Kahan's exp quotient

Percentage Accurate: 53.1% → 100.0%

Time: 10.3s

Alternatives: 15

Speedup: 8.8×

• 24.9% 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: 53.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x} - 1}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (exp x) 1.0) x))

C

double code(double x) {
	return (exp(x) - 1.0) / x;
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{expm1}\left(x\right)}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (expm1 x) x))

C

double code(double x) {
	return expm1(x) / x;
}

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.4%

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

   1. accelerator-lowering-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: 71.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{x} + -1}{x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.0026041666666666665, 0.16666666666666666\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ (exp x) -1.0) x) 2.0)
   (fma x (fma x 0.16666666666666666 0.5) 1.0)
   (*
    x
    (fma
     x
     (fma x (* (* x x) 0.0026041666666666665) 0.16666666666666666)
     0.5))))

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 * fma(x, fma(x, ((x * x) * 0.0026041666666666665), 0.16666666666666666), 0.5);
	}
	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 * fma(x, fma(x, Float64(Float64(x * x) * 0.0026041666666666665), 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], N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.0026041666666666665), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 35.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-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. accelerator-lowering-fma.f64 70.3

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

   5. Simplified 70.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 61.1

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

   5. Simplified 61.1%

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

      1. flip3-+ N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{{\left(x \cdot \frac{1}{24}\right)}^{3} + {\frac{1}{6}}^{3}}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}}, \frac{1}{2}\right), 1\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{{\left(x \cdot \frac{1}{24}\right)}^{3} + {\frac{1}{6}}^{3}}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}}, \frac{1}{2}\right), 1\right) \]

      3. unpow-prod-down N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\color{blue}{{x}^{3} \cdot {\frac{1}{24}}^{3}} + {\frac{1}{6}}^{3}}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\color{blue}{\mathsf{fma}\left({x}^{3}, {\frac{1}{24}}^{3}, {\frac{1}{6}}^{3}\right)}}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot x\right)}, {\frac{1}{24}}^{3}, {\frac{1}{6}}^{3}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot x\right)}, {\frac{1}{24}}^{3}, {\frac{1}{6}}^{3}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \color{blue}{\left(x \cdot x\right)}, {\frac{1}{24}}^{3}, {\frac{1}{6}}^{3}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \color{blue}{\frac{1}{13824}}, {\frac{1}{6}}^{3}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \color{blue}{\frac{1}{216}}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      10. swap-sqr N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left(\frac{1}{24} \cdot \frac{1}{24}\right)} + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{24} \cdot \frac{1}{24}, \frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}}, \frac{1}{2}\right), 1\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot \frac{1}{24}, \frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{576}}, \frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \frac{1}{576}, \color{blue}{\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}}\right)}, \frac{1}{2}\right), 1\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \frac{1}{576}, \color{blue}{\frac{1}{36}} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \frac{1}{576}, \frac{1}{36} - \color{blue}{x \cdot \left(\frac{1}{24} \cdot \frac{1}{6}\right)}\right)}, \frac{1}{2}\right), 1\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \frac{1}{576}, \frac{1}{36} - \color{blue}{x \cdot \left(\frac{1}{24} \cdot \frac{1}{6}\right)}\right)}, \frac{1}{2}\right), 1\right) \]

      18. metadata-eval 20.0

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), 7.233796296296296 \cdot 10^{-5}, 0.004629629629629629\right)}{\mathsf{fma}\left(x \cdot x, 0.001736111111111111, 0.027777777777777776 - x \cdot \color{blue}{0.006944444444444444}\right)}, 0.5\right), 1\right) \]

   7. Applied egg-rr 20.0%

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), 7.233796296296296 \cdot 10^{-5}, 0.004629629629629629\right)}{\mathsf{fma}\left(x \cdot x, 0.001736111111111111, 0.027777777777777776 - x \cdot 0.006944444444444444\right)}}, 0.5\right), 1\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\color{blue}{\frac{1}{36}}}, \frac{1}{2}\right), 1\right) \]
   9. Step-by-step derivation

   10. Simplified 73.1%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), 7.233796296296296 \cdot 10^{-5}, 0.004629629629629629\right)}{\color{blue}{0.027777777777777776}}, 0.5\right), 1\right) \]

   11. Taylor expanded in x around -inf

       \[\leadsto \color{blue}{-1 \cdot \left({x}^{5} \cdot \left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}}{{x}^{3}} - \frac{1}{384}\right)\right)} \]

   13. Simplified 73.1%

       \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.0026041666666666665 \cdot \left(x \cdot x\right), 0.16666666666666666\right), 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x} + -1}{x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.0026041666666666665, 0.16666666666666666\right), 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{x} + -1}{x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.0026041666666666665, 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

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 (* (* x x) 0.0026041666666666665) 0.16666666666666666)))))

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, ((x * x) * 0.0026041666666666665), 0.16666666666666666));
	}
	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, Float64(Float64(x * x) * 0.0026041666666666665), 0.16666666666666666)));
	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 * N[(N[(x * x), $MachinePrecision] * 0.0026041666666666665), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 35.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-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. accelerator-lowering-fma.f64 70.3

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

   5. Simplified 70.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 61.1

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

   5. Simplified 61.1%

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

      1. flip3-+ N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{{\left(x \cdot \frac{1}{24}\right)}^{3} + {\frac{1}{6}}^{3}}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}}, \frac{1}{2}\right), 1\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{{\left(x \cdot \frac{1}{24}\right)}^{3} + {\frac{1}{6}}^{3}}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}}, \frac{1}{2}\right), 1\right) \]

      3. unpow-prod-down N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\color{blue}{{x}^{3} \cdot {\frac{1}{24}}^{3}} + {\frac{1}{6}}^{3}}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\color{blue}{\mathsf{fma}\left({x}^{3}, {\frac{1}{24}}^{3}, {\frac{1}{6}}^{3}\right)}}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot x\right)}, {\frac{1}{24}}^{3}, {\frac{1}{6}}^{3}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot x\right)}, {\frac{1}{24}}^{3}, {\frac{1}{6}}^{3}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \color{blue}{\left(x \cdot x\right)}, {\frac{1}{24}}^{3}, {\frac{1}{6}}^{3}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \color{blue}{\frac{1}{13824}}, {\frac{1}{6}}^{3}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \color{blue}{\frac{1}{216}}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      10. swap-sqr N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left(\frac{1}{24} \cdot \frac{1}{24}\right)} + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{24} \cdot \frac{1}{24}, \frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}}, \frac{1}{2}\right), 1\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot \frac{1}{24}, \frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{576}}, \frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \frac{1}{576}, \color{blue}{\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}}\right)}, \frac{1}{2}\right), 1\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \frac{1}{576}, \color{blue}{\frac{1}{36}} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \frac{1}{576}, \frac{1}{36} - \color{blue}{x \cdot \left(\frac{1}{24} \cdot \frac{1}{6}\right)}\right)}, \frac{1}{2}\right), 1\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \frac{1}{576}, \frac{1}{36} - \color{blue}{x \cdot \left(\frac{1}{24} \cdot \frac{1}{6}\right)}\right)}, \frac{1}{2}\right), 1\right) \]

      18. metadata-eval 20.0

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), 7.233796296296296 \cdot 10^{-5}, 0.004629629629629629\right)}{\mathsf{fma}\left(x \cdot x, 0.001736111111111111, 0.027777777777777776 - x \cdot \color{blue}{0.006944444444444444}\right)}, 0.5\right), 1\right) \]

   7. Applied egg-rr 20.0%

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), 7.233796296296296 \cdot 10^{-5}, 0.004629629629629629\right)}{\mathsf{fma}\left(x \cdot x, 0.001736111111111111, 0.027777777777777776 - x \cdot 0.006944444444444444\right)}}, 0.5\right), 1\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\color{blue}{\frac{1}{36}}}, \frac{1}{2}\right), 1\right) \]
   9. Step-by-step derivation

   10. Simplified 73.1%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), 7.233796296296296 \cdot 10^{-5}, 0.004629629629629629\right)}{\color{blue}{0.027777777777777776}}, 0.5\right), 1\right) \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{{x}^{5} \cdot \left(\frac{1}{384} + \frac{1}{6} \cdot \frac{1}{{x}^{3}}\right)} \]

   13. Simplified 73.1%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x, 0.0026041666666666665 \cdot \left(x \cdot x\right), 0.16666666666666666\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x} + -1}{x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.0026041666666666665, 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{x} + -1}{x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.0026041666666666665 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ (exp x) -1.0) x) 2.0)
   (fma x (fma x 0.16666666666666666 0.5) 1.0)
   (* x (* 0.0026041666666666665 (* x (* x (* x x)))))))

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 * (0.0026041666666666665 * (x * (x * (x * x))));
	}
	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(0.0026041666666666665 * Float64(x * Float64(x * Float64(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[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(0.0026041666666666665 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $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 35.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-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. accelerator-lowering-fma.f64 70.3

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

   5. Simplified 70.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 61.1

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

   5. Simplified 61.1%

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

      1. flip3-+ N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{{\left(x \cdot \frac{1}{24}\right)}^{3} + {\frac{1}{6}}^{3}}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}}, \frac{1}{2}\right), 1\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{{\left(x \cdot \frac{1}{24}\right)}^{3} + {\frac{1}{6}}^{3}}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}}, \frac{1}{2}\right), 1\right) \]

      3. unpow-prod-down N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\color{blue}{{x}^{3} \cdot {\frac{1}{24}}^{3}} + {\frac{1}{6}}^{3}}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\color{blue}{\mathsf{fma}\left({x}^{3}, {\frac{1}{24}}^{3}, {\frac{1}{6}}^{3}\right)}}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot x\right)}, {\frac{1}{24}}^{3}, {\frac{1}{6}}^{3}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot x\right)}, {\frac{1}{24}}^{3}, {\frac{1}{6}}^{3}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \color{blue}{\left(x \cdot x\right)}, {\frac{1}{24}}^{3}, {\frac{1}{6}}^{3}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \color{blue}{\frac{1}{13824}}, {\frac{1}{6}}^{3}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \color{blue}{\frac{1}{216}}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      10. swap-sqr N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left(\frac{1}{24} \cdot \frac{1}{24}\right)} + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{24} \cdot \frac{1}{24}, \frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}}, \frac{1}{2}\right), 1\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot \frac{1}{24}, \frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{576}}, \frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \frac{1}{576}, \color{blue}{\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}}\right)}, \frac{1}{2}\right), 1\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \frac{1}{576}, \color{blue}{\frac{1}{36}} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \frac{1}{576}, \frac{1}{36} - \color{blue}{x \cdot \left(\frac{1}{24} \cdot \frac{1}{6}\right)}\right)}, \frac{1}{2}\right), 1\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \frac{1}{576}, \frac{1}{36} - \color{blue}{x \cdot \left(\frac{1}{24} \cdot \frac{1}{6}\right)}\right)}, \frac{1}{2}\right), 1\right) \]

      18. metadata-eval 20.0

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), 7.233796296296296 \cdot 10^{-5}, 0.004629629629629629\right)}{\mathsf{fma}\left(x \cdot x, 0.001736111111111111, 0.027777777777777776 - x \cdot \color{blue}{0.006944444444444444}\right)}, 0.5\right), 1\right) \]

   7. Applied egg-rr 20.0%

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), 7.233796296296296 \cdot 10^{-5}, 0.004629629629629629\right)}{\mathsf{fma}\left(x \cdot x, 0.001736111111111111, 0.027777777777777776 - x \cdot 0.006944444444444444\right)}}, 0.5\right), 1\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\color{blue}{\frac{1}{36}}}, \frac{1}{2}\right), 1\right) \]
   9. Step-by-step derivation

   10. Simplified 73.1%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), 7.233796296296296 \cdot 10^{-5}, 0.004629629629629629\right)}{\color{blue}{0.027777777777777776}}, 0.5\right), 1\right) \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{384} \cdot {x}^{5}} \]
   12. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1}{384} \cdot {x}^{\color{blue}{\left(4 + 1\right)}} \]

       2. pow-plus N/A

          \[\leadsto \frac{1}{384} \cdot \color{blue}{\left({x}^{4} \cdot x\right)} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{384} \cdot {x}^{4}\right) \cdot x} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{\left({x}^{4} \cdot \frac{1}{384}\right)} \cdot x \]

       5. associate-*l* N/A

          \[\leadsto \color{blue}{{x}^{4} \cdot \left(\frac{1}{384} \cdot x\right)} \]

       6. *-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{384} \cdot x\right) \cdot {x}^{4}} \]

       7. *-commutative N/A

          \[\leadsto \color{blue}{\left(x \cdot \frac{1}{384}\right)} \cdot {x}^{4} \]

       8. metadata-eval N/A

          \[\leadsto \left(x \cdot \frac{1}{384}\right) \cdot {x}^{\color{blue}{\left(3 + 1\right)}} \]

       9. pow-plus N/A

          \[\leadsto \left(x \cdot \frac{1}{384}\right) \cdot \color{blue}{\left({x}^{3} \cdot x\right)} \]

       10. associate-*l* N/A

           \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{384}\right) \cdot {x}^{3}\right) \cdot x} \]

       11. associate-*r* N/A

           \[\leadsto \color{blue}{\left(x \cdot \left(\frac{1}{384} \cdot {x}^{3}\right)\right)} \cdot x \]

       12. associate-*r* N/A

           \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{384} \cdot {x}^{3}\right) \cdot x\right)} \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{384} \cdot {x}^{3}\right) \cdot x\right)} \]

       14. associate-*l* N/A

           \[\leadsto x \cdot \color{blue}{\left(\frac{1}{384} \cdot \left({x}^{3} \cdot x\right)\right)} \]

       15. pow-plus N/A

           \[\leadsto x \cdot \left(\frac{1}{384} \cdot \color{blue}{{x}^{\left(3 + 1\right)}}\right) \]

       16. metadata-eval N/A

           \[\leadsto x \cdot \left(\frac{1}{384} \cdot {x}^{\color{blue}{4}}\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \color{blue}{\left(\frac{1}{384} \cdot {x}^{4}\right)} \]

       18. metadata-eval N/A

           \[\leadsto x \cdot \left(\frac{1}{384} \cdot {x}^{\color{blue}{\left(3 + 1\right)}}\right) \]

       19. pow-plus N/A

           \[\leadsto x \cdot \left(\frac{1}{384} \cdot \color{blue}{\left({x}^{3} \cdot x\right)}\right) \]

       20. *-commutative N/A

           \[\leadsto x \cdot \left(\frac{1}{384} \cdot \color{blue}{\left(x \cdot {x}^{3}\right)}\right) \]

       21. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \left(\frac{1}{384} \cdot \color{blue}{\left(x \cdot {x}^{3}\right)}\right) \]

       22. cube-mult N/A

           \[\leadsto x \cdot \left(\frac{1}{384} \cdot \left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \]

       23. unpow2 N/A

           \[\leadsto x \cdot \left(\frac{1}{384} \cdot \left(x \cdot \left(x \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

       24. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \left(\frac{1}{384} \cdot \left(x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right)\right) \]

       25. unpow2 N/A

           \[\leadsto x \cdot \left(\frac{1}{384} \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

       26. *-lowering-*.f64 73.1

           \[\leadsto x \cdot \left(0.0026041666666666665 \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

   13. Simplified 73.1%

       \[\leadsto \color{blue}{x \cdot \left(0.0026041666666666665 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x} + -1}{x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.0026041666666666665 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 35.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-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. accelerator-lowering-fma.f64 70.3

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

   5. Simplified 70.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 61.1

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

   5. Simplified 61.1%

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

   6. Taylor expanded in x around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. distribute-lft-in N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. associate-/r/ N/A

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

      14. metadata-eval N/A

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

      15. *-rgt-identity N/A

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

      16. associate-*r/ N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. rgt-mult-inverse N/A

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

      20. *-rgt-identity N/A

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

   8. Simplified 61.1%

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

4. Final simplification 68.3%

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

Alternative 6: 67.9% accurate, 0.8× speedup

Math

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

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 0.041666666666666664 0.16666666666666666)))))

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, 0.041666666666666664, 0.16666666666666666));
	}
	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, 0.041666666666666664, 0.16666666666666666)));
	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 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 35.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-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. accelerator-lowering-fma.f64 70.3

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

   5. Simplified 70.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 61.1

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

   5. Simplified 61.1%

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

   6. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*l* N/A

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

      8. lft-mult-inverse N/A

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

      9. metadata-eval N/A

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

      10. associate-*r* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lft-mult-inverse N/A

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

      14. associate-*l* N/A

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

      15. distribute-rgt-in N/A

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

      16. +-commutative N/A

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

      17. *-lowering-*.f64 N/A

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

      18. distribute-rgt-in N/A

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

      19. associate-*l* N/A

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

      20. lft-mult-inverse N/A

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

      21. metadata-eval N/A

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

   8. Simplified 61.1%

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

4. Final simplification 68.3%

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

Alternative 7: 72.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 7.233796296296296 \cdot 10^{-5}, 0.004629629629629629\right)\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;x \leq 2 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \mathsf{fma}\left(t\_1 \cdot t\_1, 1296, -0.25\right), \frac{1}{\mathsf{fma}\left(t\_0, x \cdot 36, -0.5\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.0026041666666666665 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x (* (* x x) 7.233796296296296e-5) 0.004629629629629629))
        (t_1 (* x t_0)))
   (if (<= x 2e+60)
     (fma
      (* x (fma (* t_1 t_1) 1296.0 -0.25))
      (/ 1.0 (fma t_0 (* x 36.0) -0.5))
      1.0)
     (* x (* 0.0026041666666666665 (* x (* x (* x x))))))))

C

double code(double x) {
	double t_0 = fma(x, ((x * x) * 7.233796296296296e-5), 0.004629629629629629);
	double t_1 = x * t_0;
	double tmp;
	if (x <= 2e+60) {
		tmp = fma((x * fma((t_1 * t_1), 1296.0, -0.25)), (1.0 / fma(t_0, (x * 36.0), -0.5)), 1.0);
	} else {
		tmp = x * (0.0026041666666666665 * (x * (x * (x * x))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(x, Float64(Float64(x * x) * 7.233796296296296e-5), 0.004629629629629629)
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if (x <= 2e+60)
		tmp = fma(Float64(x * fma(Float64(t_1 * t_1), 1296.0, -0.25)), Float64(1.0 / fma(t_0, Float64(x * 36.0), -0.5)), 1.0);
	else
		tmp = Float64(x * Float64(0.0026041666666666665 * Float64(x * Float64(x * Float64(x * x)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(N[(x * x), $MachinePrecision] * 7.233796296296296e-5), $MachinePrecision] + 0.004629629629629629), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[LessEqual[x, 2e+60], N[(N[(x * N[(N[(t$95$1 * t$95$1), $MachinePrecision] * 1296.0 + -0.25), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(t$95$0 * N[(x * 36.0), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(0.0026041666666666665 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9999999999999999e60

   1. Initial program 40.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 65.0

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

   5. Simplified 65.0%

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

      1. flip3-+ N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{{\left(x \cdot \frac{1}{24}\right)}^{3} + {\frac{1}{6}}^{3}}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}}, \frac{1}{2}\right), 1\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{{\left(x \cdot \frac{1}{24}\right)}^{3} + {\frac{1}{6}}^{3}}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}}, \frac{1}{2}\right), 1\right) \]

      3. unpow-prod-down N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\color{blue}{{x}^{3} \cdot {\frac{1}{24}}^{3}} + {\frac{1}{6}}^{3}}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\color{blue}{\mathsf{fma}\left({x}^{3}, {\frac{1}{24}}^{3}, {\frac{1}{6}}^{3}\right)}}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot x\right)}, {\frac{1}{24}}^{3}, {\frac{1}{6}}^{3}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot x\right)}, {\frac{1}{24}}^{3}, {\frac{1}{6}}^{3}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \color{blue}{\left(x \cdot x\right)}, {\frac{1}{24}}^{3}, {\frac{1}{6}}^{3}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \color{blue}{\frac{1}{13824}}, {\frac{1}{6}}^{3}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \color{blue}{\frac{1}{216}}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      10. swap-sqr N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left(\frac{1}{24} \cdot \frac{1}{24}\right)} + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{24} \cdot \frac{1}{24}, \frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}}, \frac{1}{2}\right), 1\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot \frac{1}{24}, \frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{576}}, \frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \frac{1}{576}, \color{blue}{\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}}\right)}, \frac{1}{2}\right), 1\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \frac{1}{576}, \color{blue}{\frac{1}{36}} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \frac{1}{576}, \frac{1}{36} - \color{blue}{x \cdot \left(\frac{1}{24} \cdot \frac{1}{6}\right)}\right)}, \frac{1}{2}\right), 1\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \frac{1}{576}, \frac{1}{36} - \color{blue}{x \cdot \left(\frac{1}{24} \cdot \frac{1}{6}\right)}\right)}, \frac{1}{2}\right), 1\right) \]

      18. metadata-eval 64.8

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), 7.233796296296296 \cdot 10^{-5}, 0.004629629629629629\right)}{\mathsf{fma}\left(x \cdot x, 0.001736111111111111, 0.027777777777777776 - x \cdot \color{blue}{0.006944444444444444}\right)}, 0.5\right), 1\right) \]

   7. Applied egg-rr 64.8%

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), 7.233796296296296 \cdot 10^{-5}, 0.004629629629629629\right)}{\mathsf{fma}\left(x \cdot x, 0.001736111111111111, 0.027777777777777776 - x \cdot 0.006944444444444444\right)}}, 0.5\right), 1\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\color{blue}{\frac{1}{36}}}, \frac{1}{2}\right), 1\right) \]
   9. Step-by-step derivation

   10. Simplified 65.0%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), 7.233796296296296 \cdot 10^{-5}, 0.004629629629629629\right)}{\color{blue}{0.027777777777777776}}, 0.5\right), 1\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(x \cdot \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{13824} + \frac{1}{216}}{\frac{1}{36}} + \frac{1}{2}\right) \cdot x} + 1 \]

       2. flip-+ N/A

          \[\leadsto \color{blue}{\frac{\left(x \cdot \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{13824} + \frac{1}{216}}{\frac{1}{36}}\right) \cdot \left(x \cdot \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{13824} + \frac{1}{216}}{\frac{1}{36}}\right) - \frac{1}{2} \cdot \frac{1}{2}}{x \cdot \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{13824} + \frac{1}{216}}{\frac{1}{36}} - \frac{1}{2}}} \cdot x + 1 \]

       3. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\left(\left(x \cdot \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{13824} + \frac{1}{216}}{\frac{1}{36}}\right) \cdot \left(x \cdot \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{13824} + \frac{1}{216}}{\frac{1}{36}}\right) - \frac{1}{2} \cdot \frac{1}{2}\right) \cdot x}{x \cdot \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{13824} + \frac{1}{216}}{\frac{1}{36}} - \frac{1}{2}}} + 1 \]

       4. div-inv N/A

          \[\leadsto \color{blue}{\left(\left(\left(x \cdot \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{13824} + \frac{1}{216}}{\frac{1}{36}}\right) \cdot \left(x \cdot \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{13824} + \frac{1}{216}}{\frac{1}{36}}\right) - \frac{1}{2} \cdot \frac{1}{2}\right) \cdot x\right) \cdot \frac{1}{x \cdot \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{13824} + \frac{1}{216}}{\frac{1}{36}} - \frac{1}{2}}} + 1 \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(x \cdot \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{13824} + \frac{1}{216}}{\frac{1}{36}}\right) \cdot \left(x \cdot \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{13824} + \frac{1}{216}}{\frac{1}{36}}\right) - \frac{1}{2} \cdot \frac{1}{2}\right) \cdot x, \frac{1}{x \cdot \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{13824} + \frac{1}{216}}{\frac{1}{36}} - \frac{1}{2}}, 1\right)} \]

   12. Applied egg-rr 67.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 7.233796296296296 \cdot 10^{-5}, 0.004629629629629629\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 7.233796296296296 \cdot 10^{-5}, 0.004629629629629629\right)\right), 1296, -0.25\right) \cdot x, \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 7.233796296296296 \cdot 10^{-5}, 0.004629629629629629\right), 36 \cdot x, -0.5\right)}, 1\right)} \]

   if 1.9999999999999999e60 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 83.8

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

   5. Simplified 83.8%

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

      1. flip3-+ N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{{\left(x \cdot \frac{1}{24}\right)}^{3} + {\frac{1}{6}}^{3}}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}}, \frac{1}{2}\right), 1\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{{\left(x \cdot \frac{1}{24}\right)}^{3} + {\frac{1}{6}}^{3}}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}}, \frac{1}{2}\right), 1\right) \]

      3. unpow-prod-down N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\color{blue}{{x}^{3} \cdot {\frac{1}{24}}^{3}} + {\frac{1}{6}}^{3}}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\color{blue}{\mathsf{fma}\left({x}^{3}, {\frac{1}{24}}^{3}, {\frac{1}{6}}^{3}\right)}}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot x\right)}, {\frac{1}{24}}^{3}, {\frac{1}{6}}^{3}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot x\right)}, {\frac{1}{24}}^{3}, {\frac{1}{6}}^{3}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \color{blue}{\left(x \cdot x\right)}, {\frac{1}{24}}^{3}, {\frac{1}{6}}^{3}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \color{blue}{\frac{1}{13824}}, {\frac{1}{6}}^{3}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \color{blue}{\frac{1}{216}}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      10. swap-sqr N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left(\frac{1}{24} \cdot \frac{1}{24}\right)} + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{24} \cdot \frac{1}{24}, \frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}}, \frac{1}{2}\right), 1\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot \frac{1}{24}, \frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{576}}, \frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \frac{1}{576}, \color{blue}{\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}}\right)}, \frac{1}{2}\right), 1\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \frac{1}{576}, \color{blue}{\frac{1}{36}} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \frac{1}{576}, \frac{1}{36} - \color{blue}{x \cdot \left(\frac{1}{24} \cdot \frac{1}{6}\right)}\right)}, \frac{1}{2}\right), 1\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \frac{1}{576}, \frac{1}{36} - \color{blue}{x \cdot \left(\frac{1}{24} \cdot \frac{1}{6}\right)}\right)}, \frac{1}{2}\right), 1\right) \]

      18. metadata-eval 26.3

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), 7.233796296296296 \cdot 10^{-5}, 0.004629629629629629\right)}{\mathsf{fma}\left(x \cdot x, 0.001736111111111111, 0.027777777777777776 - x \cdot \color{blue}{0.006944444444444444}\right)}, 0.5\right), 1\right) \]

   7. Applied egg-rr 26.3%

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), 7.233796296296296 \cdot 10^{-5}, 0.004629629629629629\right)}{\mathsf{fma}\left(x \cdot x, 0.001736111111111111, 0.027777777777777776 - x \cdot 0.006944444444444444\right)}}, 0.5\right), 1\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\color{blue}{\frac{1}{36}}}, \frac{1}{2}\right), 1\right) \]
   9. Step-by-step derivation

   10. Simplified 100.0%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), 7.233796296296296 \cdot 10^{-5}, 0.004629629629629629\right)}{\color{blue}{0.027777777777777776}}, 0.5\right), 1\right) \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{384} \cdot {x}^{5}} \]
   12. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1}{384} \cdot {x}^{\color{blue}{\left(4 + 1\right)}} \]

       2. pow-plus N/A

          \[\leadsto \frac{1}{384} \cdot \color{blue}{\left({x}^{4} \cdot x\right)} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{384} \cdot {x}^{4}\right) \cdot x} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{\left({x}^{4} \cdot \frac{1}{384}\right)} \cdot x \]

       5. associate-*l* N/A

          \[\leadsto \color{blue}{{x}^{4} \cdot \left(\frac{1}{384} \cdot x\right)} \]

       6. *-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{384} \cdot x\right) \cdot {x}^{4}} \]

       7. *-commutative N/A

          \[\leadsto \color{blue}{\left(x \cdot \frac{1}{384}\right)} \cdot {x}^{4} \]

       8. metadata-eval N/A

          \[\leadsto \left(x \cdot \frac{1}{384}\right) \cdot {x}^{\color{blue}{\left(3 + 1\right)}} \]

       9. pow-plus N/A

          \[\leadsto \left(x \cdot \frac{1}{384}\right) \cdot \color{blue}{\left({x}^{3} \cdot x\right)} \]

       10. associate-*l* N/A

           \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{384}\right) \cdot {x}^{3}\right) \cdot x} \]

       11. associate-*r* N/A

           \[\leadsto \color{blue}{\left(x \cdot \left(\frac{1}{384} \cdot {x}^{3}\right)\right)} \cdot x \]

       12. associate-*r* N/A

           \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{384} \cdot {x}^{3}\right) \cdot x\right)} \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{384} \cdot {x}^{3}\right) \cdot x\right)} \]

       14. associate-*l* N/A

           \[\leadsto x \cdot \color{blue}{\left(\frac{1}{384} \cdot \left({x}^{3} \cdot x\right)\right)} \]

       15. pow-plus N/A

           \[\leadsto x \cdot \left(\frac{1}{384} \cdot \color{blue}{{x}^{\left(3 + 1\right)}}\right) \]

       16. metadata-eval N/A

           \[\leadsto x \cdot \left(\frac{1}{384} \cdot {x}^{\color{blue}{4}}\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \color{blue}{\left(\frac{1}{384} \cdot {x}^{4}\right)} \]

       18. metadata-eval N/A

           \[\leadsto x \cdot \left(\frac{1}{384} \cdot {x}^{\color{blue}{\left(3 + 1\right)}}\right) \]

       19. pow-plus N/A

           \[\leadsto x \cdot \left(\frac{1}{384} \cdot \color{blue}{\left({x}^{3} \cdot x\right)}\right) \]

       20. *-commutative N/A

           \[\leadsto x \cdot \left(\frac{1}{384} \cdot \color{blue}{\left(x \cdot {x}^{3}\right)}\right) \]

       21. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \left(\frac{1}{384} \cdot \color{blue}{\left(x \cdot {x}^{3}\right)}\right) \]

       22. cube-mult N/A

           \[\leadsto x \cdot \left(\frac{1}{384} \cdot \left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \]

       23. unpow2 N/A

           \[\leadsto x \cdot \left(\frac{1}{384} \cdot \left(x \cdot \left(x \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

       24. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \left(\frac{1}{384} \cdot \left(x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right)\right) \]

       25. unpow2 N/A

           \[\leadsto x \cdot \left(\frac{1}{384} \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

       26. *-lowering-*.f64 100.0

           \[\leadsto x \cdot \left(0.0026041666666666665 \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{x \cdot \left(0.0026041666666666665 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \mathsf{fma}\left(\left(x \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 7.233796296296296 \cdot 10^{-5}, 0.004629629629629629\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 7.233796296296296 \cdot 10^{-5}, 0.004629629629629629\right)\right), 1296, -0.25\right), \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 7.233796296296296 \cdot 10^{-5}, 0.004629629629629629\right), x \cdot 36, -0.5\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.0026041666666666665 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.0026041666666666665, 0.16666666666666666\right), 0.5\right), 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  x
  (fma x (fma x (* (* x x) 0.0026041666666666665) 0.16666666666666666) 0.5)
  1.0))

C

double code(double x) {
	return fma(x, fma(x, fma(x, ((x * x) * 0.0026041666666666665), 0.16666666666666666), 0.5), 1.0);
}

Julia

function code(x)
	return fma(x, fma(x, fma(x, Float64(Float64(x * x) * 0.0026041666666666665), 0.16666666666666666), 0.5), 1.0)
end

Wolfram

code[x_] := N[(x * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.0026041666666666665), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 49.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} \]
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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 67.9

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

5. Simplified 67.9%

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

   1. flip3-+ N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{{\left(x \cdot \frac{1}{24}\right)}^{3} + {\frac{1}{6}}^{3}}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}}, \frac{1}{2}\right), 1\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{{\left(x \cdot \frac{1}{24}\right)}^{3} + {\frac{1}{6}}^{3}}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}}, \frac{1}{2}\right), 1\right) \]

   3. unpow-prod-down N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\color{blue}{{x}^{3} \cdot {\frac{1}{24}}^{3}} + {\frac{1}{6}}^{3}}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\color{blue}{\mathsf{fma}\left({x}^{3}, {\frac{1}{24}}^{3}, {\frac{1}{6}}^{3}\right)}}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

   5. cube-mult N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot x\right)}, {\frac{1}{24}}^{3}, {\frac{1}{6}}^{3}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot x\right)}, {\frac{1}{24}}^{3}, {\frac{1}{6}}^{3}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \color{blue}{\left(x \cdot x\right)}, {\frac{1}{24}}^{3}, {\frac{1}{6}}^{3}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \color{blue}{\frac{1}{13824}}, {\frac{1}{6}}^{3}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \color{blue}{\frac{1}{216}}\right)}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

   10. swap-sqr N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left(\frac{1}{24} \cdot \frac{1}{24}\right)} + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

   11. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{24} \cdot \frac{1}{24}, \frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}}, \frac{1}{2}\right), 1\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot \frac{1}{24}, \frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{576}}, \frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

   14. --lowering--.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \frac{1}{576}, \color{blue}{\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}}\right)}, \frac{1}{2}\right), 1\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \frac{1}{576}, \color{blue}{\frac{1}{36}} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \frac{1}{576}, \frac{1}{36} - \color{blue}{x \cdot \left(\frac{1}{24} \cdot \frac{1}{6}\right)}\right)}, \frac{1}{2}\right), 1\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\mathsf{fma}\left(x \cdot x, \frac{1}{576}, \frac{1}{36} - \color{blue}{x \cdot \left(\frac{1}{24} \cdot \frac{1}{6}\right)}\right)}, \frac{1}{2}\right), 1\right) \]

   18. metadata-eval 58.8

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), 7.233796296296296 \cdot 10^{-5}, 0.004629629629629629\right)}{\mathsf{fma}\left(x \cdot x, 0.001736111111111111, 0.027777777777777776 - x \cdot \color{blue}{0.006944444444444444}\right)}, 0.5\right), 1\right) \]

7. Applied egg-rr 58.8%

   \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), 7.233796296296296 \cdot 10^{-5}, 0.004629629629629629\right)}{\mathsf{fma}\left(x \cdot x, 0.001736111111111111, 0.027777777777777776 - x \cdot 0.006944444444444444\right)}}, 0.5\right), 1\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{13824}, \frac{1}{216}\right)}{\color{blue}{\frac{1}{36}}}, \frac{1}{2}\right), 1\right) \]
9. Step-by-step derivation

10. Simplified 70.4%

    \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), 7.233796296296296 \cdot 10^{-5}, 0.004629629629629629\right)}{\color{blue}{0.027777777777777776}}, 0.5\right), 1\right) \]

11. Taylor expanded in x around 0

    \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{6} + \frac{1}{384} \cdot {x}^{3}}, \frac{1}{2}\right), 1\right) \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{384} \cdot {x}^{3} + \frac{1}{6}}, \frac{1}{2}\right), 1\right) \]

    2. unpow3 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{384} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} + \frac{1}{6}, \frac{1}{2}\right), 1\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{384} \cdot \left(\color{blue}{{x}^{2}} \cdot x\right) + \frac{1}{6}, \frac{1}{2}\right), 1\right) \]

    4. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\left(\frac{1}{384} \cdot {x}^{2}\right) \cdot x} + \frac{1}{6}, \frac{1}{2}\right), 1\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{384} \cdot {x}^{2}\right)} + \frac{1}{6}, \frac{1}{2}\right), 1\right) \]

    6. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{384} \cdot {x}^{2}, \frac{1}{6}\right)}, \frac{1}{2}\right), 1\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{384} \cdot {x}^{2}}, \frac{1}{6}\right), \frac{1}{2}\right), 1\right) \]

    8. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{384} \cdot \color{blue}{\left(x \cdot x\right)}, \frac{1}{6}\right), \frac{1}{2}\right), 1\right) \]

    9. *-lowering-*.f64 70.4

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.0026041666666666665 \cdot \color{blue}{\left(x \cdot x\right)}, 0.16666666666666666\right), 0.5\right), 1\right) \]

13. Simplified 70.4%

    \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.0026041666666666665 \cdot \left(x \cdot x\right), 0.16666666666666666\right)}, 0.5\right), 1\right) \]

14. Final simplification 70.4%

    \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.0026041666666666665, 0.16666666666666666\right), 0.5\right), 1\right) \]
15. Add Preprocessing

Alternative 9: 68.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.041666666666666664\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 6.5)
   (fma x (fma x 0.16666666666666666 0.5) 1.0)
   (* (* x (* x x)) 0.041666666666666664)))

C

double code(double x) {
	double tmp;
	if (x <= 6.5) {
		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 (x <= 6.5)
		tmp = fma(x, fma(x, 0.16666666666666666, 0.5), 1.0);
	else
		tmp = Float64(Float64(x * Float64(x * x)) * 0.041666666666666664);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 6.5], N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.5

   1. Initial program 35.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-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. accelerator-lowering-fma.f64 70.3

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

   5. Simplified 70.3%

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

   if 6.5 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} \]
   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 61.1

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

   5. Simplified 61.1%

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

   6. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{3}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 61.1

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

   8. Simplified 61.1%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.041666666666666664\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.5% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} \]
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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 67.9

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

5. Simplified 67.9%

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

Alternative 11: 63.4% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.5

   1. Initial program 35.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 69.6

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

   5. Simplified 69.6%

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

   if 2.5 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-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. accelerator-lowering-fma.f64 44.4

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

   5. Simplified 44.4%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 44.4%

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

Alternative 12: 63.4% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.5) (fma x 0.5 1.0) (* (* x x) 0.16666666666666666)))

C

double code(double x) {
	double tmp;
	if (x <= 4.5) {
		tmp = fma(x, 0.5, 1.0);
	} else {
		tmp = (x * x) * 0.16666666666666666;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.5)
		tmp = fma(x, 0.5, 1.0);
	else
		tmp = Float64(Float64(x * x) * 0.16666666666666666);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4.5], N[(x * 0.5 + 1.0), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.5

   1. Initial program 35.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 69.6

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

   5. Simplified 69.6%

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

   if 4.5 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-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. accelerator-lowering-fma.f64 44.4

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

   5. Simplified 44.4%

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

   6. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 44.4

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

   8. Simplified 44.4%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.0% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.4%

   \[\frac{e^{x} - 1}{x} \]
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. accelerator-lowering-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. accelerator-lowering-fma.f64 64.6

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

5. Simplified 64.6%

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

Alternative 14: 51.4% accurate, 16.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.4%

   \[\frac{e^{x} - 1}{x} \]
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. accelerator-lowering-fma.f64 55.5

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

5. Simplified 55.5%

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

Alternative 15: 51.2% accurate, 115.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 49.4%

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

3. Taylor expanded in x around 0

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

5. Simplified 55.1%

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

Developer Target 1: 52.6% 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 2024204 
(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))