Kahan's exp quotient

Percentage Accurate: 52.4% → 100.0%

Time: 9.8s

Alternatives: 13

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: 52.4% 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 55.5%

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

   1. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{e^{x} - 1}}{x} \]

   2. lift-exp.f64 N/A

      \[\leadsto \frac{\color{blue}{e^{x}} - 1}{x} \]

   3. lower-expm1.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 73.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{x} - 1}{x} \leq 0.005:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(0.5, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0625, \left(x \cdot x\right) \cdot \left(x \cdot x\right), -1\right)}{\mathsf{fma}\left(0.5, x, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (- (exp x) 1.0) x) 0.005)
   (/ -1.0 (fma 0.5 x -1.0))
   (/ (fma 0.0625 (* (* x x) (* x x)) -1.0) (fma 0.5 x -1.0))))

C

double code(double x) {
	double tmp;
	if (((exp(x) - 1.0) / x) <= 0.005) {
		tmp = -1.0 / fma(0.5, x, -1.0);
	} else {
		tmp = fma(0.0625, ((x * x) * (x * x)), -1.0) / fma(0.5, x, -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) - 1.0) / x) <= 0.005)
		tmp = Float64(-1.0 / fma(0.5, x, -1.0));
	else
		tmp = Float64(fma(0.0625, Float64(Float64(x * x) * Float64(x * x)), -1.0) / fma(0.5, x, -1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision] / x), $MachinePrecision], 0.005], N[(-1.0 / N[(0.5 * x + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(0.5 * x + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 39.7%

      \[\frac{e^{x} - 1}{x} \]
   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 62.8

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

   5. Applied rewrites 62.8%

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

   7. Applied rewrites 62.4%

      \[\leadsto \frac{\mathsf{fma}\left(0.0625, \left(x \cdot x\right) \cdot \left(x \cdot x\right), -1\right) \cdot 1}{\color{blue}{\mathsf{fma}\left(x, x \cdot 0.25, 1\right) \cdot \mathsf{fma}\left(x, 0.5, -1\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), -1\right) \cdot 1}{\frac{1}{2} \cdot x - \color{blue}{1}} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.7%

       \[\leadsto \frac{\mathsf{fma}\left(0.0625, \left(x \cdot x\right) \cdot \left(x \cdot x\right), -1\right) \cdot 1}{\mathsf{fma}\left(0.5, \color{blue}{x}, -1\right)} \]

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 69.3%

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

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

   1. Initial program 99.2%

      \[\frac{e^{x} - 1}{x} \]
   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 8.3

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

   5. Applied rewrites 8.3%

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

   7. Applied rewrites 12.5%

      \[\leadsto \frac{\mathsf{fma}\left(0.0625, \left(x \cdot x\right) \cdot \left(x \cdot x\right), -1\right) \cdot 1}{\color{blue}{\mathsf{fma}\left(x, x \cdot 0.25, 1\right) \cdot \mathsf{fma}\left(x, 0.5, -1\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), -1\right) \cdot 1}{\frac{1}{2} \cdot x - \color{blue}{1}} \]
   9. Step-by-step derivation

   10. Applied rewrites 81.8%

       \[\leadsto \frac{\mathsf{fma}\left(0.0625, \left(x \cdot x\right) \cdot \left(x \cdot x\right), -1\right) \cdot 1}{\mathsf{fma}\left(0.5, \color{blue}{x}, -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x} - 1}{x} \leq 0.005:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(0.5, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0625, \left(x \cdot x\right) \cdot \left(x \cdot x\right), -1\right)}{\mathsf{fma}\left(0.5, x, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{x} - 1}{x} \leq 2:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(0.5, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.0625 \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right)}{\mathsf{fma}\left(0.5, x, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (- (exp x) 1.0) x) 2.0)
   (/ -1.0 (fma 0.5 x -1.0))
   (/ (* 0.0625 (* (* x (* x x)) x)) (fma 0.5 x -1.0))))

C

double code(double x) {
	double tmp;
	if (((exp(x) - 1.0) / x) <= 2.0) {
		tmp = -1.0 / fma(0.5, x, -1.0);
	} else {
		tmp = (0.0625 * ((x * (x * x)) * x)) / fma(0.5, x, -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) - 1.0) / x) <= 2.0)
		tmp = Float64(-1.0 / fma(0.5, x, -1.0));
	else
		tmp = Float64(Float64(0.0625 * Float64(Float64(x * Float64(x * x)) * x)) / fma(0.5, x, -1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision] / x), $MachinePrecision], 2.0], N[(-1.0 / N[(0.5 * x + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 * N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[(0.5 * x + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 40.4%

      \[\frac{e^{x} - 1}{x} \]
   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. lower-fma.f64 62.9

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

   5. Applied rewrites 62.9%

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

   7. Applied rewrites 62.5%

      \[\leadsto \frac{\mathsf{fma}\left(0.0625, \left(x \cdot x\right) \cdot \left(x \cdot x\right), -1\right) \cdot 1}{\color{blue}{\mathsf{fma}\left(x, x \cdot 0.25, 1\right) \cdot \mathsf{fma}\left(x, 0.5, -1\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), -1\right) \cdot 1}{\frac{1}{2} \cdot x - \color{blue}{1}} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.8%

       \[\leadsto \frac{\mathsf{fma}\left(0.0625, \left(x \cdot x\right) \cdot \left(x \cdot x\right), -1\right) \cdot 1}{\mathsf{fma}\left(0.5, \color{blue}{x}, -1\right)} \]

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 69.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 5.6

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

   5. Applied rewrites 5.6%

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

   7. Applied rewrites 10.0%

      \[\leadsto \frac{\mathsf{fma}\left(0.0625, \left(x \cdot x\right) \cdot \left(x \cdot x\right), -1\right) \cdot 1}{\color{blue}{\mathsf{fma}\left(x, x \cdot 0.25, 1\right) \cdot \mathsf{fma}\left(x, 0.5, -1\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), -1\right) \cdot 1}{\frac{1}{2} \cdot x - \color{blue}{1}} \]
   9. Step-by-step derivation

   10. Applied rewrites 82.5%

       \[\leadsto \frac{\mathsf{fma}\left(0.0625, \left(x \cdot x\right) \cdot \left(x \cdot x\right), -1\right) \cdot 1}{\mathsf{fma}\left(0.5, \color{blue}{x}, -1\right)} \]

   11. Taylor expanded in x around inf

       \[\leadsto \frac{\frac{1}{16} \cdot {x}^{4}}{\mathsf{fma}\left(\color{blue}{\frac{1}{2}}, x, -1\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 82.5%

       \[\leadsto \frac{0.0625 \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right)}{\mathsf{fma}\left(\color{blue}{0.5}, x, -1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 73.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 40.4%

      \[\frac{e^{x} - 1}{x} \]
   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. lower-fma.f64 62.9

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

   5. Applied rewrites 62.9%

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

   7. Applied rewrites 62.5%

      \[\leadsto \frac{\mathsf{fma}\left(0.0625, \left(x \cdot x\right) \cdot \left(x \cdot x\right), -1\right) \cdot 1}{\color{blue}{\mathsf{fma}\left(x, x \cdot 0.25, 1\right) \cdot \mathsf{fma}\left(x, 0.5, -1\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), -1\right) \cdot 1}{\frac{1}{2} \cdot x - \color{blue}{1}} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.8%

       \[\leadsto \frac{\mathsf{fma}\left(0.0625, \left(x \cdot x\right) \cdot \left(x \cdot x\right), -1\right) \cdot 1}{\mathsf{fma}\left(0.5, \color{blue}{x}, -1\right)} \]

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 69.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 81.0

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

   5. Applied rewrites 81.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 82.5%

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

Alternative 5: 72.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 39.7%

      \[\frac{e^{x} - 1}{x} \]
   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 62.8

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

   5. Applied rewrites 62.8%

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

   7. Applied rewrites 62.4%

      \[\leadsto \frac{\mathsf{fma}\left(0.0625, \left(x \cdot x\right) \cdot \left(x \cdot x\right), -1\right) \cdot 1}{\color{blue}{\mathsf{fma}\left(x, x \cdot 0.25, 1\right) \cdot \mathsf{fma}\left(x, 0.5, -1\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), -1\right) \cdot 1}{\frac{1}{2} \cdot x - \color{blue}{1}} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.7%

       \[\leadsto \frac{\mathsf{fma}\left(0.0625, \left(x \cdot x\right) \cdot \left(x \cdot x\right), -1\right) \cdot 1}{\mathsf{fma}\left(0.5, \color{blue}{x}, -1\right)} \]

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 69.3%

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

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

   1. Initial program 99.2%

      \[\frac{e^{x} - 1}{x} \]
   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 74.3

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

   5. Applied rewrites 74.3%

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

Alternative 6: 72.0% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (- (exp x) 1.0) x) 2.0)
   (/ -1.0 (fma 0.5 x -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 = -1.0 / fma(0.5, x, -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 = Float64(-1.0 / fma(0.5, x, -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[(-1.0 / N[(0.5 * x + -1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 40.4%

      \[\frac{e^{x} - 1}{x} \]
   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. lower-fma.f64 62.9

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

   5. Applied rewrites 62.9%

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

   7. Applied rewrites 62.5%

      \[\leadsto \frac{\mathsf{fma}\left(0.0625, \left(x \cdot x\right) \cdot \left(x \cdot x\right), -1\right) \cdot 1}{\color{blue}{\mathsf{fma}\left(x, x \cdot 0.25, 1\right) \cdot \mathsf{fma}\left(x, 0.5, -1\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), -1\right) \cdot 1}{\frac{1}{2} \cdot x - \color{blue}{1}} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.8%

       \[\leadsto \frac{\mathsf{fma}\left(0.0625, \left(x \cdot x\right) \cdot \left(x \cdot x\right), -1\right) \cdot 1}{\mathsf{fma}\left(0.5, \color{blue}{x}, -1\right)} \]

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 69.3%

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

   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 73.9

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

   5. Applied rewrites 73.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 73.9%

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

Alternative 7: 68.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (((exp(x) - 1.0) / x) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = fma(x, 0.041666666666666664, 0.16666666666666666) * (x * x);
	}
	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(fma(x, 0.041666666666666664, 0.16666666666666666) * 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], 1.0, N[(N[(x * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 40.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 63.9%

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

   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

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} \]
   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 73.9

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

   5. Applied rewrites 73.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 73.9%

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

Alternative 8: 68.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (((exp(x) - 1.0) / x) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = 0.041666666666666664 * (x * (x * x));
	}
	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 = 0.041666666666666664d0 * (x * (x * x))
    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 = 0.041666666666666664 * (x * (x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((math.exp(x) - 1.0) / x) <= 2.0:
		tmp = 1.0
	else:
		tmp = 0.041666666666666664 * (x * (x * x))
	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(0.041666666666666664 * Float64(x * Float64(x * x)));
	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 = 0.041666666666666664 * (x * (x * x));
	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[(0.041666666666666664 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 40.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 63.9%

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

   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

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} \]
   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 73.9

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

   5. Applied rewrites 73.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 73.9%

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

Alternative 9: 64.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	tmp = 0
	if ((math.exp(x) - 1.0) / x) <= 2.0:
		tmp = 1.0
	else:
		tmp = 0.16666666666666666 * (x * x)
	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(0.16666666666666666 * Float64(x * x));
	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 = 0.16666666666666666 * (x * x);
	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[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 40.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 63.9%

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

   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 55.4

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

   5. Applied rewrites 55.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 55.4%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 56.6%

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

Alternative 10: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right)\\ t_1 := \mathsf{fma}\left(x, x \cdot t\_0, -x\right)\\ \mathbf{if}\;x \leq 5 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(0.5, x, -1\right)}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_0, x \cdot x, x\right) \cdot t\_1}{t\_1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x (fma x 0.041666666666666664 0.16666666666666666) 0.5))
        (t_1 (fma x (* x t_0) (- x))))
   (if (<= x 5e-32)
     (/ -1.0 (fma 0.5 x -1.0))
     (if (<= x 1.2e+77)
       (/ (/ (* (fma t_0 (* x x) x) t_1) t_1) x)
       (/ (* 0.041666666666666664 (* (* x (* x x)) x)) x)))))

C

double code(double x) {
	double t_0 = fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5);
	double t_1 = fma(x, (x * t_0), -x);
	double tmp;
	if (x <= 5e-32) {
		tmp = -1.0 / fma(0.5, x, -1.0);
	} else if (x <= 1.2e+77) {
		tmp = ((fma(t_0, (x * x), x) * t_1) / t_1) / x;
	} else {
		tmp = (0.041666666666666664 * ((x * (x * x)) * x)) / x;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5)
	t_1 = fma(x, Float64(x * t_0), Float64(-x))
	tmp = 0.0
	if (x <= 5e-32)
		tmp = Float64(-1.0 / fma(0.5, x, -1.0));
	elseif (x <= 1.2e+77)
		tmp = Float64(Float64(Float64(fma(t_0, Float64(x * x), x) * t_1) / t_1) / x);
	else
		tmp = Float64(Float64(0.041666666666666664 * Float64(Float64(x * Float64(x * x)) * x)) / x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * t$95$0), $MachinePrecision] + (-x)), $MachinePrecision]}, If[LessEqual[x, 5e-32], N[(-1.0 / N[(0.5 * x + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2e+77], N[(N[(N[(N[(t$95$0 * N[(x * x), $MachinePrecision] + x), $MachinePrecision] * t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision] / x), $MachinePrecision], N[(N[(0.041666666666666664 * N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 5e-32

   1. Initial program 39.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 62.6

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

   5. Applied rewrites 62.6%

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

   7. Applied rewrites 62.2%

      \[\leadsto \frac{\mathsf{fma}\left(0.0625, \left(x \cdot x\right) \cdot \left(x \cdot x\right), -1\right) \cdot 1}{\color{blue}{\mathsf{fma}\left(x, x \cdot 0.25, 1\right) \cdot \mathsf{fma}\left(x, 0.5, -1\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), -1\right) \cdot 1}{\frac{1}{2} \cdot x - \color{blue}{1}} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.5%

       \[\leadsto \frac{\mathsf{fma}\left(0.0625, \left(x \cdot x\right) \cdot \left(x \cdot x\right), -1\right) \cdot 1}{\mathsf{fma}\left(0.5, \color{blue}{x}, -1\right)} \]

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 69.2%

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

   if 5e-32 < x < 1.1999999999999999e77

   1. Initial program 90.5%

      \[\frac{e^{x} - 1}{x} \]
   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 25.6

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

   5. Applied rewrites 25.6%

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

   7. Applied rewrites 49.3%

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

   if 1.1999999999999999e77 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

Alternative 11: 67.3% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.5%

   \[\frac{e^{x} - 1}{x} \]
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 65.8

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

5. Applied rewrites 65.8%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 65.3%

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

Alternative 12: 64.5% 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 55.5%

   \[\frac{e^{x} - 1}{x} \]
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 61.6

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

5. Applied rewrites 61.6%

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

Alternative 13: 51.8% 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 55.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 48.5%

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

Developer Target 1: 51.9% 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 2024254 
(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))