Hyperbolic sine

Percentage Accurate: 53.4% → 100.0%

Time: 12.0s

Alternatives: 13

Speedup: 12.8×

• 48.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{e^{x} - e^{-x}}{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (exp x) (exp (- x))) 2.0))

C

double code(double x) {
	return (exp(x) - exp(-x)) / 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (exp(x) - exp(-x)) / 2.0d0
end function

Java

public static double code(double x) {
	return (Math.exp(x) - Math.exp(-x)) / 2.0;
}

Python

def code(x):
	return (math.exp(x) - math.exp(-x)) / 2.0

Julia

function code(x)
	return Float64(Float64(exp(x) - exp(Float64(-x))) / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - exp(-x)) / 2.0;
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Initial Program: 53.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x} - e^{-x}}{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (exp x) (exp (- x))) 2.0))

C

double code(double x) {
	return (exp(x) - exp(-x)) / 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (exp(x) - exp(-x)) / 2.0d0
end function

Java

public static double code(double x) {
	return (Math.exp(x) - Math.exp(-x)) / 2.0;
}

Python

def code(x):
	return (math.exp(x) - math.exp(-x)) / 2.0

Julia

function code(x)
	return Float64(Float64(exp(x) - exp(Float64(-x))) / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - exp(-x)) / 2.0;
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Alternative 1: 100.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sinh x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sinh x))

C

double code(double x) {
	return sinh(x);
}

Fortran

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

Java

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

Python

def code(x):
	return math.sinh(x)

Julia

function code(x)
	return sinh(x)
end

MATLAB

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

Wolfram

code[x_] := N[Sinh[x], $MachinePrecision]

Derivation

1. Initial program 56.4%

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

   1. sinh-def N/A

      \[\leadsto \color{blue}{\sinh x} \]

   2. sinh-lowering-sinh.f64 100.0

      \[\leadsto \color{blue}{\sinh x} \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\sinh x} \]
5. Add Preprocessing

Alternative 2: 88.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.5:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (exp x) (exp (- x))) 0.5)
   (* x (fma x (* x 0.16666666666666666) 1.0))
   (* (* x (* x x)) (fma x (* x 0.008333333333333333) 0.16666666666666666))))

C

double code(double x) {
	double tmp;
	if ((exp(x) - exp(-x)) <= 0.5) {
		tmp = x * fma(x, (x * 0.16666666666666666), 1.0);
	} else {
		tmp = (x * (x * x)) * fma(x, (x * 0.008333333333333333), 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) - exp(Float64(-x))) <= 0.5)
		tmp = Float64(x * fma(x, Float64(x * 0.16666666666666666), 1.0));
	else
		tmp = Float64(Float64(x * Float64(x * x)) * fma(x, Float64(x * 0.008333333333333333), 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 0.5], N[(x * N[(x * N[(x * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 0.5

   1. Initial program 40.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + 1\right)} \]

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

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)} \]

      4. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right) \]

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

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right) \]

      6. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}, 1\right) \]

      8. associate-*l* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)} + \frac{1}{6}, 1\right) \]

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

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), \frac{1}{6}\right)}, 1\right) \]

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

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)}, \frac{1}{6}\right), 1\right) \]

      11. +-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{5040} \cdot {x}^{2} + \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      13. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{5040} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      14. associate-*l* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{5040}\right)} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

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

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

      16. *-lowering-*.f64 94.9

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.0001984126984126984}, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   5. Simplified 94.9%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
   6. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\frac{\left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) - \frac{1}{120} \cdot \frac{1}{120}}{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}}, \frac{1}{6}\right), 1\right) \]

      2. clear-num N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\frac{1}{\frac{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}{\left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) - \frac{1}{120} \cdot \frac{1}{120}}}}, \frac{1}{6}\right), 1\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\frac{1}{\frac{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}{\left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) - \frac{1}{120} \cdot \frac{1}{120}}}}, \frac{1}{6}\right), 1\right) \]

      4. clear-num N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{\color{blue}{\frac{1}{\frac{\left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) - \frac{1}{120} \cdot \frac{1}{120}}{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}}}}, \frac{1}{6}\right), 1\right) \]

      5. flip-+ N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{\frac{1}{\color{blue}{x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}}}}, \frac{1}{6}\right), 1\right) \]

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

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{\color{blue}{\frac{1}{x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}}}}, \frac{1}{6}\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{\frac{1}{\color{blue}{\left(x \cdot x\right) \cdot \frac{1}{5040}} + \frac{1}{120}}}, \frac{1}{6}\right), 1\right) \]

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

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{\frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{5040}, \frac{1}{120}\right)}}}, \frac{1}{6}\right), 1\right) \]

      9. *-lowering-*.f64 94.9

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{\frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.0001984126984126984, 0.008333333333333333\right)}}, 0.16666666666666666\right), 1\right) \]

   7. Applied egg-rr 94.9%

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right)}}}, 0.16666666666666666\right), 1\right) \]

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 88.7

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

   10. Simplified 88.7%

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

   if 0.5 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + 1\right)} \]

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

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)} \]

      4. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right) \]

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

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right) \]

      6. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}, 1\right) \]

      8. associate-*l* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)} + \frac{1}{6}, 1\right) \]

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

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), \frac{1}{6}\right)}, 1\right) \]

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

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)}, \frac{1}{6}\right), 1\right) \]

      11. +-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{5040} \cdot {x}^{2} + \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      13. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{5040} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      14. associate-*l* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{5040}\right)} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

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

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

      16. *-lowering-*.f64 76.1

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.0001984126984126984}, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   5. Simplified 76.1%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{120} \cdot x}, \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{120}}, \frac{1}{6}\right), 1\right) \]

      2. *-lowering-*.f64 74.4

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.008333333333333333}, 0.16666666666666666\right), 1\right) \]

   8. Simplified 74.4%

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.008333333333333333}, 0.16666666666666666\right), 1\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{5} \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{{x}^{5} \cdot \frac{1}{120} + {x}^{5} \cdot \left(\frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)} \]

       2. metadata-eval N/A

          \[\leadsto {x}^{\color{blue}{\left(4 + 1\right)}} \cdot \frac{1}{120} + {x}^{5} \cdot \left(\frac{1}{6} \cdot \frac{1}{{x}^{2}}\right) \]

       3. pow-plus N/A

          \[\leadsto \color{blue}{\left({x}^{4} \cdot x\right)} \cdot \frac{1}{120} + {x}^{5} \cdot \left(\frac{1}{6} \cdot \frac{1}{{x}^{2}}\right) \]

       4. metadata-eval N/A

          \[\leadsto \left({x}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot x\right) \cdot \frac{1}{120} + {x}^{5} \cdot \left(\frac{1}{6} \cdot \frac{1}{{x}^{2}}\right) \]

       5. pow-sqr N/A

          \[\leadsto \left(\color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \cdot x\right) \cdot \frac{1}{120} + {x}^{5} \cdot \left(\frac{1}{6} \cdot \frac{1}{{x}^{2}}\right) \]

       6. associate-*r* N/A

          \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot x\right)\right)} \cdot \frac{1}{120} + {x}^{5} \cdot \left(\frac{1}{6} \cdot \frac{1}{{x}^{2}}\right) \]

       7. unpow2 N/A

          \[\leadsto \left({x}^{2} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right)\right) \cdot \frac{1}{120} + {x}^{5} \cdot \left(\frac{1}{6} \cdot \frac{1}{{x}^{2}}\right) \]

       8. unpow3 N/A

          \[\leadsto \left({x}^{2} \cdot \color{blue}{{x}^{3}}\right) \cdot \frac{1}{120} + {x}^{5} \cdot \left(\frac{1}{6} \cdot \frac{1}{{x}^{2}}\right) \]

       9. *-commutative N/A

          \[\leadsto \color{blue}{\left({x}^{3} \cdot {x}^{2}\right)} \cdot \frac{1}{120} + {x}^{5} \cdot \left(\frac{1}{6} \cdot \frac{1}{{x}^{2}}\right) \]

       10. associate-*r* N/A

           \[\leadsto \color{blue}{{x}^{3} \cdot \left({x}^{2} \cdot \frac{1}{120}\right)} + {x}^{5} \cdot \left(\frac{1}{6} \cdot \frac{1}{{x}^{2}}\right) \]

       11. *-commutative N/A

           \[\leadsto {x}^{3} \cdot \color{blue}{\left(\frac{1}{120} \cdot {x}^{2}\right)} + {x}^{5} \cdot \left(\frac{1}{6} \cdot \frac{1}{{x}^{2}}\right) \]

       12. associate-*r/ N/A

           \[\leadsto {x}^{3} \cdot \left(\frac{1}{120} \cdot {x}^{2}\right) + {x}^{5} \cdot \color{blue}{\frac{\frac{1}{6} \cdot 1}{{x}^{2}}} \]

       13. metadata-eval N/A

           \[\leadsto {x}^{3} \cdot \left(\frac{1}{120} \cdot {x}^{2}\right) + {x}^{5} \cdot \frac{\color{blue}{\frac{1}{6}}}{{x}^{2}} \]

       14. associate-*r/ N/A

           \[\leadsto {x}^{3} \cdot \left(\frac{1}{120} \cdot {x}^{2}\right) + \color{blue}{\frac{{x}^{5} \cdot \frac{1}{6}}{{x}^{2}}} \]

       15. associate-*l/ N/A

           \[\leadsto {x}^{3} \cdot \left(\frac{1}{120} \cdot {x}^{2}\right) + \color{blue}{\frac{{x}^{5}}{{x}^{2}} \cdot \frac{1}{6}} \]

   11. Simplified 74.4%

       \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 88.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.5:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.008333333333333333 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (exp x) (exp (- x))) 0.5)
   (* x (fma x (* x 0.16666666666666666) 1.0))
   (* 0.008333333333333333 (* x (* x (* x (* x x)))))))

C

double code(double x) {
	double tmp;
	if ((exp(x) - exp(-x)) <= 0.5) {
		tmp = x * fma(x, (x * 0.16666666666666666), 1.0);
	} else {
		tmp = 0.008333333333333333 * (x * (x * (x * (x * x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) - exp(Float64(-x))) <= 0.5)
		tmp = Float64(x * fma(x, Float64(x * 0.16666666666666666), 1.0));
	else
		tmp = Float64(0.008333333333333333 * Float64(x * Float64(x * Float64(x * Float64(x * x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 0.5], N[(x * N[(x * N[(x * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.008333333333333333 * N[(x * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 0.5

   1. Initial program 40.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + 1\right)} \]

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

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)} \]

      4. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right) \]

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

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right) \]

      6. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}, 1\right) \]

      8. associate-*l* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)} + \frac{1}{6}, 1\right) \]

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

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), \frac{1}{6}\right)}, 1\right) \]

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

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)}, \frac{1}{6}\right), 1\right) \]

      11. +-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{5040} \cdot {x}^{2} + \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      13. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{5040} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      14. associate-*l* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{5040}\right)} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

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

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

      16. *-lowering-*.f64 94.9

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.0001984126984126984}, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   5. Simplified 94.9%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
   6. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\frac{\left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) - \frac{1}{120} \cdot \frac{1}{120}}{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}}, \frac{1}{6}\right), 1\right) \]

      2. clear-num N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\frac{1}{\frac{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}{\left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) - \frac{1}{120} \cdot \frac{1}{120}}}}, \frac{1}{6}\right), 1\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\frac{1}{\frac{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}{\left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) - \frac{1}{120} \cdot \frac{1}{120}}}}, \frac{1}{6}\right), 1\right) \]

      4. clear-num N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{\color{blue}{\frac{1}{\frac{\left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) - \frac{1}{120} \cdot \frac{1}{120}}{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}}}}, \frac{1}{6}\right), 1\right) \]

      5. flip-+ N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{\frac{1}{\color{blue}{x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}}}}, \frac{1}{6}\right), 1\right) \]

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

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{\color{blue}{\frac{1}{x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}}}}, \frac{1}{6}\right), 1\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{\frac{1}{\color{blue}{\left(x \cdot x\right) \cdot \frac{1}{5040}} + \frac{1}{120}}}, \frac{1}{6}\right), 1\right) \]

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

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{\frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{5040}, \frac{1}{120}\right)}}}, \frac{1}{6}\right), 1\right) \]

      9. *-lowering-*.f64 94.9

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{\frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.0001984126984126984, 0.008333333333333333\right)}}, 0.16666666666666666\right), 1\right) \]

   7. Applied egg-rr 94.9%

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right)}}}, 0.16666666666666666\right), 1\right) \]

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 88.7

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

   10. Simplified 88.7%

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

   if 0.5 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) \cdot x} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + 1\right)} \cdot x \]

      3. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) \cdot x + x} \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x\right)} + x \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x, x\right)} \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x, x\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x, x\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}, x\right) \]

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

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}, x\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} + \frac{1}{6}\right)}, x\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{120}} + \frac{1}{6}\right), x\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} + \frac{1}{6}\right), x\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{120}\right)} + \frac{1}{6}\right), x\right) \]

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

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{120}, \frac{1}{6}\right)}, x\right) \]

      15. *-lowering-*.f64 74.4

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.008333333333333333}, 0.16666666666666666\right), x\right) \]

   5. Simplified 74.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), x\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\frac{1}{120} \cdot {x}^{2}\right)}, x\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{120}\right)}, x\right) \]

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

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{120}\right)}, x\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120}\right), x\right) \]

      4. *-lowering-*.f64 74.4

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot 0.008333333333333333\right), x\right) \]

   8. Simplified 74.4%

      \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot 0.008333333333333333\right)}, x\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{120} \cdot {x}^{5}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{1}{120} \cdot {x}^{5}} \]

       2. metadata-eval N/A

          \[\leadsto \frac{1}{120} \cdot {x}^{\color{blue}{\left(4 + 1\right)}} \]

       3. pow-plus N/A

          \[\leadsto \frac{1}{120} \cdot \color{blue}{\left({x}^{4} \cdot x\right)} \]

       4. *-commutative N/A

          \[\leadsto \frac{1}{120} \cdot \color{blue}{\left(x \cdot {x}^{4}\right)} \]

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

          \[\leadsto \frac{1}{120} \cdot \color{blue}{\left(x \cdot {x}^{4}\right)} \]

       6. metadata-eval N/A

          \[\leadsto \frac{1}{120} \cdot \left(x \cdot {x}^{\color{blue}{\left(3 + 1\right)}}\right) \]

       7. pow-plus N/A

          \[\leadsto \frac{1}{120} \cdot \left(x \cdot \color{blue}{\left({x}^{3} \cdot x\right)}\right) \]

       8. *-commutative N/A

          \[\leadsto \frac{1}{120} \cdot \left(x \cdot \color{blue}{\left(x \cdot {x}^{3}\right)}\right) \]

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

          \[\leadsto \frac{1}{120} \cdot \left(x \cdot \color{blue}{\left(x \cdot {x}^{3}\right)}\right) \]

       10. cube-mult N/A

           \[\leadsto \frac{1}{120} \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \frac{1}{120} \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

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

           \[\leadsto \frac{1}{120} \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \frac{1}{120} \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

       14. *-lowering-*.f64 74.4

           \[\leadsto 0.008333333333333333 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

   11. Simplified 74.4%

       \[\leadsto \color{blue}{0.008333333333333333 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (exp x) (exp (- x))) 0.5) x (* 0.16666666666666666 (* x (* x x)))))

C

double code(double x) {
	double tmp;
	if ((exp(x) - exp(-x)) <= 0.5) {
		tmp = x;
	} else {
		tmp = 0.16666666666666666 * (x * (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((exp(x) - exp(-x)) <= 0.5d0) then
        tmp = x
    else
        tmp = 0.16666666666666666d0 * (x * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((Math.exp(x) - Math.exp(-x)) <= 0.5) {
		tmp = x;
	} else {
		tmp = 0.16666666666666666 * (x * (x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (math.exp(x) - math.exp(-x)) <= 0.5:
		tmp = x
	else:
		tmp = 0.16666666666666666 * (x * (x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) - exp(Float64(-x))) <= 0.5)
		tmp = x;
	else
		tmp = Float64(0.16666666666666666 * Float64(x * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((exp(x) - exp(-x)) <= 0.5)
		tmp = x;
	else
		tmp = 0.16666666666666666 * (x * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 0.5

   1. Initial program 40.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 66.5%

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

   if 0.5 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 60.4

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

   5. Simplified 60.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{3}} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{3}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 60.4

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

   8. Simplified 60.4%

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

Alternative 5: 86.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right)\\ t_2 := x \cdot t\_1\\ \mathbf{if}\;x \leq 5:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \mathsf{fma}\left(-x, \frac{x}{\mathsf{fma}\left(x, x \cdot 2.857142857142857, -120\right)}, 0.16666666666666666\right), x \cdot x, x\right)\\ \mathbf{elif}\;x \leq 10^{+61}:\\ \;\;\;\;\frac{t\_0 \cdot \mathsf{fma}\left(t\_2, \left(x \cdot x\right) \cdot t\_2, -0.027777777777777776\right)}{\mathsf{fma}\left(x \cdot x, t\_1, -0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;0.008333333333333333 \cdot \left(x \cdot \left(x \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x)))
        (t_1 (fma x (* x 0.0001984126984126984) 0.008333333333333333))
        (t_2 (* x t_1)))
   (if (<= x 5.0)
     (fma
      (*
       x
       (fma
        (- x)
        (/ x (fma x (* x 2.857142857142857) -120.0))
        0.16666666666666666))
      (* x x)
      x)
     (if (<= x 1e+61)
       (/
        (* t_0 (fma t_2 (* (* x x) t_2) -0.027777777777777776))
        (fma (* x x) t_1 -0.16666666666666666))
       (* 0.008333333333333333 (* x (* x t_0)))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = fma(x, (x * 0.0001984126984126984), 0.008333333333333333);
	double t_2 = x * t_1;
	double tmp;
	if (x <= 5.0) {
		tmp = fma((x * fma(-x, (x / fma(x, (x * 2.857142857142857), -120.0)), 0.16666666666666666)), (x * x), x);
	} else if (x <= 1e+61) {
		tmp = (t_0 * fma(t_2, ((x * x) * t_2), -0.027777777777777776)) / fma((x * x), t_1, -0.16666666666666666);
	} else {
		tmp = 0.008333333333333333 * (x * (x * t_0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	t_1 = fma(x, Float64(x * 0.0001984126984126984), 0.008333333333333333)
	t_2 = Float64(x * t_1)
	tmp = 0.0
	if (x <= 5.0)
		tmp = fma(Float64(x * fma(Float64(-x), Float64(x / fma(x, Float64(x * 2.857142857142857), -120.0)), 0.16666666666666666)), Float64(x * x), x);
	elseif (x <= 1e+61)
		tmp = Float64(Float64(t_0 * fma(t_2, Float64(Float64(x * x) * t_2), -0.027777777777777776)) / fma(Float64(x * x), t_1, -0.16666666666666666));
	else
		tmp = Float64(0.008333333333333333 * Float64(x * Float64(x * t_0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]}, Block[{t$95$2 = N[(x * t$95$1), $MachinePrecision]}, If[LessEqual[x, 5.0], N[(N[(x * N[((-x) * N[(x / N[(x * N[(x * 2.857142857142857), $MachinePrecision] + -120.0), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[x, 1e+61], N[(N[(t$95$0 * N[(t$95$2 * N[(N[(x * x), $MachinePrecision] * t$95$2), $MachinePrecision] + -0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * t$95$1 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(0.008333333333333333 * N[(x * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 5

   1. Initial program 40.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + 1\right)} \]

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

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)} \]

      4. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right) \]

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

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right) \]

      6. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}, 1\right) \]

      8. associate-*l* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)} + \frac{1}{6}, 1\right) \]

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

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), \frac{1}{6}\right)}, 1\right) \]

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

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)}, \frac{1}{6}\right), 1\right) \]

      11. +-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{5040} \cdot {x}^{2} + \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      13. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{5040} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      14. associate-*l* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{5040}\right)} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

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

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

      16. *-lowering-*.f64 94.9

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.0001984126984126984}, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   5. Simplified 94.9%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right) + x \cdot 1} \]

      2. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) + \frac{1}{6}\right) \cdot \left(x \cdot x\right)\right)} + x \cdot 1 \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right) \cdot \left(x \cdot x\right)} + x \cdot 1 \]

      4. *-rgt-identity N/A

         \[\leadsto \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right) \cdot \left(x \cdot x\right) + \color{blue}{x} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) + \frac{1}{6}\right), x \cdot x, x\right)} \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) + \frac{1}{6}\right)}, x \cdot x, x\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)} + \frac{1}{6}\right), x \cdot x, x\right) \]

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

         \[\leadsto \mathsf{fma}\left(x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}, \frac{1}{6}\right)}, x \cdot x, x\right) \]

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

         \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}, \frac{1}{6}\right), x \cdot x, x\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right) \cdot \frac{1}{5040}} + \frac{1}{120}, \frac{1}{6}\right), x \cdot x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), x \cdot x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), x \cdot x, x\right) \]

      13. *-lowering-*.f64 94.9

          \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \color{blue}{x \cdot x}, x\right) \]

   7. Applied egg-rr 94.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), x \cdot x, x\right)} \]
   8. Step-by-step derivation

      1. remove-double-div N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\frac{1}{\frac{1}{\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}}}} + \frac{1}{6}\right), x \cdot x, x\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\color{blue}{\frac{x \cdot x}{\frac{1}{\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}}}} + \frac{1}{6}\right), x \cdot x, x\right) \]

      3. remove-double-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{x \cdot x}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}}\right)\right)\right)}} + \frac{1}{6}\right), x \cdot x, x\right) \]

      4. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{x \cdot x}{\color{blue}{-1 \cdot \left(\mathsf{neg}\left(\frac{1}{\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}}\right)\right)}} + \frac{1}{6}\right), x \cdot x, x\right) \]

      5. times-frac N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\color{blue}{\frac{x}{-1} \cdot \frac{x}{\mathsf{neg}\left(\frac{1}{\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}}\right)}} + \frac{1}{6}\right), x \cdot x, x\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\color{blue}{\left(x \cdot \frac{1}{-1}\right)} \cdot \frac{x}{\mathsf{neg}\left(\frac{1}{\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}}\right)} + \frac{1}{6}\right), x \cdot x, x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\left(x \cdot \color{blue}{-1}\right) \cdot \frac{x}{\mathsf{neg}\left(\frac{1}{\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}}\right)} + \frac{1}{6}\right), x \cdot x, x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\left(x \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \cdot \frac{x}{\mathsf{neg}\left(\frac{1}{\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}}\right)} + \frac{1}{6}\right), x \cdot x, x\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(x \cdot 1\right)\right)} \cdot \frac{x}{\mathsf{neg}\left(\frac{1}{\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}}\right)} + \frac{1}{6}\right), x \cdot x, x\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \left(\left(\mathsf{neg}\left(\color{blue}{x}\right)\right) \cdot \frac{x}{\mathsf{neg}\left(\frac{1}{\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}}\right)} + \frac{1}{6}\right), x \cdot x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{x}{\mathsf{neg}\left(\frac{1}{\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}}\right)}, \frac{1}{6}\right)}, x \cdot x, x\right) \]

   9. Applied egg-rr 94.9%

      \[\leadsto \mathsf{fma}\left(x \cdot \color{blue}{\mathsf{fma}\left(-x, \frac{x}{\frac{1}{\mathsf{fma}\left(x \cdot x, -0.0001984126984126984, -0.008333333333333333\right)}}, 0.16666666666666666\right)}, x \cdot x, x\right) \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{x}{\color{blue}{\frac{20}{7} \cdot {x}^{2} - 120}}, \frac{1}{6}\right), x \cdot x, x\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{x}{\color{blue}{\frac{20}{7} \cdot {x}^{2} + \left(\mathsf{neg}\left(120\right)\right)}}, \frac{1}{6}\right), x \cdot x, x\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{x}{\frac{20}{7} \cdot \color{blue}{\left(x \cdot x\right)} + \left(\mathsf{neg}\left(120\right)\right)}, \frac{1}{6}\right), x \cdot x, x\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{x}{\color{blue}{\left(\frac{20}{7} \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(120\right)\right)}, \frac{1}{6}\right), x \cdot x, x\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{x}{\color{blue}{x \cdot \left(\frac{20}{7} \cdot x\right)} + \left(\mathsf{neg}\left(120\right)\right)}, \frac{1}{6}\right), x \cdot x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{x}{\color{blue}{\mathsf{fma}\left(x, \frac{20}{7} \cdot x, \mathsf{neg}\left(120\right)\right)}}, \frac{1}{6}\right), x \cdot x, x\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{x}{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{20}{7}}, \mathsf{neg}\left(120\right)\right)}, \frac{1}{6}\right), x \cdot x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(\mathsf{neg}\left(x\right), \frac{x}{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{20}{7}}, \mathsf{neg}\left(120\right)\right)}, \frac{1}{6}\right), x \cdot x, x\right) \]

       8. metadata-eval 82.5

          \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(-x, \frac{x}{\mathsf{fma}\left(x, x \cdot 2.857142857142857, \color{blue}{-120}\right)}, 0.16666666666666666\right), x \cdot x, x\right) \]

   12. Simplified 82.5%

       \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(-x, \frac{x}{\color{blue}{\mathsf{fma}\left(x, x \cdot 2.857142857142857, -120\right)}}, 0.16666666666666666\right), x \cdot x, x\right) \]

   if 5 < x < 9.99999999999999949e60

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + 1\right)} \]

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

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)} \]

      4. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right) \]

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

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right) \]

      6. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}, 1\right) \]

      8. associate-*l* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)} + \frac{1}{6}, 1\right) \]

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

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), \frac{1}{6}\right)}, 1\right) \]

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

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)}, \frac{1}{6}\right), 1\right) \]

      11. +-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{5040} \cdot {x}^{2} + \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      13. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{5040} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      14. associate-*l* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{5040}\right)} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

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

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

      16. *-lowering-*.f64 11.1

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.0001984126984126984}, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   5. Simplified 11.1%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left({x}^{6} \cdot \left(\frac{1}{5040} + \left(\frac{\frac{1}{6}}{{x}^{4}} + \frac{1}{120} \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

   8. Simplified 11.1%

      \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right)} \]

   10. Applied egg-rr 57.5%

       \[\leadsto \color{blue}{\frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), \left(x \cdot x\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right)\right), -0.027777777777777776\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right)}} \]

   if 9.99999999999999949e60 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) \cdot x} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + 1\right)} \cdot x \]

      3. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) \cdot x + x} \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x\right)} + x \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x, x\right)} \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x, x\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x, x\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}, x\right) \]

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

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}, x\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} + \frac{1}{6}\right)}, x\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{120}} + \frac{1}{6}\right), x\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} + \frac{1}{6}\right), x\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{120}\right)} + \frac{1}{6}\right), x\right) \]

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

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{120}, \frac{1}{6}\right)}, x\right) \]

      15. *-lowering-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.008333333333333333}, 0.16666666666666666\right), x\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), x\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\frac{1}{120} \cdot {x}^{2}\right)}, x\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{120}\right)}, x\right) \]

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

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{120}\right)}, x\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120}\right), x\right) \]

      4. *-lowering-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot 0.008333333333333333\right), x\right) \]

   8. Simplified 100.0%

      \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot 0.008333333333333333\right)}, x\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{120} \cdot {x}^{5}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{1}{120} \cdot {x}^{5}} \]

       2. metadata-eval N/A

          \[\leadsto \frac{1}{120} \cdot {x}^{\color{blue}{\left(4 + 1\right)}} \]

       3. pow-plus N/A

          \[\leadsto \frac{1}{120} \cdot \color{blue}{\left({x}^{4} \cdot x\right)} \]

       4. *-commutative N/A

          \[\leadsto \frac{1}{120} \cdot \color{blue}{\left(x \cdot {x}^{4}\right)} \]

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

          \[\leadsto \frac{1}{120} \cdot \color{blue}{\left(x \cdot {x}^{4}\right)} \]

       6. metadata-eval N/A

          \[\leadsto \frac{1}{120} \cdot \left(x \cdot {x}^{\color{blue}{\left(3 + 1\right)}}\right) \]

       7. pow-plus N/A

          \[\leadsto \frac{1}{120} \cdot \left(x \cdot \color{blue}{\left({x}^{3} \cdot x\right)}\right) \]

       8. *-commutative N/A

          \[\leadsto \frac{1}{120} \cdot \left(x \cdot \color{blue}{\left(x \cdot {x}^{3}\right)}\right) \]

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

          \[\leadsto \frac{1}{120} \cdot \left(x \cdot \color{blue}{\left(x \cdot {x}^{3}\right)}\right) \]

       10. cube-mult N/A

           \[\leadsto \frac{1}{120} \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \frac{1}{120} \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

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

           \[\leadsto \frac{1}{120} \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \frac{1}{120} \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

       14. *-lowering-*.f64 100.0

           \[\leadsto 0.008333333333333333 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{0.008333333333333333 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 76.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, 0.16666666666666666\right)\\ \mathbf{if}\;x \leq 10^{+61}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot t\_0, t\_0, -1\right)}{\mathsf{fma}\left(x, t\_0, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.008333333333333333 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (fma (* x x) 0.008333333333333333 0.16666666666666666))))
   (if (<= x 1e+61)
     (/ (* x (fma (* (* x x) t_0) t_0 -1.0)) (fma x t_0 -1.0))
     (* 0.008333333333333333 (* x (* x (* x (* x x))))))))

C

double code(double x) {
	double t_0 = x * fma((x * x), 0.008333333333333333, 0.16666666666666666);
	double tmp;
	if (x <= 1e+61) {
		tmp = (x * fma(((x * x) * t_0), t_0, -1.0)) / fma(x, t_0, -1.0);
	} else {
		tmp = 0.008333333333333333 * (x * (x * (x * (x * x))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * fma(Float64(x * x), 0.008333333333333333, 0.16666666666666666))
	tmp = 0.0
	if (x <= 1e+61)
		tmp = Float64(Float64(x * fma(Float64(Float64(x * x) * t_0), t_0, -1.0)) / fma(x, t_0, -1.0));
	else
		tmp = Float64(0.008333333333333333 * Float64(x * Float64(x * Float64(x * Float64(x * x)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(N[(x * x), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1e+61], N[(N[(x * N[(N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] / N[(x * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(0.008333333333333333 * N[(x * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999949e60

   1. Initial program 46.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + 1\right)} \]

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

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)} \]

      4. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right) \]

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

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right) \]

      6. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}, 1\right) \]

      8. associate-*l* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)} + \frac{1}{6}, 1\right) \]

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

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), \frac{1}{6}\right)}, 1\right) \]

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

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)}, \frac{1}{6}\right), 1\right) \]

      11. +-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{5040} \cdot {x}^{2} + \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      13. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{5040} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      14. associate-*l* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{5040}\right)} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

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

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

      16. *-lowering-*.f64 87.6

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.0001984126984126984}, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   5. Simplified 87.6%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{120} \cdot x}, \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{120}}, \frac{1}{6}\right), 1\right) \]

      2. *-lowering-*.f64 85.6

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.008333333333333333}, 0.16666666666666666\right), 1\right) \]

   8. Simplified 85.6%

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.008333333333333333}, 0.16666666666666666\right), 1\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right) + 1\right) \cdot x} \]

      2. flip-+ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right) - 1 \cdot 1}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right) - 1}} \cdot x \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right) - 1 \cdot 1\right) \cdot x}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right) - 1}} \]

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

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right) - 1 \cdot 1\right) \cdot x}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right) - 1}} \]

   10. Applied egg-rr 65.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, 0.16666666666666666\right)\right), x \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, 0.16666666666666666\right), -1\right) \cdot x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, 0.16666666666666666\right), -1\right)}} \]

   if 9.99999999999999949e60 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) \cdot x} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + 1\right)} \cdot x \]

      3. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) \cdot x + x} \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x\right)} + x \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x, x\right)} \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x, x\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x, x\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}, x\right) \]

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

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}, x\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} + \frac{1}{6}\right)}, x\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{120}} + \frac{1}{6}\right), x\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} + \frac{1}{6}\right), x\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{120}\right)} + \frac{1}{6}\right), x\right) \]

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

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{120}, \frac{1}{6}\right)}, x\right) \]

      15. *-lowering-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.008333333333333333}, 0.16666666666666666\right), x\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), x\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\frac{1}{120} \cdot {x}^{2}\right)}, x\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{120}\right)}, x\right) \]

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

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{120}\right)}, x\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120}\right), x\right) \]

      4. *-lowering-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot 0.008333333333333333\right), x\right) \]

   8. Simplified 100.0%

      \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot 0.008333333333333333\right)}, x\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{120} \cdot {x}^{5}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{1}{120} \cdot {x}^{5}} \]

       2. metadata-eval N/A

          \[\leadsto \frac{1}{120} \cdot {x}^{\color{blue}{\left(4 + 1\right)}} \]

       3. pow-plus N/A

          \[\leadsto \frac{1}{120} \cdot \color{blue}{\left({x}^{4} \cdot x\right)} \]

       4. *-commutative N/A

          \[\leadsto \frac{1}{120} \cdot \color{blue}{\left(x \cdot {x}^{4}\right)} \]

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

          \[\leadsto \frac{1}{120} \cdot \color{blue}{\left(x \cdot {x}^{4}\right)} \]

       6. metadata-eval N/A

          \[\leadsto \frac{1}{120} \cdot \left(x \cdot {x}^{\color{blue}{\left(3 + 1\right)}}\right) \]

       7. pow-plus N/A

          \[\leadsto \frac{1}{120} \cdot \left(x \cdot \color{blue}{\left({x}^{3} \cdot x\right)}\right) \]

       8. *-commutative N/A

          \[\leadsto \frac{1}{120} \cdot \left(x \cdot \color{blue}{\left(x \cdot {x}^{3}\right)}\right) \]

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

          \[\leadsto \frac{1}{120} \cdot \left(x \cdot \color{blue}{\left(x \cdot {x}^{3}\right)}\right) \]

       10. cube-mult N/A

           \[\leadsto \frac{1}{120} \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \frac{1}{120} \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

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

           \[\leadsto \frac{1}{120} \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \frac{1}{120} \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

       14. *-lowering-*.f64 100.0

           \[\leadsto 0.008333333333333333 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{0.008333333333333333 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+61}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, 0.16666666666666666\right)\right), x \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, 0.16666666666666666\right), -1\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, 0.16666666666666666\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.008333333333333333 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), x \cdot x, x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (*
   x
   (fma
    (* x x)
    (fma (* x x) 0.0001984126984126984 0.008333333333333333)
    0.16666666666666666))
  (* x x)
  x))

C

double code(double x) {
	return fma((x * fma((x * x), fma((x * x), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), (x * x), x);
}

Julia

function code(x)
	return fma(Float64(x * fma(Float64(x * x), fma(Float64(x * x), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), Float64(x * x), x)
end

Wolfram

code[x_] := N[(N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 56.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

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

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + 1\right)} \]

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

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)} \]

   4. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right) \]

   6. +-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}}, 1\right) \]

   7. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}, 1\right) \]

   8. associate-*l* N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)} + \frac{1}{6}, 1\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), \frac{1}{6}\right)}, 1\right) \]

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

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)}, \frac{1}{6}\right), 1\right) \]

   11. +-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{5040} \cdot {x}^{2} + \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

   12. *-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

   13. unpow2 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{5040} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

   14. associate-*l* N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{5040}\right)} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

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

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

   16. *-lowering-*.f64 90.0

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.0001984126984126984}, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

5. Simplified 90.0%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
6. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto \color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right) + x \cdot 1} \]

   2. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) + \frac{1}{6}\right) \cdot \left(x \cdot x\right)\right)} + x \cdot 1 \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right) \cdot \left(x \cdot x\right)} + x \cdot 1 \]

   4. *-rgt-identity N/A

      \[\leadsto \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right) \cdot \left(x \cdot x\right) + \color{blue}{x} \]

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) + \frac{1}{6}\right), x \cdot x, x\right)} \]

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

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) + \frac{1}{6}\right)}, x \cdot x, x\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \left(\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)} + \frac{1}{6}\right), x \cdot x, x\right) \]

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

      \[\leadsto \mathsf{fma}\left(x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}, \frac{1}{6}\right)}, x \cdot x, x\right) \]

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

      \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}, \frac{1}{6}\right), x \cdot x, x\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right) \cdot \frac{1}{5040}} + \frac{1}{120}, \frac{1}{6}\right), x \cdot x, x\right) \]

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

       \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), x \cdot x, x\right) \]

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

       \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), x \cdot x, x\right) \]

   13. *-lowering-*.f64 90.0

       \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \color{blue}{x \cdot x}, x\right) \]

7. Applied egg-rr 90.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), x \cdot x, x\right)} \]
8. Add Preprocessing

Alternative 8: 93.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (fma
   (* x x)
   (fma
    x
    (* x (fma x (* x 0.0001984126984126984) 0.008333333333333333))
    0.16666666666666666)
   1.0)))

C

double code(double x) {
	return x * fma((x * x), fma(x, (x * fma(x, (x * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666), 1.0);
}

Julia

function code(x)
	return Float64(x * fma(Float64(x * x), fma(x, Float64(x * fma(x, Float64(x * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666), 1.0))
end

Wolfram

code[x_] := N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

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

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + 1\right)} \]

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

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)} \]

   4. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right) \]

   6. +-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}}, 1\right) \]

   7. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}, 1\right) \]

   8. associate-*l* N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)} + \frac{1}{6}, 1\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), \frac{1}{6}\right)}, 1\right) \]

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

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)}, \frac{1}{6}\right), 1\right) \]

   11. +-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{5040} \cdot {x}^{2} + \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

   12. *-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

   13. unpow2 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{5040} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

   14. associate-*l* N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{5040}\right)} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

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

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

   16. *-lowering-*.f64 90.0

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.0001984126984126984}, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

5. Simplified 90.0%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
6. Add Preprocessing

Alternative 9: 93.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right)\right), x \cdot x, x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (* (* x x) (* x (fma (* x x) 0.0001984126984126984 0.008333333333333333)))
  (* x x)
  x))

C

double code(double x) {
	return fma(((x * x) * (x * fma((x * x), 0.0001984126984126984, 0.008333333333333333))), (x * x), x);
}

Julia

function code(x)
	return fma(Float64(Float64(x * x) * Float64(x * fma(Float64(x * x), 0.0001984126984126984, 0.008333333333333333))), Float64(x * x), x)
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 56.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

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

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + 1\right)} \]

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

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)} \]

   4. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right) \]

   6. +-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}}, 1\right) \]

   7. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}, 1\right) \]

   8. associate-*l* N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)} + \frac{1}{6}, 1\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), \frac{1}{6}\right)}, 1\right) \]

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

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)}, \frac{1}{6}\right), 1\right) \]

   11. +-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{5040} \cdot {x}^{2} + \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

   12. *-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

   13. unpow2 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{5040} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

   14. associate-*l* N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{5040}\right)} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

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

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

   16. *-lowering-*.f64 90.0

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.0001984126984126984}, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

5. Simplified 90.0%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
6. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto \color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right) + x \cdot 1} \]

   2. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) + \frac{1}{6}\right) \cdot \left(x \cdot x\right)\right)} + x \cdot 1 \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right) \cdot \left(x \cdot x\right)} + x \cdot 1 \]

   4. *-rgt-identity N/A

      \[\leadsto \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right) \cdot \left(x \cdot x\right) + \color{blue}{x} \]

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) + \frac{1}{6}\right), x \cdot x, x\right)} \]

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

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) + \frac{1}{6}\right)}, x \cdot x, x\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \left(\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)} + \frac{1}{6}\right), x \cdot x, x\right) \]

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

      \[\leadsto \mathsf{fma}\left(x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}, \frac{1}{6}\right)}, x \cdot x, x\right) \]

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

      \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}, \frac{1}{6}\right), x \cdot x, x\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right) \cdot \frac{1}{5040}} + \frac{1}{120}, \frac{1}{6}\right), x \cdot x, x\right) \]

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

       \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), x \cdot x, x\right) \]

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

       \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), x \cdot x, x\right) \]

   13. *-lowering-*.f64 90.0

       \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \color{blue}{x \cdot x}, x\right) \]

7. Applied egg-rr 90.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), x \cdot x, x\right)} \]

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{5} \cdot \left(\frac{1}{5040} + \frac{1}{120} \cdot \frac{1}{{x}^{2}}\right)}, x \cdot x, x\right) \]
9. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{5040} \cdot {x}^{5} + \left(\frac{1}{120} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{5}}, x \cdot x, x\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{120} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{5} + \frac{1}{5040} \cdot {x}^{5}}, x \cdot x, x\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot \left(\frac{1}{{x}^{2}} \cdot {x}^{5}\right)} + \frac{1}{5040} \cdot {x}^{5}, x \cdot x, x\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot \color{blue}{\frac{1 \cdot {x}^{5}}{{x}^{2}}} + \frac{1}{5040} \cdot {x}^{5}, x \cdot x, x\right) \]

   5. *-lft-identity N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot \frac{\color{blue}{{x}^{5}}}{{x}^{2}} + \frac{1}{5040} \cdot {x}^{5}, x \cdot x, x\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot \frac{{x}^{\color{blue}{\left(4 + 1\right)}}}{{x}^{2}} + \frac{1}{5040} \cdot {x}^{5}, x \cdot x, x\right) \]

   7. pow-plus N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot \frac{\color{blue}{{x}^{4} \cdot x}}{{x}^{2}} + \frac{1}{5040} \cdot {x}^{5}, x \cdot x, x\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot \frac{\color{blue}{x \cdot {x}^{4}}}{{x}^{2}} + \frac{1}{5040} \cdot {x}^{5}, x \cdot x, x\right) \]

   9. associate-/l* N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot \color{blue}{\left(x \cdot \frac{{x}^{4}}{{x}^{2}}\right)} + \frac{1}{5040} \cdot {x}^{5}, x \cdot x, x\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot \frac{{x}^{\color{blue}{\left(2 \cdot 2\right)}}}{{x}^{2}}\right) + \frac{1}{5040} \cdot {x}^{5}, x \cdot x, x\right) \]

   11. pow-sqr N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot \frac{\color{blue}{{x}^{2} \cdot {x}^{2}}}{{x}^{2}}\right) + \frac{1}{5040} \cdot {x}^{5}, x \cdot x, x\right) \]

   12. associate-/l* N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \frac{{x}^{2}}{{x}^{2}}\right)}\right) + \frac{1}{5040} \cdot {x}^{5}, x \cdot x, x\right) \]

   13. *-rgt-identity N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot \left({x}^{2} \cdot \frac{\color{blue}{{x}^{2} \cdot 1}}{{x}^{2}}\right)\right) + \frac{1}{5040} \cdot {x}^{5}, x \cdot x, x\right) \]

   14. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot \left({x}^{2} \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{{x}^{2}}\right)}\right)\right) + \frac{1}{5040} \cdot {x}^{5}, x \cdot x, x\right) \]

   15. rgt-mult-inverse N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot \left({x}^{2} \cdot \color{blue}{1}\right)\right) + \frac{1}{5040} \cdot {x}^{5}, x \cdot x, x\right) \]

   16. *-rgt-identity N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot \color{blue}{{x}^{2}}\right) + \frac{1}{5040} \cdot {x}^{5}, x \cdot x, x\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) + \frac{1}{5040} \cdot {x}^{5}, x \cdot x, x\right) \]

   18. cube-mult N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot \color{blue}{{x}^{3}} + \frac{1}{5040} \cdot {x}^{5}, x \cdot x, x\right) \]

   19. unpow3 N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} + \frac{1}{5040} \cdot {x}^{5}, x \cdot x, x\right) \]

   20. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot \left(\color{blue}{{x}^{2}} \cdot x\right) + \frac{1}{5040} \cdot {x}^{5}, x \cdot x, x\right) \]

   21. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{120} \cdot {x}^{2}\right) \cdot x} + \frac{1}{5040} \cdot {x}^{5}, x \cdot x, x\right) \]

10. Simplified 89.6%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right)\right)}, x \cdot x, x\right) \]
11. Add Preprocessing

Alternative 10: 91.3% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (fma (* x x) (fma x (* x 0.008333333333333333) 0.16666666666666666) 1.0)))

C

double code(double x) {
	return x * fma((x * x), fma(x, (x * 0.008333333333333333), 0.16666666666666666), 1.0);
}

Julia

function code(x)
	return Float64(x * fma(Float64(x * x), fma(x, Float64(x * 0.008333333333333333), 0.16666666666666666), 1.0))
end

Wolfram

code[x_] := N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

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

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + 1\right)} \]

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

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)} \]

   4. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right) \]

   6. +-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}}, 1\right) \]

   7. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}, 1\right) \]

   8. associate-*l* N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)} + \frac{1}{6}, 1\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), \frac{1}{6}\right)}, 1\right) \]

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

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)}, \frac{1}{6}\right), 1\right) \]

   11. +-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{5040} \cdot {x}^{2} + \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

   12. *-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

   13. unpow2 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{5040} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

   14. associate-*l* N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{5040}\right)} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

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

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

   16. *-lowering-*.f64 90.0

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.0001984126984126984}, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

5. Simplified 90.0%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{120} \cdot x}, \frac{1}{6}\right), 1\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{120}}, \frac{1}{6}\right), 1\right) \]

   2. *-lowering-*.f64 88.4

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.008333333333333333}, 0.16666666666666666\right), 1\right) \]

8. Simplified 88.4%

   \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.008333333333333333}, 0.16666666666666666\right), 1\right) \]
9. Add Preprocessing

Alternative 11: 90.9% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot x, \left(x \cdot \left(x \cdot x\right)\right) \cdot 0.008333333333333333, x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (* x x) (* (* x (* x x)) 0.008333333333333333) x))

C

double code(double x) {
	return fma((x * x), ((x * (x * x)) * 0.008333333333333333), x);
}

Julia

function code(x)
	return fma(Float64(x * x), Float64(Float64(x * Float64(x * x)) * 0.008333333333333333), x)
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 56.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) \cdot x} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + 1\right)} \cdot x \]

   3. distribute-lft1-in N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) \cdot x + x} \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x\right)} + x \]

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

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x, x\right)} \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x, x\right) \]

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

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x, x\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}, x\right) \]

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

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}, x\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} + \frac{1}{6}\right)}, x\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{120}} + \frac{1}{6}\right), x\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} + \frac{1}{6}\right), x\right) \]

   13. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{120}\right)} + \frac{1}{6}\right), x\right) \]

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

       \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{120}, \frac{1}{6}\right)}, x\right) \]

   15. *-lowering-*.f64 88.4

       \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.008333333333333333}, 0.16666666666666666\right), x\right) \]

5. Simplified 88.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), x\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{120} \cdot {x}^{3}}, x\right) \]
7. Step-by-step derivation

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

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{120} \cdot {x}^{3}}, x\right) \]

   2. cube-mult N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{120} \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}, x\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{120} \cdot \left(x \cdot \color{blue}{{x}^{2}}\right), x\right) \]

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

      \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{120} \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}, x\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{120} \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right), x\right) \]

   6. *-lowering-*.f64 88.1

      \[\leadsto \mathsf{fma}\left(x \cdot x, 0.008333333333333333 \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right), x\right) \]

8. Simplified 88.1%

   \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{0.008333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)}, x\right) \]

9. Final simplification 88.1%

   \[\leadsto \mathsf{fma}\left(x \cdot x, \left(x \cdot \left(x \cdot x\right)\right) \cdot 0.008333333333333333, x\right) \]
10. Add Preprocessing

Alternative 12: 85.1% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

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

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + 1\right)} \]

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

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)} \]

   4. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right) \]

   6. +-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}}, 1\right) \]

   7. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}, 1\right) \]

   8. associate-*l* N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)} + \frac{1}{6}, 1\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), \frac{1}{6}\right)}, 1\right) \]

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

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)}, \frac{1}{6}\right), 1\right) \]

   11. +-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{5040} \cdot {x}^{2} + \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

   12. *-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

   13. unpow2 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{5040} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

   14. associate-*l* N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{5040}\right)} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

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

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

   16. *-lowering-*.f64 90.0

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.0001984126984126984}, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

5. Simplified 90.0%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
6. Step-by-step derivation

   1. flip-+ N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\frac{\left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) - \frac{1}{120} \cdot \frac{1}{120}}{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}}, \frac{1}{6}\right), 1\right) \]

   2. clear-num N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\frac{1}{\frac{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}{\left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) - \frac{1}{120} \cdot \frac{1}{120}}}}, \frac{1}{6}\right), 1\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\frac{1}{\frac{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}{\left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) - \frac{1}{120} \cdot \frac{1}{120}}}}, \frac{1}{6}\right), 1\right) \]

   4. clear-num N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{\color{blue}{\frac{1}{\frac{\left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) - \frac{1}{120} \cdot \frac{1}{120}}{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}}}}, \frac{1}{6}\right), 1\right) \]

   5. flip-+ N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{\frac{1}{\color{blue}{x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}}}}, \frac{1}{6}\right), 1\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{\color{blue}{\frac{1}{x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}}}}, \frac{1}{6}\right), 1\right) \]

   7. associate-*r* N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{\frac{1}{\color{blue}{\left(x \cdot x\right) \cdot \frac{1}{5040}} + \frac{1}{120}}}, \frac{1}{6}\right), 1\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{\frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{5040}, \frac{1}{120}\right)}}}, \frac{1}{6}\right), 1\right) \]

   9. *-lowering-*.f64 90.0

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{\frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.0001984126984126984, 0.008333333333333333\right)}}, 0.16666666666666666\right), 1\right) \]

7. Applied egg-rr 90.0%

   \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right)}}}, 0.16666666666666666\right), 1\right) \]

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 81.3

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

10. Simplified 81.3%

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

Alternative 13: 53.2% accurate, 217.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 56.4%

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

3. Taylor expanded in x around 0

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

5. Simplified 50.5%

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

Reproduce

herbie shell --seed 2024203 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))