Kahan's exp quotient

Percentage Accurate: 53.8% → 100.0%

Time: 11.2s

Alternatives: 16

Speedup: 8.8×

• 24.7% 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: 53.8% 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 56.8%

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

   1. 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: 69.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-1 + e^{x}}{x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.05555555555555555, -0.6666666666666666\right), 2\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ -1.0 (exp x)) x) 2.0)
   (fma
    x
    (/ 1.0 (fma x (fma x 0.05555555555555555 -0.6666666666666666) 2.0))
    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 = fma(x, (1.0 / fma(x, fma(x, 0.05555555555555555, -0.6666666666666666), 2.0)), 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 = fma(x, Float64(1.0 / fma(x, fma(x, 0.05555555555555555, -0.6666666666666666), 2.0)), 1.0);
	else
		tmp = Float64(Float64(0.041666666666666664 * Float64(Float64(x * 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[(x * N[(1.0 / N[(x * N[(x * 0.05555555555555555 + -0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(0.041666666666666664 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $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 40.9%

      \[\frac{e^{x} - 1}{x} \]
   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 66.6

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

   5. Applied rewrites 66.6%

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

      1. lift-fma.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-fma.f64 N/A

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

      8. lower-/.f64 66.6

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

   7. Applied rewrites 66.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x, \frac{1}{\color{blue}{2 + x \cdot \left(\frac{1}{18} \cdot x - \frac{2}{3}\right)}}, 1\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{1}{\color{blue}{x \cdot \left(\frac{1}{18} \cdot x - \frac{2}{3}\right) + 2}}, 1\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{1}{\color{blue}{\mathsf{fma}\left(x, \frac{1}{18} \cdot x - \frac{2}{3}, 2\right)}}, 1\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{1}{\mathsf{fma}\left(x, \color{blue}{\frac{1}{18} \cdot x + \left(\mathsf{neg}\left(\frac{2}{3}\right)\right)}, 2\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{1}{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{18}} + \left(\mathsf{neg}\left(\frac{2}{3}\right)\right), 2\right)}, 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{1}{\mathsf{fma}\left(x, x \cdot \frac{1}{18} + \color{blue}{\frac{-2}{3}}, 2\right)}, 1\right) \]

      6. lower-fma.f64 68.2

         \[\leadsto \mathsf{fma}\left(x, \frac{1}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.05555555555555555, -0.6666666666666666\right)}, 2\right)}, 1\right) \]

   10. Applied rewrites 68.2%

       \[\leadsto \mathsf{fma}\left(x, \frac{1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.05555555555555555, -0.6666666666666666\right), 2\right)}}, 1\right) \]

   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

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}}{x} \]
   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 72.4

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot {x}^{4}}}{x} \]

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 72.4

         \[\leadsto \frac{0.041666666666666664 \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)}{x} \]

   8. Applied rewrites 72.4%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-1 + e^{x}}{x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.05555555555555555, -0.6666666666666666\right), 2\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.2% 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 40.9%

      \[\frac{e^{x} - 1}{x} \]
   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 66.8

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

   5. Applied rewrites 66.8%

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

   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 69.8

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in x around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. distribute-lft-in N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

   8. Applied rewrites 69.8%

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

4. Final simplification 67.6%

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

Alternative 4: 67.2% accurate, 0.8× speedup

Math

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

FPCore

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

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 = fma(x, 0.041666666666666664, 0.16666666666666666) * (x * x);
	}
	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(fma(x, 0.041666666666666664, 0.16666666666666666) * Float64(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[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] * N[(x * x), $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 40.9%

      \[\frac{e^{x} - 1}{x} \]
   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 66.8

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

   5. Applied rewrites 66.8%

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

   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 69.8

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lft-mult-inverse N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 69.8

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

   8. Applied rewrites 69.8%

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

4. Final simplification 67.6%

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

Alternative 5: 67.2% accurate, 0.8× speedup

Math

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

FPCore

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

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 * (0.041666666666666664 * (x * x));
	}
	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(0.041666666666666664 * Float64(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[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(0.041666666666666664 * N[(x * x), $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 40.9%

      \[\frac{e^{x} - 1}{x} \]
   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 66.8

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

   5. Applied rewrites 66.8%

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

   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 69.8

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 69.8

         \[\leadsto x \cdot \left(0.041666666666666664 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   8. Applied rewrites 69.8%

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

4. Final simplification 67.6%

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

Alternative 6: 62.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(-1.0 + exp(x)) / x) <= 2.0)
		tmp = fma(x, 0.5, 1.0);
	else
		tmp = Float64(0.16666666666666666 * Float64(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[(x * 0.5 + 1.0), $MachinePrecision], N[(0.16666666666666666 * N[(x * x), $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 40.9%

      \[\frac{e^{x} - 1}{x} \]
   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 65.4

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

   5. Applied rewrites 65.4%

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

   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 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2}} \]

      2. unpow2 N/A

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

      3. lower-*.f64 52.4

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

   8. Applied rewrites 52.4%

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

4. Final simplification 61.9%

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

Alternative 7: 71.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right)\\ t_1 := \mathsf{fma}\left(t\_0, x \cdot x, -x\right)\\ \mathbf{if}\;x \leq 4.5:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.05555555555555555, -0.6666666666666666\right), 2\right)}, 1\right)\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, x \cdot x, x\right) \cdot t\_1}{x \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{x}\\ \end{array} \end{array} \]

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 4.5)
     (fma
      x
      (/ 1.0 (fma x (fma x 0.05555555555555555 -0.6666666666666666) 2.0))
      1.0)
     (if (<= x 8.5e+61)
       (/ (* (fma t_0 (* x x) x) t_1) (* x t_1))
       (/ (* 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 <= 4.5) {
		tmp = fma(x, (1.0 / fma(x, fma(x, 0.05555555555555555, -0.6666666666666666), 2.0)), 1.0);
	} else if (x <= 8.5e+61) {
		tmp = (fma(t_0, (x * x), x) * t_1) / (x * t_1);
	} 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 <= 4.5)
		tmp = fma(x, Float64(1.0 / fma(x, fma(x, 0.05555555555555555, -0.6666666666666666), 2.0)), 1.0);
	elseif (x <= 8.5e+61)
		tmp = Float64(Float64(fma(t_0, Float64(x * x), x) * t_1) / Float64(x * t_1));
	else
		tmp = Float64(Float64(0.041666666666666664 * Float64(Float64(x * 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, 4.5], N[(x * N[(1.0 / N[(x * N[(x * 0.05555555555555555 + -0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 8.5e+61], N[(N[(N[(t$95$0 * N[(x * x), $MachinePrecision] + x), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(x * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(0.041666666666666664 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.5

   1. Initial program 40.9%

      \[\frac{e^{x} - 1}{x} \]
   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 66.6

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

   5. Applied rewrites 66.6%

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

      1. lift-fma.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-fma.f64 N/A

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

      8. lower-/.f64 66.6

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

   7. Applied rewrites 66.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x, \frac{1}{\color{blue}{2 + x \cdot \left(\frac{1}{18} \cdot x - \frac{2}{3}\right)}}, 1\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{1}{\color{blue}{x \cdot \left(\frac{1}{18} \cdot x - \frac{2}{3}\right) + 2}}, 1\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{1}{\color{blue}{\mathsf{fma}\left(x, \frac{1}{18} \cdot x - \frac{2}{3}, 2\right)}}, 1\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{1}{\mathsf{fma}\left(x, \color{blue}{\frac{1}{18} \cdot x + \left(\mathsf{neg}\left(\frac{2}{3}\right)\right)}, 2\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{1}{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{18}} + \left(\mathsf{neg}\left(\frac{2}{3}\right)\right), 2\right)}, 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{1}{\mathsf{fma}\left(x, x \cdot \frac{1}{18} + \color{blue}{\frac{-2}{3}}, 2\right)}, 1\right) \]

      6. lower-fma.f64 68.2

         \[\leadsto \mathsf{fma}\left(x, \frac{1}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.05555555555555555, -0.6666666666666666\right)}, 2\right)}, 1\right) \]

   10. Applied rewrites 68.2%

       \[\leadsto \mathsf{fma}\left(x, \frac{1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.05555555555555555, -0.6666666666666666\right), 2\right)}}, 1\right) \]

   if 4.5 < x < 8.50000000000000035e61

   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 4.8

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

   5. Applied rewrites 4.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   7. Applied rewrites 60.8%

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

   if 8.50000000000000035e61 < 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 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot {x}^{4}}}{x} \]

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 94.6

         \[\leadsto \frac{0.041666666666666664 \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)}{x} \]

   8. Applied rewrites 94.6%

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

Alternative 8: 70.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right)\\ \mathbf{if}\;x \leq 1.65 \cdot 10^{+103}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0 \cdot \left(x \cdot x\right), -1\right) \cdot \frac{1}{\mathsf{fma}\left(x, t\_0, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.041666666666666664 \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x (fma x 0.041666666666666664 0.16666666666666666) 0.5)))
   (if (<= x 1.65e+103)
     (* (fma t_0 (* t_0 (* x x)) -1.0) (/ 1.0 (fma x t_0 -1.0)))
     (* x (* 0.041666666666666664 (* x x))))))

C

double code(double x) {
	double t_0 = fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5);
	double tmp;
	if (x <= 1.65e+103) {
		tmp = fma(t_0, (t_0 * (x * x)), -1.0) * (1.0 / fma(x, t_0, -1.0));
	} else {
		tmp = x * (0.041666666666666664 * (x * x));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5)
	tmp = 0.0
	if (x <= 1.65e+103)
		tmp = Float64(fma(t_0, Float64(t_0 * Float64(x * x)), -1.0) * Float64(1.0 / fma(x, t_0, -1.0)));
	else
		tmp = Float64(x * Float64(0.041666666666666664 * Float64(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]}, If[LessEqual[x, 1.65e+103], N[(N[(t$95$0 * N[(t$95$0 * N[(x * x), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * N[(1.0 / N[(x * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.65000000000000004e103

   1. Initial program 47.1%

      \[\frac{e^{x} - 1}{x} \]
   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 60.2

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

   5. Applied rewrites 60.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 64.9%

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

   if 1.65000000000000004e103 < 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 100.0

         \[\leadsto x \cdot \left(0.041666666666666664 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   8. Applied rewrites 100.0%

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

Alternative 9: 70.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right)\\ \mathbf{if}\;x \leq 1.65 \cdot 10^{+103}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, t\_0 \cdot \left(x \cdot x\right), -1\right)}{\mathsf{fma}\left(x, t\_0, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.041666666666666664 \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x (fma x 0.041666666666666664 0.16666666666666666) 0.5)))
   (if (<= x 1.65e+103)
     (/ (fma t_0 (* t_0 (* x x)) -1.0) (fma x t_0 -1.0))
     (* x (* 0.041666666666666664 (* x x))))))

C

double code(double x) {
	double t_0 = fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5);
	double tmp;
	if (x <= 1.65e+103) {
		tmp = fma(t_0, (t_0 * (x * x)), -1.0) / fma(x, t_0, -1.0);
	} else {
		tmp = x * (0.041666666666666664 * (x * x));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5)
	tmp = 0.0
	if (x <= 1.65e+103)
		tmp = Float64(fma(t_0, Float64(t_0 * Float64(x * x)), -1.0) / fma(x, t_0, -1.0));
	else
		tmp = Float64(x * Float64(0.041666666666666664 * Float64(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]}, If[LessEqual[x, 1.65e+103], N[(N[(t$95$0 * N[(t$95$0 * N[(x * x), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(x * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.65000000000000004e103

   1. Initial program 47.1%

      \[\frac{e^{x} - 1}{x} \]
   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 60.2

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

   5. Applied rewrites 60.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 64.9%

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

   if 1.65000000000000004e103 < 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 100.0

         \[\leadsto x \cdot \left(0.041666666666666664 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   8. Applied rewrites 100.0%

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

Alternative 10: 68.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.001736111111111111, -0.027777777777777776\right), \frac{-1}{\mathsf{fma}\left(x, -0.041666666666666664, 0.16666666666666666\right)}, \mathsf{fma}\left(x, 0.5, 1\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (* (* x x) (fma x (* x 0.001736111111111111) -0.027777777777777776))
  (/ -1.0 (fma x -0.041666666666666664 0.16666666666666666))
  (fma x 0.5 1.0)))

C

double code(double x) {
	return fma(((x * x) * fma(x, (x * 0.001736111111111111), -0.027777777777777776)), (-1.0 / fma(x, -0.041666666666666664, 0.16666666666666666)), fma(x, 0.5, 1.0));
}

Julia

function code(x)
	return fma(Float64(Float64(x * x) * fma(x, Float64(x * 0.001736111111111111), -0.027777777777777776)), Float64(-1.0 / fma(x, -0.041666666666666664, 0.16666666666666666)), fma(x, 0.5, 1.0))
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.001736111111111111), $MachinePrecision] + -0.027777777777777776), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(x * -0.041666666666666664 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(x * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.8%

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

3. Taylor expanded in x around 0

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

   1. +-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 67.5

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

5. Applied rewrites 67.5%

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

   1. lift-fma.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. distribute-lft-in N/A

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

   5. associate-+l+ N/A

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

   6. associate-*r* N/A

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

   7. lift-fma.f64 N/A

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

   8. flip-+ N/A

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

   9. div-inv N/A

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

   10. associate-*r* N/A

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

   11. lower-fma.f64 N/A

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

7. Applied rewrites 68.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.001736111111111111, -0.027777777777777776\right), \frac{-1}{\mathsf{fma}\left(x, -0.041666666666666664, 0.16666666666666666\right)}, \mathsf{fma}\left(x, 0.5, 1\right)\right)} \]
8. Add Preprocessing

Alternative 11: 68.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fma x (* x (fma x (fma x 0.041666666666666664 0.16666666666666666) 0.5)) x)
  x))

C

double code(double x) {
	return fma(x, (x * fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5)), x) / x;
}

Julia

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

Wolfram

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

Derivation

1. Initial program 56.8%

   \[\frac{e^{x} - 1}{x} \]
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 68.2

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

5. Applied rewrites 68.2%

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

Alternative 12: 66.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma x (fma x (fma x 0.041666666666666664 0.16666666666666666) 0.5) 1.0))

C

double code(double x) {
	return fma(x, fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5), 1.0);
}

Julia

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

Wolfram

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

Derivation

1. Initial program 56.8%

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

3. Taylor expanded in x around 0

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

   1. +-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 67.5

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

5. Applied rewrites 67.5%

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

Alternative 13: 62.7% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.5) (fma x 0.5 1.0) (* x (fma x 0.16666666666666666 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 2.5) {
		tmp = fma(x, 0.5, 1.0);
	} else {
		tmp = x * fma(x, 0.16666666666666666, 0.5);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.5)
		tmp = fma(x, 0.5, 1.0);
	else
		tmp = Float64(x * fma(x, 0.16666666666666666, 0.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.5], N[(x * 0.5 + 1.0), $MachinePrecision], N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5

   1. Initial program 40.9%

      \[\frac{e^{x} - 1}{x} \]
   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 65.4

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

   5. Applied rewrites 65.4%

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

   if 2.5 < 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 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lft-mult-inverse N/A

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

      6. associate-*l* N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-/r* N/A

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

      11. unpow2 N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. rgt-mult-inverse N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-in N/A

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

      18. *-commutative N/A

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

      19. associate-*l* N/A

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

   8. Applied rewrites 52.4%

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

Alternative 14: 63.3% 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 56.8%

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

3. Taylor expanded in x around 0

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

   1. +-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 62.9

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

5. Applied rewrites 62.9%

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

Alternative 15: 50.6% 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 56.8%

   \[\frac{e^{x} - 1}{x} \]
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 49.4

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

5. Applied rewrites 49.4%

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

Alternative 16: 50.4% 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 56.8%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 48.3%

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

Developer Target 1: 53.3% 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 2024214 
(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))