Kahan's exp quotient

Percentage Accurate: 52.5% → 100.0%

Time: 10.1s

Alternatives: 14

Speedup: 8.8×

• 24.2% 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: 52.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.0%

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

   1. lower-expm1.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 70.6% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ (exp x) -1.0) x) 2.0)
   (fma (* x x) 0.16666666666666666 (fma x 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 * x), 0.16666666666666666, fma(x, 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(Float64(x * x), 0.16666666666666666, fma(x, 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[(N[(x * x), $MachinePrecision] * 0.16666666666666666 + N[(x * 0.5 + 1.0), $MachinePrecision]), $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 37.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 67.7

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

   5. Applied rewrites 67.7%

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

      1. distribute-lft-in N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 67.7

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

   7. Applied rewrites 67.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 75.2

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

   5. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)} \]
   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. lower-/.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. lower-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. lower-*.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. lower-*.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. associate-+r- N/A

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

      11. lower--.f64 N/A

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

      12. swap-sqr N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. associate-*l* N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval 12.0

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

   7. Applied rewrites 12.0%

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

   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. Applied rewrites 90.0%

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

   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

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. metadata-eval N/A

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

       8. pow-plus N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. cube-mult N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 90.0

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

   13. Applied rewrites 90.0%

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

4. Final simplification 72.7%

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

Alternative 3: 67.5% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ (exp x) -1.0) x) 2.0)
   (fma (* x x) 0.16666666666666666 (fma x 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 * x), 0.16666666666666666, fma(x, 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(Float64(x * x), 0.16666666666666666, fma(x, 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[(N[(x * x), $MachinePrecision] * 0.16666666666666666 + N[(x * 0.5 + 1.0), $MachinePrecision]), $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 37.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 67.7

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

   5. Applied rewrites 67.7%

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

      1. distribute-lft-in N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 67.7

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

   7. Applied rewrites 67.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 84.9

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{1}{24} + \frac{1}{6} \cdot \frac{1}{x}\right)} \]
   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. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 75.2

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

   8. Applied rewrites 75.2%

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

4. Final simplification 69.4%

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

Alternative 4: 67.5% 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 37.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 67.7

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

   5. Applied rewrites 67.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 84.9

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{1}{24} + \frac{1}{6} \cdot \frac{1}{x}\right)} \]
   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. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 75.2

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

   8. Applied rewrites 75.2%

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

4. Final simplification 69.4%

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

Alternative 5: 67.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision], 2.0], N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $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 37.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 67.7

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

   5. Applied rewrites 67.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 75.2

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

   5. Applied rewrites 75.2%

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

   6. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 75.2

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

   8. Applied rewrites 75.2%

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

4. Final simplification 69.4%

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

Alternative 6: 63.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 37.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 67.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 65.0

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

   5. Applied rewrites 65.0%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lft-mult-inverse N/A

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

      6. associate-*l* N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-/r* N/A

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

      11. unpow2 N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. rgt-mult-inverse N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-in N/A

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

      18. *-commutative N/A

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

      19. associate-*l* N/A

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

   8. Applied rewrites 65.0%

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

4. Final simplification 66.9%

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

Alternative 7: 63.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 37.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 67.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 65.0

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

   5. Applied rewrites 65.0%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 65.0

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

   8. Applied rewrites 65.0%

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

4. Final simplification 66.9%

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

Alternative 8: 70.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 7.233796296296296 \cdot 10^{-5}, 0.004629629629629629\right)\right), 36, \mathsf{fma}\left(x, 0.5, 1\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (* x (* x (fma x (* (* x x) 7.233796296296296e-5) 0.004629629629629629)))
  36.0
  (fma x 0.5 1.0)))

C

double code(double x) {
	return fma((x * (x * fma(x, ((x * x) * 7.233796296296296e-5), 0.004629629629629629))), 36.0, fma(x, 0.5, 1.0));
}

Julia

function code(x)
	return fma(Float64(x * Float64(x * fma(x, Float64(Float64(x * x) * 7.233796296296296e-5), 0.004629629629629629))), 36.0, fma(x, 0.5, 1.0))
end

Wolfram

code[x_] := N[(N[(x * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 7.233796296296296e-5), $MachinePrecision] + 0.004629629629629629), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 36.0 + N[(x * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 69.0

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

5. Applied rewrites 69.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)} \]
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. lower-/.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. lower-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. lower-*.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. lower-*.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. associate-+r- N/A

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

   11. lower--.f64 N/A

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

   12. swap-sqr N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. associate-*l* N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval 54.7

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

7. Applied rewrites 54.7%

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

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. Applied rewrites 72.2%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-fma.f64 N/A

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

    4. frac-2neg N/A

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

    5. frac-2neg N/A

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

    6. lift-/.f64 N/A

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

    7. lift-fma.f64 N/A

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

12. Applied rewrites 72.2%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 7.233796296296296 \cdot 10^{-5}, 0.004629629629629629\right)\right), 36, \mathsf{fma}\left(x, 0.5, 1\right)\right)} \]
13. Add Preprocessing

Alternative 9: 69.9% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 69.0

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

5. Applied rewrites 69.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)} \]
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. lower-/.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. lower-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. lower-*.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. lower-*.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. associate-+r- N/A

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

   11. lower--.f64 N/A

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

   12. swap-sqr N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. associate-*l* N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval 54.7

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

7. Applied rewrites 54.7%

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

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. Applied rewrites 72.2%

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

11. Taylor expanded in x around inf

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

    1. *-commutative N/A

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

    2. cube-mult N/A

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

    3. unpow2 N/A

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

    4. associate-*l* N/A

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

    5. lower-*.f64 N/A

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

    6. unpow2 N/A

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

    7. associate-*l* N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 N/A

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

    10. *-commutative N/A

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

    11. lower-*.f64 72.1

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

13. Applied rewrites 72.1%

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

Alternative 10: 67.1% 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 51.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 69.0

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

5. Applied rewrites 69.0%

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

Alternative 11: 63.8% 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 51.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 67.1

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

5. Applied rewrites 67.1%

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

Alternative 12: 51.9% accurate, 16.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 53.4

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

5. Applied rewrites 53.4%

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

Alternative 13: 51.7% accurate, 115.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 51.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 53.1%

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

Alternative 14: 3.3% accurate, 115.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 51.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 3.6%

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

   1. metadata-eval N/A

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

   2. div0 3.6

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

7. Applied rewrites 3.6%

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

Developer Target 1: 52.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x} - 1\\ \mathbf{if}\;x < 1 \land x > -1:\\ \;\;\;\;\frac{t\_0}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (exp x) 1.0)))
   (if (and (< x 1.0) (> x -1.0)) (/ t_0 (log (exp x))) (/ t_0 x))))

C

double code(double x) {
	double t_0 = exp(x) - 1.0;
	double tmp;
	if ((x < 1.0) && (x > -1.0)) {
		tmp = t_0 / log(exp(x));
	} else {
		tmp = t_0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(x) - 1.0d0
    if ((x < 1.0d0) .and. (x > (-1.0d0))) then
        tmp = t_0 / log(exp(x))
    else
        tmp = t_0 / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.exp(x) - 1.0;
	double tmp;
	if ((x < 1.0) && (x > -1.0)) {
		tmp = t_0 / Math.log(Math.exp(x));
	} else {
		tmp = t_0 / x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.exp(x) - 1.0
	tmp = 0
	if (x < 1.0) and (x > -1.0):
		tmp = t_0 / math.log(math.exp(x))
	else:
		tmp = t_0 / x
	return tmp

Julia

function code(x)
	t_0 = Float64(exp(x) - 1.0)
	tmp = 0.0
	if ((x < 1.0) && (x > -1.0))
		tmp = Float64(t_0 / log(exp(x)));
	else
		tmp = Float64(t_0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = exp(x) - 1.0;
	tmp = 0.0;
	if ((x < 1.0) && (x > -1.0))
		tmp = t_0 / log(exp(x));
	else
		tmp = t_0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision]}, If[And[Less[x, 1.0], Greater[x, -1.0]], N[(t$95$0 / N[Log[N[Exp[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 / x), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024221 
(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))