Hyperbolic sine

Percentage Accurate: 53.8% → 100.0%

Time: 11.4s

Alternatives: 16

Speedup: 12.8×

• 49.2% 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: 53.8% 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 57.9%

   \[\frac{e^{x} - e^{-x}}{2} \]
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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \left(x \cdot 0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;0.0001984126984126984 \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (exp x) (exp (- x))) 2e-5)
   (fma x (* x (* x 0.16666666666666666)) x)
   (* 0.0001984126984126984 (* x (* (* x x) (* x (* x (* x x))))))))

C

double code(double x) {
	double tmp;
	if ((exp(x) - exp(-x)) <= 2e-5) {
		tmp = fma(x, (x * (x * 0.16666666666666666)), x);
	} else {
		tmp = 0.0001984126984126984 * (x * ((x * x) * (x * (x * (x * x)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) - exp(Float64(-x))) <= 2e-5)
		tmp = fma(x, Float64(x * Float64(x * 0.16666666666666666)), x);
	else
		tmp = Float64(0.0001984126984126984 * Float64(x * Float64(Float64(x * 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], 2e-5], N[(x * N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(0.0001984126984126984 * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 2.00000000000000016e-5

   1. Initial program 45.0%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 95.2

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

   5. Simplified 95.2%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
   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 95.2

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

   7. Applied egg-rr 95.2%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 89.9

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

   10. Simplified 89.9%

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

   if 2.00000000000000016e-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 90.6

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

   5. Simplified 90.6%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
   6. 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 90.6

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

   7. Applied egg-rr 90.6%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{5040} \cdot {x}^{7}} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{1}{5040} \cdot {x}^{7}} \]

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

      7. pow-plus N/A

         \[\leadsto \frac{1}{5040} \cdot \left(x \cdot \color{blue}{\left({x}^{5} \cdot x\right)}\right) \]

      8. metadata-eval N/A

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

      9. pow-plus N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

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

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

      16. metadata-eval N/A

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

      17. pow-plus N/A

          \[\leadsto \frac{1}{5040} \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left({x}^{3} \cdot x\right)}\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \frac{1}{5040} \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot {x}^{3}\right)}\right)\right) \]

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

          \[\leadsto \frac{1}{5040} \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot {x}^{3}\right)}\right)\right) \]

      20. cube-mult N/A

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

      21. unpow2 N/A

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

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

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

      23. unpow2 N/A

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

      24. *-lowering-*.f64 90.6

          \[\leadsto 0.0001984126984126984 \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

   10. Simplified 90.6%

       \[\leadsto \color{blue}{0.0001984126984126984 \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 86.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \left(x \cdot 0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (exp x) (exp (- x))) 2e-5)
   (fma x (* x (* x 0.16666666666666666)) x)
   (* (* x (* x x)) (fma (* x x) 0.008333333333333333 0.16666666666666666))))

C

double code(double x) {
	double tmp;
	if ((exp(x) - exp(-x)) <= 2e-5) {
		tmp = fma(x, (x * (x * 0.16666666666666666)), x);
	} 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))) <= 2e-5)
		tmp = fma(x, Float64(x * Float64(x * 0.16666666666666666)), x);
	else
		tmp = Float64(Float64(x * Float64(x * x)) * fma(Float64(x * x), 0.008333333333333333, 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 2e-5], N[(x * N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 2.00000000000000016e-5

   1. Initial program 45.0%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 95.2

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

   5. Simplified 95.2%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
   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 95.2

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

   7. Applied egg-rr 95.2%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 89.9

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

   10. Simplified 89.9%

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

   if 2.00000000000000016e-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 82.8

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

   5. Simplified 82.8%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 82.8%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \left(x \cdot 0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \left(x \cdot 0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;0.008333333333333333 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (exp x) (exp (- x))) 2e-5)
   (fma x (* x (* x 0.16666666666666666)) x)
   (* 0.008333333333333333 (* x (* x (* x (* x x)))))))

C

double code(double x) {
	double tmp;
	if ((exp(x) - exp(-x)) <= 2e-5) {
		tmp = fma(x, (x * (x * 0.16666666666666666)), x);
	} 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))) <= 2e-5)
		tmp = fma(x, Float64(x * Float64(x * 0.16666666666666666)), x);
	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], 2e-5], N[(x * N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $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))) < 2.00000000000000016e-5

   1. Initial program 45.0%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 95.2

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

   5. Simplified 95.2%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
   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 95.2

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

   7. Applied egg-rr 95.2%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 89.9

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

   10. Simplified 89.9%

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

   if 2.00000000000000016e-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 82.8

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

   5. Simplified 82.8%

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

   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 82.8

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

   8. Simplified 82.8%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

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

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

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

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

      17. associate-*r* N/A

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

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

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

      19. *-lowering-*.f64 82.8

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

   10. Applied egg-rr 82.8%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{120} \cdot {x}^{5}} \]
   12. 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 82.8

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

   13. Simplified 82.8%

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

Alternative 5: 94.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq 1.75 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), x \cdot x, x\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\frac{t\_0 \cdot \left(6.944444444444444 \cdot 10^{-5} \cdot \left(\left(x \cdot x\right) \cdot t\_0\right)\right) - x \cdot x}{\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(x \cdot 0.008333333333333333\right)\right), -x\right)}\\ \mathbf{else}:\\ \;\;\;\;0.008333333333333333 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) (* x x))))
   (if (<= x 1.75e+31)
     (fma
      (*
       x
       (fma
        (* x x)
        (fma (* x x) 0.0001984126984126984 0.008333333333333333)
        0.16666666666666666))
      (* x x)
      x)
     (if (<= x 5e+61)
       (/
        (- (* t_0 (* 6.944444444444444e-5 (* (* x x) t_0))) (* x x))
        (fma (* x x) (* x (* x (* x 0.008333333333333333))) (- x)))
       (* 0.008333333333333333 (* x (* x (* x (* x x)))))))))

C

double code(double x) {
	double t_0 = (x * x) * (x * x);
	double tmp;
	if (x <= 1.75e+31) {
		tmp = fma((x * fma((x * x), fma((x * x), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), (x * x), x);
	} else if (x <= 5e+61) {
		tmp = ((t_0 * (6.944444444444444e-5 * ((x * x) * t_0))) - (x * x)) / fma((x * x), (x * (x * (x * 0.008333333333333333))), -x);
	} else {
		tmp = 0.008333333333333333 * (x * (x * (x * (x * x))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(x * x))
	tmp = 0.0
	if (x <= 1.75e+31)
		tmp = fma(Float64(x * fma(Float64(x * x), fma(Float64(x * x), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), Float64(x * x), x);
	elseif (x <= 5e+61)
		tmp = Float64(Float64(Float64(t_0 * Float64(6.944444444444444e-5 * Float64(Float64(x * x) * t_0))) - Float64(x * x)) / fma(Float64(x * x), Float64(x * Float64(x * Float64(x * 0.008333333333333333))), Float64(-x)));
	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[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.75e+31], 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], If[LessEqual[x, 5e+61], N[(N[(N[(t$95$0 * N[(6.944444444444444e-5 * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-x)), $MachinePrecision]), $MachinePrecision], N[(0.008333333333333333 * N[(x * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.75e31

   1. Initial program 45.6%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 94.3

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

   5. Simplified 94.3%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
   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 94.3

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

   7. Applied egg-rr 94.3%

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

   if 1.75e31 < x < 5.00000000000000018e61

   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 6.4

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

   5. Simplified 6.4%

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

   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 6.4

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

   8. Simplified 6.4%

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

      1. flip-+ N/A

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

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

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

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(6.944444444444444 \cdot 10^{-5} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) - x \cdot x}{\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(x \cdot 0.008333333333333333\right)\right), -x\right)}} \]

   if 5.00000000000000018e61 < 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

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{120} \cdot {x}^{3}}, x\right) \]
   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 100.0

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

   8. Simplified 100.0%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

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

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

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

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

      17. associate-*r* N/A

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

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

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

      19. *-lowering-*.f64 100.0

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

   10. Applied egg-rr 100.0%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{120} \cdot {x}^{5}} \]
   12. 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

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

   13. 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: 75.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \mathsf{fma}\left(t\_0, t\_0 \cdot 5.787037037037037 \cdot 10^{-7}, 0.004629629629629629\right), \frac{1}{\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 6.944444444444444 \cdot 10^{-5}, 0.027777777777777776\right) - \left(x \cdot x\right) \cdot 0.001388888888888889}, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.008333333333333333 \cdot \left(x \cdot \left(x \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= x 5e+61)
     (fma
      (* t_0 (fma t_0 (* t_0 5.787037037037037e-7) 0.004629629629629629))
      (/
       1.0
       (-
        (fma (* x x) (* (* x x) 6.944444444444444e-5) 0.027777777777777776)
        (* (* x x) 0.001388888888888889)))
      x)
     (* 0.008333333333333333 (* x (* x t_0))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= 5e+61) {
		tmp = fma((t_0 * fma(t_0, (t_0 * 5.787037037037037e-7), 0.004629629629629629)), (1.0 / (fma((x * x), ((x * x) * 6.944444444444444e-5), 0.027777777777777776) - ((x * x) * 0.001388888888888889))), x);
	} else {
		tmp = 0.008333333333333333 * (x * (x * t_0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= 5e+61)
		tmp = fma(Float64(t_0 * fma(t_0, Float64(t_0 * 5.787037037037037e-7), 0.004629629629629629)), Float64(1.0 / Float64(fma(Float64(x * x), Float64(Float64(x * x) * 6.944444444444444e-5), 0.027777777777777776) - Float64(Float64(x * x) * 0.001388888888888889))), x);
	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]}, If[LessEqual[x, 5e+61], N[(N[(t$95$0 * N[(t$95$0 * N[(t$95$0 * 5.787037037037037e-7), $MachinePrecision] + 0.004629629629629629), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 6.944444444444444e-5), $MachinePrecision] + 0.027777777777777776), $MachinePrecision] - N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(0.008333333333333333 * N[(x * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000018e61

   1. Initial program 48.0%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 89.1

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

   5. Simplified 89.1%

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

      1. associate-*r* N/A

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

      2. pow3 N/A

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

      3. flip3-+ N/A

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

      4. associate-*r/ N/A

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

      5. div-inv N/A

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

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

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3} \cdot \left({\left(x \cdot \left(x \cdot \frac{1}{120}\right)\right)}^{3} + {\frac{1}{6}}^{3}\right), \frac{1}{\left(x \cdot \left(x \cdot \frac{1}{120}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{120}\right)\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \left(x \cdot \frac{1}{120}\right)\right) \cdot \frac{1}{6}\right)}, x\right)} \]

   7. Applied egg-rr 64.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot 5.787037037037037 \cdot 10^{-7}, 0.004629629629629629\right), \frac{1}{\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 6.944444444444444 \cdot 10^{-5}, 0.027777777777777776\right) - \left(x \cdot x\right) \cdot 0.001388888888888889}, x\right)} \]

   if 5.00000000000000018e61 < 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

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{120} \cdot {x}^{3}}, x\right) \]
   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 100.0

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

   8. Simplified 100.0%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

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

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

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

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

      17. associate-*r* N/A

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

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

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

      19. *-lowering-*.f64 100.0

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

   10. Applied egg-rr 100.0%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{120} \cdot {x}^{5}} \]
   12. 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

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

   13. 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. Add Preprocessing

Alternative 7: 93.0% 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 57.9%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 94.2

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

5. Simplified 94.2%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
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 94.2

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

7. Applied egg-rr 94.2%

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

Alternative 8: 93.0% 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 57.9%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 94.2

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

5. Simplified 94.2%

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

Alternative 9: 92.5% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 57.9%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 94.2

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

5. Simplified 94.2%

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

   1. distribute-lft-in N/A

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

   2. associate-+l+ N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

   9. associate-*r* N/A

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

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

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

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

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

   12. associate-*r* N/A

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

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

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

   14. *-lowering-*.f64 94.2

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

7. Applied egg-rr 94.2%

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

8. Taylor expanded in x around 0

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

10. Simplified 93.9%

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

Alternative 10: 92.5% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 57.9%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 94.2

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

5. Simplified 94.2%

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

   1. distribute-lft-in N/A

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

   2. associate-+l+ N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

   9. associate-*r* N/A

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

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

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

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

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

   12. associate-*r* N/A

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

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

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

   14. *-lowering-*.f64 94.2

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

7. Applied egg-rr 94.2%

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

   1. distribute-rgt-in N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

9. Applied egg-rr 94.2%

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

10. Taylor expanded in x around 0

    \[\leadsto x \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right), x \cdot \frac{1}{5040}, \color{blue}{1}\right) \]
11. Step-by-step derivation

12. Simplified 93.9%

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

Alternative 11: 90.0% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
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 91.2

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

5. Simplified 91.2%

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

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

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

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

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

   5. associate-*r* N/A

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

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

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

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

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

   8. *-lowering-*.f64 91.2

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

7. Applied egg-rr 91.2%

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

8. Final simplification 91.2%

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

Alternative 12: 90.0% 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 57.9%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 94.2

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

5. Simplified 94.2%

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

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 91.2

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

8. Simplified 91.2%

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

Alternative 13: 89.7% 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 57.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
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 91.2

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

5. Simplified 91.2%

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

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 90.9

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

8. Simplified 90.9%

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

9. Final simplification 90.9%

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

Alternative 14: 67.6% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.4) x (* (* x (* x x)) 0.16666666666666666)))

C

double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = x;
	} else {
		tmp = (x * (x * x)) * 0.16666666666666666;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.4d0) then
        tmp = x
    else
        tmp = (x * (x * x)) * 0.16666666666666666d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = x;
	} else {
		tmp = (x * (x * x)) * 0.16666666666666666;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.4:
		tmp = x
	else:
		tmp = (x * (x * x)) * 0.16666666666666666
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.4)
		tmp = x;
	else
		tmp = Float64(Float64(x * Float64(x * x)) * 0.16666666666666666);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.4)
		tmp = x;
	else
		tmp = (x * (x * x)) * 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.4], x, N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 45.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 61.1%

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

   if 2.39999999999999991 < 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 68.5

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

   5. Simplified 68.5%

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

   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 68.5

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

   8. Simplified 68.5%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 83.7% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 57.9%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 94.2

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

5. Simplified 94.2%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
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 94.2

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

7. Applied egg-rr 94.2%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

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

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

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

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 84.9

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

10. Simplified 84.9%

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

Alternative 16: 52.7% 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 57.9%

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

3. Taylor expanded in x around 0

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

5. Simplified 48.1%

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

Reproduce

herbie shell --seed 2024198 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))