Kahan's exp quotient

Percentage Accurate: 53.6% → 100.0%

Time: 9.9s

Alternatives: 16

Speedup: 8.8×

• 24.4% 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: 53.6% 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 59.0%

   \[\frac{e^{x} - 1}{x} \]
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 egg-rr 100.0%

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

Alternative 2: 72.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{x} + -1}{x} \leq 0.005:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x \cdot x, x\right) \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ (exp x) -1.0) x) 0.005)
   (/ 1.0 (fma x -0.5 1.0))
   (*
    (fma
     (fma x (fma x 0.041666666666666664 0.16666666666666666) 0.5)
     (* x x)
     x)
    (/ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (((exp(x) + -1.0) / x) <= 0.005) {
		tmp = 1.0 / fma(x, -0.5, 1.0);
	} else {
		tmp = fma(fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5), (x * x), x) * (1.0 / x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) + -1.0) / x) <= 0.005)
		tmp = Float64(1.0 / fma(x, -0.5, 1.0));
	else
		tmp = Float64(fma(fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5), Float64(x * x), x) * Float64(1.0 / x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision], 0.005], N[(1.0 / N[(x * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(x * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (exp.f64 x) #s(literal 1 binary64)) x) < 0.0050000000000000001

   1. Initial program 40.9%

      \[\frac{e^{x} - 1}{x} \]
   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 61.5

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

   5. Simplified 61.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}}{x} \]
   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. lift-fma.f64 N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 61.5

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 61.5

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

   7. Applied egg-rr 61.5%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 68.4

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

   10. Simplified 68.4%

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

   if 0.0050000000000000001 < (/.f64 (-.f64 (exp.f64 x) #s(literal 1 binary64)) x)

   1. Initial program 96.3%

      \[\frac{e^{x} - 1}{x} \]
   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 82.8

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

   5. Simplified 82.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}}{x} \]
   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. lift-fma.f64 N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 82.9

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

   7. Applied egg-rr 82.9%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x} + -1}{x} \leq 0.005:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x \cdot x, x\right) \cdot \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{x} + -1}{x} \leq 0.999:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ (exp x) -1.0) x) 0.999)
   (/ 1.0 (fma x -0.5 1.0))
   (/
    (fma
     x
     (* x (fma x (fma x 0.041666666666666664 0.16666666666666666) 0.5))
     x)
    x)))

C

double code(double x) {
	double tmp;
	if (((exp(x) + -1.0) / x) <= 0.999) {
		tmp = 1.0 / fma(x, -0.5, 1.0);
	} else {
		tmp = fma(x, (x * fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5)), x) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) + -1.0) / x) <= 0.999)
		tmp = Float64(1.0 / fma(x, -0.5, 1.0));
	else
		tmp = Float64(fma(x, Float64(x * fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5)), x) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision], 0.999], N[(1.0 / N[(x * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * N[(x * N[(x * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (exp.f64 x) #s(literal 1 binary64)) x) < 0.998999999999999999

   1. Initial program 40.8%

      \[\frac{e^{x} - 1}{x} \]
   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 61.7

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

   5. Simplified 61.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}}{x} \]
   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. lift-fma.f64 N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 61.7

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 61.7

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

   7. Applied egg-rr 61.7%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 68.5

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

   10. Simplified 68.5%

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

   if 0.998999999999999999 < (/.f64 (-.f64 (exp.f64 x) #s(literal 1 binary64)) x)

   1. Initial program 97.0%

      \[\frac{e^{x} - 1}{x} \]
   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 82.6

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

   5. Simplified 82.6%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x} + -1}{x} \leq 0.999:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 41.4%

      \[\frac{e^{x} - 1}{x} \]
   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 63.0

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

   5. Simplified 63.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}}{x} \]
   6. 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. lift-fma.f64 N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 63.0

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 63.0

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

   7. Applied egg-rr 63.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 69.3

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

   10. Simplified 69.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 81.3

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

   5. Simplified 81.3%

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

   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

         \[\leadsto \frac{\frac{1}{24} \cdot {x}^{\color{blue}{\left(3 + 1\right)}}}{x} \]

      3. pow-plus N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \color{blue}{\left({x}^{3} \cdot x\right)}}{x} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \color{blue}{\left(x \cdot {x}^{3}\right)}}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \color{blue}{\left(x \cdot {x}^{3}\right)}}{x} \]

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 81.3

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

   8. Simplified 81.3%

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

4. Final simplification 72.9%

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

Alternative 5: 71.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{x} + -1}{x} \leq 0.005:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ (exp x) -1.0) x) 0.005)
   (/ 1.0 (fma x -0.5 1.0))
   (fma x (fma x (fma x 0.041666666666666664 0.16666666666666666) 0.5) 1.0)))

C

double code(double x) {
	double tmp;
	if (((exp(x) + -1.0) / x) <= 0.005) {
		tmp = 1.0 / fma(x, -0.5, 1.0);
	} else {
		tmp = fma(x, fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) + -1.0) / x) <= 0.005)
		tmp = Float64(1.0 / fma(x, -0.5, 1.0));
	else
		tmp = fma(x, fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5), 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision], 0.005], N[(1.0 / N[(x * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (exp.f64 x) #s(literal 1 binary64)) x) < 0.0050000000000000001

   1. Initial program 40.9%

      \[\frac{e^{x} - 1}{x} \]
   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 61.5

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

   5. Simplified 61.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}}{x} \]
   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. lift-fma.f64 N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 61.5

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 61.5

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

   7. Applied egg-rr 61.5%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 68.4

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

   10. Simplified 68.4%

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

   if 0.0050000000000000001 < (/.f64 (-.f64 (exp.f64 x) #s(literal 1 binary64)) x)

   1. Initial program 96.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 75.2

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

   5. Simplified 75.2%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x} + -1}{x} \leq 0.005:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{x} + -1}{x} \leq 2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(x, 0.125, 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ (exp x) -1.0) x) 2.0)
   (/ 1.0 (fma x -0.5 1.0))
   (* x (* x (fma x 0.125 0.25)))))

C

double code(double x) {
	double tmp;
	if (((exp(x) + -1.0) / x) <= 2.0) {
		tmp = 1.0 / fma(x, -0.5, 1.0);
	} else {
		tmp = x * (x * fma(x, 0.125, 0.25));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) + -1.0) / x) <= 2.0)
		tmp = Float64(1.0 / fma(x, -0.5, 1.0));
	else
		tmp = Float64(x * Float64(x * fma(x, 0.125, 0.25)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision], 2.0], N[(1.0 / N[(x * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * 0.125 + 0.25), $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 41.4%

      \[\frac{e^{x} - 1}{x} \]
   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 63.0

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

   5. Simplified 63.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}}{x} \]
   6. 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. lift-fma.f64 N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 63.0

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 63.0

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

   7. Applied egg-rr 63.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 69.3

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

   10. Simplified 69.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot x} \]
   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 5.8

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

   5. Simplified 5.8%

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

      1. flip-+ N/A

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

      2. metadata-eval N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. flip3-- N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied egg-rr 1.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \color{blue}{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot x\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot x\right)\right) + 1}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot x\right), 1\right)}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot x\right) + \frac{1}{2}}, 1\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{4} + \frac{1}{8} \cdot x, \frac{1}{2}\right)}, 1\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{8} \cdot x + \frac{1}{4}}, \frac{1}{2}\right), 1\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{8}} + \frac{1}{4}, \frac{1}{2}\right), 1\right)\right) \]

      7. lower-fma.f64 15.8

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

   10. Simplified 15.8%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right)} \]
   12. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right) \]

       2. unpow2 N/A

          \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right)\right)\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right)\right)\right)} \]

       8. distribute-lft-in N/A

          \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{8} + x \cdot \left(\frac{1}{4} \cdot \frac{1}{x}\right)\right)}\right) \]

       9. *-commutative N/A

          \[\leadsto x \cdot \left(x \cdot \left(x \cdot \frac{1}{8} + x \cdot \color{blue}{\left(\frac{1}{x} \cdot \frac{1}{4}\right)}\right)\right) \]

       10. associate-*r* N/A

           \[\leadsto x \cdot \left(x \cdot \left(x \cdot \frac{1}{8} + \color{blue}{\left(x \cdot \frac{1}{x}\right) \cdot \frac{1}{4}}\right)\right) \]

       11. rgt-mult-inverse N/A

           \[\leadsto x \cdot \left(x \cdot \left(x \cdot \frac{1}{8} + \color{blue}{1} \cdot \frac{1}{4}\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto x \cdot \left(x \cdot \left(x \cdot \frac{1}{8} + \color{blue}{\frac{1}{4}}\right)\right) \]

       13. lower-fma.f64 72.9

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

   13. Simplified 72.9%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x, 0.125, 0.25\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x} + -1}{x} \leq 2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(x, 0.125, 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{x} + -1}{x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(x, 0.125, 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ (exp x) -1.0) x) 2.0)
   (fma x (fma x 0.16666666666666666 0.5) 1.0)
   (* x (* x (fma x 0.125 0.25)))))

C

double code(double x) {
	double tmp;
	if (((exp(x) + -1.0) / x) <= 2.0) {
		tmp = fma(x, fma(x, 0.16666666666666666, 0.5), 1.0);
	} else {
		tmp = x * (x * fma(x, 0.125, 0.25));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) + -1.0) / x) <= 2.0)
		tmp = fma(x, fma(x, 0.16666666666666666, 0.5), 1.0);
	else
		tmp = Float64(x * Float64(x * fma(x, 0.125, 0.25)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision], 2.0], N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * N[(x * 0.125 + 0.25), $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 41.4%

      \[\frac{e^{x} - 1}{x} \]
   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 63.6

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

   5. Simplified 63.6%

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

   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

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot x} \]
   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 5.8

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

   5. Simplified 5.8%

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

      1. flip-+ N/A

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

      2. metadata-eval N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. flip3-- N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied egg-rr 1.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \color{blue}{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot x\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot x\right)\right) + 1}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot x\right), 1\right)}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot x\right) + \frac{1}{2}}, 1\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{4} + \frac{1}{8} \cdot x, \frac{1}{2}\right)}, 1\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{8} \cdot x + \frac{1}{4}}, \frac{1}{2}\right), 1\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{8}} + \frac{1}{4}, \frac{1}{2}\right), 1\right)\right) \]

      7. lower-fma.f64 15.8

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

   10. Simplified 15.8%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right)} \]
   12. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right) \]

       2. unpow2 N/A

          \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right)\right)\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right)\right)\right)} \]

       8. distribute-lft-in N/A

          \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{8} + x \cdot \left(\frac{1}{4} \cdot \frac{1}{x}\right)\right)}\right) \]

       9. *-commutative N/A

          \[\leadsto x \cdot \left(x \cdot \left(x \cdot \frac{1}{8} + x \cdot \color{blue}{\left(\frac{1}{x} \cdot \frac{1}{4}\right)}\right)\right) \]

       10. associate-*r* N/A

           \[\leadsto x \cdot \left(x \cdot \left(x \cdot \frac{1}{8} + \color{blue}{\left(x \cdot \frac{1}{x}\right) \cdot \frac{1}{4}}\right)\right) \]

       11. rgt-mult-inverse N/A

           \[\leadsto x \cdot \left(x \cdot \left(x \cdot \frac{1}{8} + \color{blue}{1} \cdot \frac{1}{4}\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto x \cdot \left(x \cdot \left(x \cdot \frac{1}{8} + \color{blue}{\frac{1}{4}}\right)\right) \]

       13. lower-fma.f64 72.9

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

   13. Simplified 72.9%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x, 0.125, 0.25\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x} + -1}{x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(x, 0.125, 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{x} + -1}{x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot 0.125\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ (exp x) -1.0) x) 2.0)
   (fma x (fma x 0.16666666666666666 0.5) 1.0)
   (* x (* (* x x) 0.125))))

C

double code(double x) {
	double tmp;
	if (((exp(x) + -1.0) / x) <= 2.0) {
		tmp = fma(x, fma(x, 0.16666666666666666, 0.5), 1.0);
	} else {
		tmp = x * ((x * x) * 0.125);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) + -1.0) / x) <= 2.0)
		tmp = fma(x, fma(x, 0.16666666666666666, 0.5), 1.0);
	else
		tmp = Float64(x * Float64(Float64(x * x) * 0.125));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision], 2.0], N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(N[(x * x), $MachinePrecision] * 0.125), $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 41.4%

      \[\frac{e^{x} - 1}{x} \]
   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 63.6

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

   5. Simplified 63.6%

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

   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

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot x} \]
   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 5.8

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

   5. Simplified 5.8%

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

      1. flip-+ N/A

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

      2. metadata-eval N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. flip3-- N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied egg-rr 1.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \color{blue}{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot x\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot x\right)\right) + 1}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot x\right), 1\right)}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot x\right) + \frac{1}{2}}, 1\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{4} + \frac{1}{8} \cdot x, \frac{1}{2}\right)}, 1\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{8} \cdot x + \frac{1}{4}}, \frac{1}{2}\right), 1\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{8}} + \frac{1}{4}, \frac{1}{2}\right), 1\right)\right) \]

      7. lower-fma.f64 15.8

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

   10. Simplified 15.8%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{8} \cdot {x}^{3}} \]
   12. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \frac{1}{8} \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \]

       2. unpow2 N/A

          \[\leadsto \frac{1}{8} \cdot \left(x \cdot \color{blue}{{x}^{2}}\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot x\right) \cdot {x}^{2}} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} \cdot x\right)} \]

       5. unpow2 N/A

          \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{8} \cdot x\right) \]

       6. associate-*r* N/A

          \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{8} \cdot x\right)\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{8} \cdot x\right)\right)} \]

       8. *-commutative N/A

          \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{8}\right)}\right) \]

       9. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \frac{1}{8}\right)} \]

       10. unpow2 N/A

           \[\leadsto x \cdot \left(\color{blue}{{x}^{2}} \cdot \frac{1}{8}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{8}\right)} \]

       12. unpow2 N/A

           \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{8}\right) \]

       13. lower-*.f64 72.9

           \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot 0.125\right) \]

   13. Simplified 72.9%

       \[\leadsto \color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot 0.125\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x} + -1}{x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot 0.125\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 41.4%

      \[\frac{e^{x} - 1}{x} \]
   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 63.6

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

   5. Simplified 63.6%

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

   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

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}}{x} \]
   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 81.3

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

   5. Simplified 81.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{3}} \]
   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. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 72.9

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

   8. Simplified 72.9%

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

4. Final simplification 66.4%

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

Alternative 10: 62.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision], 2.0], 1.0, N[(x * N[(x * 0.16666666666666666 + 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 41.4%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 63.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-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 59.3

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

   5. Simplified 59.3%

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

   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. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*l* N/A

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

      6. lft-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 59.3

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

   8. Simplified 59.3%

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

4. Final simplification 62.0%

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

Alternative 11: 62.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision], 2.0], 1.0, N[(x * N[(x * 0.16666666666666666), $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 41.4%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 63.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-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 59.3

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

   5. Simplified 59.3%

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

   6. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 59.3

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

   8. Simplified 59.3%

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

4. Final simplification 62.0%

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

Alternative 12: 74.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right)\\ t_1 := \mathsf{fma}\left(t\_0, x \cdot x, -x\right)\\ \mathbf{if}\;x \leq 10^{-164}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, -0.5, 1\right)}\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{+61}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, x \cdot x, x\right) \cdot t\_1}{x \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x (fma x 0.041666666666666664 0.16666666666666666) 0.5))
        (t_1 (fma t_0 (* x x) (- x))))
   (if (<= x 1e-164)
     (/ 1.0 (fma x -0.5 1.0))
     (if (<= x 8.4e+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 <= 1e-164) {
		tmp = 1.0 / fma(x, -0.5, 1.0);
	} else if (x <= 8.4e+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 <= 1e-164)
		tmp = Float64(1.0 / fma(x, -0.5, 1.0));
	elseif (x <= 8.4e+61)
		tmp = Float64(Float64(fma(t_0, Float64(x * x), x) * t_1) / Float64(x * t_1));
	else
		tmp = Float64(Float64(0.041666666666666664 * Float64(x * Float64(x * Float64(x * x)))) / x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(x * x), $MachinePrecision] + (-x)), $MachinePrecision]}, If[LessEqual[x, 1e-164], N[(1.0 / N[(x * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.4e+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[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 9.99999999999999962e-165

   1. Initial program 46.8%

      \[\frac{e^{x} - 1}{x} \]
   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 56.4

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

   5. Simplified 56.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x\right)}}{x} \]
   6. 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. lift-fma.f64 N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 56.4

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 56.4

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

   7. Applied egg-rr 56.4%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 64.1

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

   10. Simplified 64.1%

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

   if 9.99999999999999962e-165 < x < 8.4000000000000004e61

   1. Initial program 41.4%

      \[\frac{e^{x} - 1}{x} \]
   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 67.2

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

   5. Simplified 67.2%

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

      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 egg-rr 81.2%

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

   if 8.4000000000000004e61 < 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 98.5

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

   5. Simplified 98.5%

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

   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

         \[\leadsto \frac{\frac{1}{24} \cdot {x}^{\color{blue}{\left(3 + 1\right)}}}{x} \]

      3. pow-plus N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \color{blue}{\left({x}^{3} \cdot x\right)}}{x} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \color{blue}{\left(x \cdot {x}^{3}\right)}}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \color{blue}{\left(x \cdot {x}^{3}\right)}}{x} \]

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 98.5

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

   8. Simplified 98.5%

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

Alternative 13: 63.1% 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 59.0%

   \[\frac{e^{x} - 1}{x} \]
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 62.3

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

5. Simplified 62.3%

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

Alternative 14: 50.7% 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 59.0%

   \[\frac{e^{x} - 1}{x} \]
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 45.7

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

5. Simplified 45.7%

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

Alternative 15: 50.7% accurate, 115.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 59.0%

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

3. Taylor expanded in x around 0

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

5. Simplified 45.1%

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

Alternative 16: 3.3% accurate, 115.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 59.0%

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

3. Taylor expanded in x around 0

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

5. Simplified 3.4%

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

   1. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0}}{x} \]

   2. div0 3.4

      \[\leadsto \color{blue}{0} \]

7. Applied egg-rr 3.4%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 53.0% 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 2024207 
(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))