[(1.0 / N[(x * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] / 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 30.6%

      \[\frac{e^{x} - 1}{x} \]
   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 73.9

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

   5. Applied rewrites 73.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}}{x} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 73.9

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

   7. Applied rewrites 73.9%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 77.9

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

   10. Applied rewrites 77.9%

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

   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. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 37.1

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

   5. Applied rewrites 37.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 9.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right), \left(x \cdot \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right)\right), 1\right)}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 79.7%

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

   11. Taylor expanded in x around inf

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot {x}^{2}, \left(x \cdot \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right)\right), 1\right)}{1} \]
   12. Step-by-step derivation

   13. Applied rewrites 79.7%

       \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(x \cdot x\right), \left(x \cdot \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)\right), 1\right)}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-1 + e^{x}}{x} \leq 2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(x \cdot x\right), \left(x \cdot \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)\right), 1\right)}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-1 + e^{x}}{x} \leq 2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.125\right), 0.125\right), 1\right)}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ -1.0 (exp x)) x) 2.0)
   (/ 1.0 (fma x -0.5 1.0))
   (/
    (fma (* x (* x x)) (fma x (fma x 0.041666666666666664 0.125) 0.125) 1.0)
    1.0)))

C

double code(double x) {
	double tmp;
	if (((-1.0 + exp(x)) / x) <= 2.0) {
		tmp = 1.0 / fma(x, -0.5, 1.0);
	} else {
		tmp = fma((x * (x * x)), fma(x, fma(x, 0.041666666666666664, 0.125), 0.125), 1.0) / 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(-1.0 + exp(x)) / x) <= 2.0)
		tmp = Float64(1.0 / fma(x, -0.5, 1.0));
	else
		tmp = Float64(fma(Float64(x * Float64(x * x)), fma(x, fma(x, 0.041666666666666664, 0.125), 0.125), 1.0) / 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(-1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 2.0], N[(1.0 / N[(x * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * 0.041666666666666664 + 0.125), $MachinePrecision] + 0.125), $MachinePrecision] + 1.0), $MachinePrecision] / 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 30.6%

      \[\frac{e^{x} - 1}{x} \]
   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 73.9

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

   5. Applied rewrites 73.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}}{x} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 73.9

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

   7. Applied rewrites 73.9%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 77.9

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

   10. Applied rewrites 77.9%

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

   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. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 37.1

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

   5. Applied rewrites 37.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 9.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right), \left(x \cdot \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right)\right), 1\right)}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 79.7%

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

   11. Taylor expanded in x around 0

       \[\leadsto \frac{1 + {x}^{3} \cdot \left(\frac{1}{8} + x \cdot \left(\frac{1}{8} + \frac{1}{24} \cdot x\right)\right)}{1} \]
   12. Step-by-step derivation

   13. Applied rewrites 74.1%

       \[\leadsto \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.125\right), 0.125\right), 1\right)}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-1 + e^{x}}{x} \leq 2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.125\right), 0.125\right), 1\right)}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-1 + e^{x}}{x} \leq 10^{-5}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ -1.0 (exp x)) x) 1e-5)
   (/ 1.0 (fma x -0.5 1.0))
   (/
    (fma
     x
     (* x (fma x (fma x 0.041666666666666664 0.16666666666666666) 0.5))
     x)
    x)))

C

double code(double x) {
	double tmp;
	if (((-1.0 + exp(x)) / x) <= 1e-5) {
		tmp = 1.0 / fma(x, -0.5, 1.0);
	} else {
		tmp = fma(x, (x * fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5)), x) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(-1.0 + exp(x)) / x) <= 1e-5)
		tmp = Float64(1.0 / fma(x, -0.5, 1.0));
	else
		tmp = Float64(fma(x, Float64(x * fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5)), x) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(-1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 1e-5], N[(1.0 / N[(x * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * N[(x * N[(x * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (exp.f64 x) #s(literal 1 binary64)) x) < 1.00000000000000008e-5

   1. Initial program 28.8%

      \[\frac{e^{x} - 1}{x} \]
   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. +-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 73.8

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

   5. Applied rewrites 73.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}}{x} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 73.8

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

   7. Applied rewrites 73.8%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 78.5

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

   10. Applied rewrites 78.5%

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

   if 1.00000000000000008e-5 < (/.f64 (-.f64 (exp.f64 x) #s(literal 1 binary64)) x)

   1. Initial program 98.7%

      \[\frac{e^{x} - 1}{x} \]
   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 71.4

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

   5. Applied rewrites 71.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-1 + e^{x}}{x} \leq 10^{-5}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-1 + e^{x}}{x} \leq 2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \mathsf{fma}\left(x, 0.125, 0.125\right), 1\right)}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ -1.0 (exp x)) x) 2.0)
   (/ 1.0 (fma x -0.5 1.0))
   (/ (fma (* x (* x x)) (fma x 0.125 0.125) 1.0) 1.0)))

C

double code(double x) {
	double tmp;
	if (((-1.0 + exp(x)) / x) <= 2.0) {
		tmp = 1.0 / fma(x, -0.5, 1.0);
	} else {
		tmp = fma((x * (x * x)), fma(x, 0.125, 0.125), 1.0) / 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(-1.0 + exp(x)) / x) <= 2.0)
		tmp = Float64(1.0 / fma(x, -0.5, 1.0));
	else
		tmp = Float64(fma(Float64(x * Float64(x * x)), fma(x, 0.125, 0.125), 1.0) / 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(-1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 2.0], N[(1.0 / N[(x * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * 0.125 + 0.125), $MachinePrecision] + 1.0), $MachinePrecision] / 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 30.6%

      \[\frac{e^{x} - 1}{x} \]
   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 73.9

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

   5. Applied rewrites 73.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}}{x} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 73.9

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

   7. Applied rewrites 73.9%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 77.9

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

   10. Applied rewrites 77.9%

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

   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. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 37.1

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

   5. Applied rewrites 37.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 9.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right), \left(x \cdot \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right)\right), 1\right)}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 79.7%

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

   11. Taylor expanded in x around 0

       \[\leadsto \frac{1 + {x}^{3} \cdot \left(\frac{1}{8} + \frac{1}{8} \cdot x\right)}{1} \]
   12. Step-by-step derivation

   13. Applied rewrites 71.2%

       \[\leadsto \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \mathsf{fma}\left(x, 0.125, 0.125\right), 1\right)}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-1 + e^{x}}{x} \leq 2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \mathsf{fma}\left(x, 0.125, 0.125\right), 1\right)}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-1 + e^{x}}{x} \leq 2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ -1.0 (exp x)) x) 2.0)
   (/ 1.0 (fma x -0.5 1.0))
   (/ (* 0.041666666666666664 (* x (* x (* x x)))) x)))

C

double code(double x) {
	double tmp;
	if (((-1.0 + exp(x)) / x) <= 2.0) {
		tmp = 1.0 / fma(x, -0.5, 1.0);
	} else {
		tmp = (0.041666666666666664 * (x * (x * (x * x)))) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(-1.0 + exp(x)) / x) <= 2.0)
		tmp = Float64(1.0 / fma(x, -0.5, 1.0));
	else
		tmp = Float64(Float64(0.041666666666666664 * Float64(x * Float64(x * Float64(x * x)))) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(-1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 2.0], N[(1.0 / N[(x * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.041666666666666664 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 30.6%

      \[\frac{e^{x} - 1}{x} \]
   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 73.9

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

   5. Applied rewrites 73.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}}{x} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 73.9

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

   7. Applied rewrites 73.9%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 77.9

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

   10. Applied rewrites 77.9%

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

   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

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}}{x} \]
   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 70.9

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

   5. Applied rewrites 70.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{24} \cdot \color{blue}{{x}^{4}}}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 70.9%

      \[\leadsto \frac{0.041666666666666664 \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-1 + e^{x}}{x} \leq 2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-1 + e^{x}}{x} \leq 10^{-5}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ -1.0 (exp x)) x) 1e-5)
   (/ 1.0 (fma x -0.5 1.0))
   (fma x (fma x (fma x 0.041666666666666664 0.16666666666666666) 0.5) 1.0)))

C

double code(double x) {
	double tmp;
	if (((-1.0 + exp(x)) / x) <= 1e-5) {
		tmp = 1.0 / fma(x, -0.5, 1.0);
	} else {
		tmp = fma(x, fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(-1.0 + exp(x)) / x) <= 1e-5)
		tmp = Float64(1.0 / fma(x, -0.5, 1.0));
	else
		tmp = fma(x, fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5), 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(-1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 1e-5], N[(1.0 / N[(x * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (exp.f64 x) #s(literal 1 binary64)) x) < 1.00000000000000008e-5

   1. Initial program 28.8%

      \[\frac{e^{x} - 1}{x} \]
   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. +-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 73.8

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

   5. Applied rewrites 73.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}}{x} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 73.8

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

   7. Applied rewrites 73.8%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 78.5

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

   10. Applied rewrites 78.5%

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

   if 1.00000000000000008e-5 < (/.f64 (-.f64 (exp.f64 x) #s(literal 1 binary64)) x)

   1. Initial program 98.7%

      \[\frac{e^{x} - 1}{x} \]
   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. +-commutative N/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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 59.2

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

   5. Applied rewrites 59.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-1 + e^{x}}{x} \leq 10^{-5}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-1 + e^{x}}{x} \leq 2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ -1.0 (exp x)) x) 2.0)
   (/ 1.0 (fma x -0.5 1.0))
   (* x (fma x (fma x 0.041666666666666664 0.16666666666666666) 0.5))))

C

double code(double x) {
	double tmp;
	if (((-1.0 + exp(x)) / x) <= 2.0) {
		tmp = 1.0 / fma(x, -0.5, 1.0);
	} else {
		tmp = x * fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(-1.0 + exp(x)) / x) <= 2.0)
		tmp = Float64(1.0 / fma(x, -0.5, 1.0));
	else
		tmp = Float64(x * fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(-1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 2.0], N[(1.0 / N[(x * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 30.6%

      \[\frac{e^{x} - 1}{x} \]
   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 73.9

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

   5. Applied rewrites 73.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}}{x} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 73.9

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

   7. Applied rewrites 73.9%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 77.9

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

   10. Applied rewrites 77.9%

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

   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. +-commutative N/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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 57.6

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto {x}^{3} \cdot \color{blue}{\left(\frac{1}{24} + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{1}{6} \cdot \frac{1}{x}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 57.6%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-1 + e^{x}}{x} \leq 2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-1 + e^{x}}{x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ -1.0 (exp x)) x) 2.0)
   (fma x (fma x 0.16666666666666666 0.5) 1.0)
   (* x (fma x (fma x 0.041666666666666664 0.16666666666666666) 0.5))))

C

double code(double x) {
	double tmp;
	if (((-1.0 + exp(x)) / x) <= 2.0) {
		tmp = fma(x, fma(x, 0.16666666666666666, 0.5), 1.0);
	} else {
		tmp = x * fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(-1.0 + exp(x)) / x) <= 2.0)
		tmp = fma(x, fma(x, 0.16666666666666666, 0.5), 1.0);
	else
		tmp = Float64(x * fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(-1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 2.0], N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * N[(x * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 30.6%

      \[\frac{e^{x} - 1}{x} \]
   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. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 74.2

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

   5. Applied rewrites 74.2%

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

   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. +-commutative N/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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 57.6

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto {x}^{3} \cdot \color{blue}{\left(\frac{1}{24} + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{1}{6} \cdot \frac{1}{x}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 57.6%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-1 + e^{x}}{x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-1 + e^{x}}{x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ -1.0 (exp x)) x) 2.0)
   (fma x (fma x 0.16666666666666666 0.5) 1.0)
   (* x (* x (fma x 0.041666666666666664 0.16666666666666666)))))

C

double code(double x) {
	double tmp;
	if (((-1.0 + exp(x)) / x) <= 2.0) {
		tmp = fma(x, fma(x, 0.16666666666666666, 0.5), 1.0);
	} else {
		tmp = x * (x * fma(x, 0.041666666666666664, 0.16666666666666666));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(-1.0 + exp(x)) / x) <= 2.0)
		tmp = fma(x, fma(x, 0.16666666666666666, 0.5), 1.0);
	else
		tmp = Float64(x * Float64(x * fma(x, 0.041666666666666664, 0.16666666666666666)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(-1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 2.0], N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * N[(x * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 30.6%

      \[\frac{e^{x} - 1}{x} \]
   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. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 74.2

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

   5. Applied rewrites 74.2%

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

   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. +-commutative N/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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 57.6

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

   5. Applied rewrites 57.6%

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

   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

   8. Applied rewrites 57.6%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-1 + e^{x}}{x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-1 + e^{x}}{x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ -1.0 (exp x)) x) 2.0)
   (fma x (fma x 0.16666666666666666 0.5) 1.0)
   (* x (* x (* x 0.041666666666666664)))))

C

double code(double x) {
	double tmp;
	if (((-1.0 + exp(x)) / x) <= 2.0) {
		tmp = fma(x, fma(x, 0.16666666666666666, 0.5), 1.0);
	} else {
		tmp = x * (x * (x * 0.041666666666666664));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(-1.0 + exp(x)) / x) <= 2.0)
		tmp = fma(x, fma(x, 0.16666666666666666, 0.5), 1.0);
	else
		tmp = Float64(x * Float64(x * Float64(x * 0.041666666666666664)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(-1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 2.0], N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 30.6%

      \[\frac{e^{x} - 1}{x} \]
   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. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 74.2

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

   5. Applied rewrites 74.2%

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

   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. +-commutative N/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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 57.6

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{24} \cdot \color{blue}{{x}^{3}} \]
   7. Step-by-step derivation

   8. Applied rewrites 57.6%

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot 0.041666666666666664\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-1 + e^{x}}{x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right)\\ t_1 := \mathsf{fma}\left(t\_0, x \cdot x, -x\right)\\ \mathbf{if}\;x \leq 5 \cdot 10^{-155}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, -0.5, 1\right)}\\ \mathbf{elif}\;x \leq 10^{+77}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_0, x \cdot x, x\right) \cdot t\_1}{t\_1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x (fma x 0.041666666666666664 0.16666666666666666) 0.5))
        (t_1 (fma t_0 (* x x) (- x))))
   (if (<= x 5e-155)
     (/ 1.0 (fma x -0.5 1.0))
     (if (<= x 1e+77)
       (/ (/ (* (fma t_0 (* x x) x) t_1) t_1) x)
       (/ (* 0.041666666666666664 (* x (* x (* x x)))) x)))))

C

double code(double x) {
	double t_0 = fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5);
	double t_1 = fma(t_0, (x * x), -x);
	double tmp;
	if (x <= 5e-155) {
		tmp = 1.0 / fma(x, -0.5, 1.0);
	} else if (x <= 1e+77) {
		tmp = ((fma(t_0, (x * x), x) * t_1) / t_1) / x;
	} else {
		tmp = (0.041666666666666664 * (x * (x * (x * x)))) / x;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5)
	t_1 = fma(t_0, Float64(x * x), Float64(-x))
	tmp = 0.0
	if (x <= 5e-155)
		tmp = Float64(1.0 / fma(x, -0.5, 1.0));
	elseif (x <= 1e+77)
		tmp = Float64(Float64(Float64(fma(t_0, Float64(x * x), x) * t_1) / t_1) / x);
	else
		tmp = Float64(Float64(0.041666666666666664 * Float64(x * Float64(x * Float64(x * x)))) / x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(x * x), $MachinePrecision] + (-x)), $MachinePrecision]}, If[LessEqual[x, 5e-155], N[(1.0 / N[(x * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e+77], N[(N[(N[(N[(t$95$0 * N[(x * x), $MachinePrecision] + x), $MachinePrecision] * t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision] / x), $MachinePrecision], N[(N[(0.041666666666666664 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.9999999999999999e-155

   1. Initial program 33.6%

      \[\frac{e^{x} - 1}{x} \]
   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 69.6

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

   5. Applied rewrites 69.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}}{x} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 69.6

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

   7. Applied rewrites 69.6%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 74.7

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

   10. Applied rewrites 74.7%

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

   if 4.9999999999999999e-155 < x < 9.99999999999999983e76

   1. Initial program 50.1%

      \[\frac{e^{x} - 1}{x} \]
   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 58.2

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

   5. Applied rewrites 58.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}}{x} \]
   6. Step-by-step derivation

   7. Applied rewrites 82.9%

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

   if 9.99999999999999983e76 < 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{24} \cdot \color{blue}{{x}^{4}}}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \frac{0.041666666666666664 \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 75.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)\\ \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, t\_0 \cdot t\_0, 1\right)}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (fma x 0.16666666666666666 0.5))))
   (if (<= x 1.25)
     (/ 1.0 (fma x -0.5 1.0))
     (/ (fma t_0 (* t_0 t_0) 1.0) 1.0))))

C

double code(double x) {
	double t_0 = x * fma(x, 0.16666666666666666, 0.5);
	double tmp;
	if (x <= 1.25) {
		tmp = 1.0 / fma(x, -0.5, 1.0);
	} else {
		tmp = fma(t_0, (t_0 * t_0), 1.0) / 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * fma(x, 0.16666666666666666, 0.5))
	tmp = 0.0
	if (x <= 1.25)
		tmp = Float64(1.0 / fma(x, -0.5, 1.0));
	else
		tmp = Float64(fma(t_0, Float64(t_0 * t_0), 1.0) / 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.25], N[(1.0 / N[(x * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] / 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 30.6%

      \[\frac{e^{x} - 1}{x} \]
   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 73.9

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

   5. Applied rewrites 73.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}}{x} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 73.9

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

   7. Applied rewrites 73.9%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 77.9

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

   10. Applied rewrites 77.9%

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

   if 1.25 < 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. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 37.1

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

   5. Applied rewrites 37.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 9.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right), \left(x \cdot \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right)\right), 1\right)}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 79.7%

       \[\leadsto \frac{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), \left(x \cdot \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)\right), 1\right)}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 63.6% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35) 1.0 (* x (fma x 0.16666666666666666 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35) {
		tmp = 1.0;
	} else {
		tmp = x * fma(x, 0.16666666666666666, 0.5);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35)
		tmp = 1.0;
	else
		tmp = Float64(x * fma(x, 0.16666666666666666, 0.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35], 1.0, N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3500000000000001

   1. Initial program 30.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 73.7%

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

   if 1.3500000000000001 < 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. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 37.1

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

   5. Applied rewrites 37.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 37.1%

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

Alternative 15: 66.9% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma x (fma (* x x) 0.041666666666666664 0.5) 1.0))

C

double code(double x) {
	return fma(x, fma((x * x), 0.041666666666666664, 0.5), 1.0);
}

Julia

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

Wolfram

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

Derivation

1. Initial program 49.3%

   \[\frac{e^{x} - 1}{x} \]
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. +-commutative N/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. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 69.5

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

5. Applied rewrites 69.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 69.5%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 69.1%

    \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right) \]
11. Add Preprocessing

Alternative 16: 63.6% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.4) 1.0 (* 0.16666666666666666 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 2.4) {
		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 <= 2.4d0) 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 <= 2.4) {
		tmp = 1.0;
	} else {
		tmp = 0.16666666666666666 * (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.4:
		tmp = 1.0
	else:
		tmp = 0.16666666666666666 * (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.4)
		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 <= 2.4)
		tmp = 1.0;
	else
		tmp = 0.16666666666666666 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.4], 1.0, N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 30.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 73.7%

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

   if 2.39999999999999991 < 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. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 37.1

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

   5. Applied rewrites 37.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 37.1%

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

Alternative 17: 63.7% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x (fma x 0.16666666666666666 0.5) 1.0))

C

double code(double x) {
	return fma(x, fma(x, 0.16666666666666666, 0.5), 1.0);
}

Julia

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

Wolfram

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

Derivation

1. Initial program 49.3%

   \[\frac{e^{x} - 1}{x} \]
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. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 64.2

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

5. Applied rewrites 64.2%

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

Alternative 18: 50.9% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 0.5, 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x 0.5 1.0))

C

double code(double x) {
	return fma(x, 0.5, 1.0);
}

Julia

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

Wolfram

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

Derivation

1. Initial program 49.3%

   \[\frac{e^{x} - 1}{x} \]
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 55.0

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

5. Applied rewrites 55.0%

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

Alternative 19: 50.7% accurate, 115.0× 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 49.3%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 54.7%

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

Developer Target 1: 53.1% 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 2024223 
(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))