Hyperbolic sine

Percentage Accurate: 54.5% → 100.0%

Time: 11.5s

Alternatives: 18

Speedup: 12.8×

• 50.0% 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: 54.5% 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 53.9%

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

   1. sinh-def N/A

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

   2. lower-sinh.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 91.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(0.0001984126984126984, x \cdot x, 0.008333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (exp x) (exp (- x))) 0.1)
   (fma x (* (* x x) (fma x (* x 0.008333333333333333) 0.16666666666666666)) x)
   (*
    x
    (*
     x
     (*
      (* x (* x x))
      (fma 0.0001984126984126984 (* x x) 0.008333333333333333))))))

C

double code(double x) {
	double tmp;
	if ((exp(x) - exp(-x)) <= 0.1) {
		tmp = fma(x, ((x * x) * fma(x, (x * 0.008333333333333333), 0.16666666666666666)), x);
	} else {
		tmp = x * (x * ((x * (x * x)) * fma(0.0001984126984126984, (x * x), 0.008333333333333333)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) - exp(Float64(-x))) <= 0.1)
		tmp = fma(x, Float64(Float64(x * x) * fma(x, Float64(x * 0.008333333333333333), 0.16666666666666666)), x);
	else
		tmp = Float64(x * Float64(x * Float64(Float64(x * Float64(x * x)) * fma(0.0001984126984126984, Float64(x * x), 0.008333333333333333))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 0.1], N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x * N[(x * N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 0.10000000000000001

   1. Initial program 38.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
   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. lower-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. lower-*.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. lower-*.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. lower-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. lower-*.f64 93.0

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

   5. Applied rewrites 93.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 93.0

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

   7. Applied rewrites 93.0%

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

   if 0.10000000000000001 < (-.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. lower-*.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. lower-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. lower-*.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. lower-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. lower-*.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. lower-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. lower-*.f64 85.1

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

   5. Applied rewrites 85.1%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-+ N/A

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

      9. lift-fma.f64 N/A

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

      10. lower-/.f64 85.1

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

      11. lift-fma.f64 N/A

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

   7. Applied rewrites 85.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 85.1%

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

       1. associate-*r* N/A

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

       2. lift-*.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 85.1

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

       10. lift-fma.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*r* N/A

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-fma.f64 85.1

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

   12. Applied rewrites 85.1%

       \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{fma}\left(0.0001984126984126984, x \cdot x, 0.008333333333333333\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(0.0001984126984126984, x \cdot x, 0.008333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;e^{x} - e^{-x} \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.0001984126984126984 \cdot \left(t\_0 \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= (- (exp x) (exp (- x))) 0.1)
     (fma
      x
      (* (* x x) (fma x (* x 0.008333333333333333) 0.16666666666666666))
      x)
     (* x (* 0.0001984126984126984 (* t_0 t_0))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double tmp;
	if ((exp(x) - exp(-x)) <= 0.1) {
		tmp = fma(x, ((x * x) * fma(x, (x * 0.008333333333333333), 0.16666666666666666)), x);
	} else {
		tmp = x * (0.0001984126984126984 * (t_0 * t_0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (Float64(exp(x) - exp(Float64(-x))) <= 0.1)
		tmp = fma(x, Float64(Float64(x * x) * fma(x, Float64(x * 0.008333333333333333), 0.16666666666666666)), x);
	else
		tmp = Float64(x * Float64(0.0001984126984126984 * Float64(t_0 * t_0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 0.1], N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x * N[(0.0001984126984126984 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 0.10000000000000001

   1. Initial program 38.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
   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. lower-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. lower-*.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. lower-*.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. lower-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. lower-*.f64 93.0

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

   5. Applied rewrites 93.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 93.0

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

   7. Applied rewrites 93.0%

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

   if 0.10000000000000001 < (-.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. lower-*.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. lower-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. lower-*.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. lower-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. lower-*.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. lower-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. lower-*.f64 85.1

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

   5. Applied rewrites 85.1%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-+ N/A

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

      9. lift-fma.f64 N/A

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

      10. lower-/.f64 85.1

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

      11. lift-fma.f64 N/A

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

   7. Applied rewrites 85.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 85.1%

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

   11. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. metadata-eval N/A

          \[\leadsto x \cdot \left(\frac{1}{5040} \cdot {x}^{\color{blue}{\left(2 \cdot 3\right)}}\right) \]

       3. pow-sqr N/A

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

       4. lower-*.f64 N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{5040} \cdot \left(\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot {x}^{3}\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{5040} \cdot \left(\color{blue}{\left(x \cdot {x}^{2}\right)} \cdot {x}^{3}\right)\right) \]

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. cube-mult N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 85.1

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

   13. Applied rewrites 85.1%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.0001984126984126984 \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (exp x) (exp (- x))) 0.1)
   (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)) <= 0.1) {
		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))) <= 0.1)
		tmp = fma(x, Float64(Float64(x * 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], 0.1], N[(x * N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $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))) < 0.10000000000000001

   1. Initial program 38.6%

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

      1. sinh-def N/A

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

      2. lower-sinh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 95.0%

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

   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. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 90.4

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

   10. Applied rewrites 90.4%

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

   if 0.10000000000000001 < (-.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. lower-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. lower-*.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. lower-*.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. lower-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. lower-*.f64 79.2

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

   5. Applied rewrites 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 79.2

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

   7. Applied rewrites 79.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 79.2%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.008333333333333333 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (exp x) (exp (- x))) 0.1)
   (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)) <= 0.1) {
		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))) <= 0.1)
		tmp = fma(x, Float64(Float64(x * x) * 0.16666666666666666), x);
	else
		tmp = Float64(0.008333333333333333 * Float64(Float64(x * x) * Float64(x * Float64(x * x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 0.1], N[(x * N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + x), $MachinePrecision], N[(0.008333333333333333 * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 0.10000000000000001

   1. Initial program 38.6%

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

      1. sinh-def N/A

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

      2. lower-sinh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 95.0%

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

   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. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 90.4

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

   10. Applied rewrites 90.4%

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

   if 0.10000000000000001 < (-.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. lower-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. lower-*.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. lower-*.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. lower-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. lower-*.f64 79.2

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

   5. Applied rewrites 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 79.2

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

   7. Applied rewrites 79.2%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 79.2

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

   10. Applied rewrites 79.2%

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

   11. Taylor expanded in x around inf

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

       1. lower-*.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. *-lft-identity N/A

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

       5. lft-mult-inverse N/A

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

       6. associate-*r* N/A

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

       7. unpow2 N/A

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

       8. unpow3 N/A

          \[\leadsto \frac{1}{120} \cdot \left({x}^{4} \cdot \left(\frac{1}{{x}^{2}} \cdot \color{blue}{{x}^{3}}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \frac{1}{120} \cdot \left({x}^{4} \cdot \left(\frac{1}{\color{blue}{x \cdot x}} \cdot {x}^{3}\right)\right) \]

       10. associate-/r* N/A

           \[\leadsto \frac{1}{120} \cdot \left({x}^{4} \cdot \left(\color{blue}{\frac{\frac{1}{x}}{x}} \cdot {x}^{3}\right)\right) \]

       11. associate-*l/ N/A

           \[\leadsto \frac{1}{120} \cdot \left({x}^{4} \cdot \color{blue}{\frac{\frac{1}{x} \cdot {x}^{3}}{x}}\right) \]

       12. associate-*r/ N/A

           \[\leadsto \frac{1}{120} \cdot \color{blue}{\frac{{x}^{4} \cdot \left(\frac{1}{x} \cdot {x}^{3}\right)}{x}} \]

   13. Applied rewrites 79.2%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.008333333333333333 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right)\\ \mathbf{if}\;x \leq 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \frac{x \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \left(t\_0 \cdot t\_0\right), -0.027777777777777776\right)}{\mathsf{fma}\left(x, x \cdot t\_0, -0.16666666666666666\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.008333333333333333 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (* x x) 0.0001984126984126984 0.008333333333333333)))
   (if (<= x 1e+56)
     (fma
      x
      (*
       x
       (/
        (* x (fma (* x x) (* (* x x) (* t_0 t_0)) -0.027777777777777776))
        (fma x (* x t_0) -0.16666666666666666)))
      x)
     (* 0.008333333333333333 (* (* x x) (* x (* x x)))))))

C

double code(double x) {
	double t_0 = fma((x * x), 0.0001984126984126984, 0.008333333333333333);
	double tmp;
	if (x <= 1e+56) {
		tmp = fma(x, (x * ((x * fma((x * x), ((x * x) * (t_0 * t_0)), -0.027777777777777776)) / fma(x, (x * t_0), -0.16666666666666666))), x);
	} else {
		tmp = 0.008333333333333333 * ((x * x) * (x * (x * x)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(Float64(x * x), 0.0001984126984126984, 0.008333333333333333)
	tmp = 0.0
	if (x <= 1e+56)
		tmp = fma(x, Float64(x * Float64(Float64(x * fma(Float64(x * x), Float64(Float64(x * x) * Float64(t_0 * t_0)), -0.027777777777777776)) / fma(x, Float64(x * t_0), -0.16666666666666666))), x);
	else
		tmp = Float64(0.008333333333333333 * Float64(Float64(x * x) * Float64(x * Float64(x * x))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]}, If[LessEqual[x, 1e+56], N[(x * N[(x * N[(N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] + -0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x * t$95$0), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(0.008333333333333333 * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000009e56

   1. Initial program 42.7%

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

      1. sinh-def N/A

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

      2. lower-sinh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 90.8%

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

   9. Applied rewrites 70.2%

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

   if 1.00000000000000009e56 < 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. lower-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. lower-*.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. lower-*.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. lower-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. lower-*.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. Applied rewrites 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in x around inf

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

       1. lower-*.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. *-lft-identity N/A

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

       5. lft-mult-inverse N/A

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

       6. associate-*r* N/A

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

       7. unpow2 N/A

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

       8. unpow3 N/A

          \[\leadsto \frac{1}{120} \cdot \left({x}^{4} \cdot \left(\frac{1}{{x}^{2}} \cdot \color{blue}{{x}^{3}}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \frac{1}{120} \cdot \left({x}^{4} \cdot \left(\frac{1}{\color{blue}{x \cdot x}} \cdot {x}^{3}\right)\right) \]

       10. associate-/r* N/A

           \[\leadsto \frac{1}{120} \cdot \left({x}^{4} \cdot \left(\color{blue}{\frac{\frac{1}{x}}{x}} \cdot {x}^{3}\right)\right) \]

       11. associate-*l/ N/A

           \[\leadsto \frac{1}{120} \cdot \left({x}^{4} \cdot \color{blue}{\frac{\frac{1}{x} \cdot {x}^{3}}{x}}\right) \]

       12. associate-*r/ N/A

           \[\leadsto \frac{1}{120} \cdot \color{blue}{\frac{{x}^{4} \cdot \left(\frac{1}{x} \cdot {x}^{3}\right)}{x}} \]

   13. Applied rewrites 100.0%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \frac{x \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \left(\mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right) \cdot \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right)\right), -0.027777777777777776\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.008333333333333333 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.9% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.9%

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

   1. sinh-def N/A

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

   2. lower-sinh.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

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

7. Applied rewrites 92.6%

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

Alternative 8: 92.9% 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 53.9%

   \[\frac{e^{x} - e^{-x}}{2} \]
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. lower-*.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. lower-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. lower-*.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. lower-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. lower-*.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. lower-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. lower-*.f64 92.6

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

5. Applied rewrites 92.6%

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

Alternative 9: 92.5% accurate, 5.7× speedup

Math

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

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(x * x), Float64(Float64(x * x) * fma(x, Float64(x * 0.0001984126984126984), 0.008333333333333333)), 1.0))
end

Wolfram

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

Derivation

1. Initial program 53.9%

   \[\frac{e^{x} - e^{-x}}{2} \]
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. lower-*.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. lower-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. lower-*.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. lower-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. lower-*.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. lower-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. lower-*.f64 92.6

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

5. Applied rewrites 92.6%

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

6. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. associate-*l/ N/A

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

   5. metadata-eval N/A

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

   6. pow-sqr N/A

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

   7. associate-*l* N/A

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

   8. associate-/l* N/A

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

   9. *-rgt-identity N/A

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

   10. associate-*r/ N/A

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

   11. rgt-mult-inverse N/A

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

   12. *-rgt-identity N/A

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

   13. metadata-eval N/A

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

   14. pow-sqr N/A

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

   15. associate-*l* N/A

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

   16. distribute-rgt-in N/A

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

   17. +-commutative N/A

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

   18. lower-*.f64 N/A

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

8. Applied rewrites 92.0%

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

Alternative 10: 92.5% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.9%

   \[\frac{e^{x} - e^{-x}}{2} \]
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. lower-*.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. lower-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. lower-*.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. lower-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. lower-*.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. lower-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. lower-*.f64 92.6

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

5. Applied rewrites 92.6%

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

6. Taylor expanded in x around inf

   \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{5040} \cdot {x}^{4}}, 1\right) \]
7. Step-by-step derivation

   1. metadata-eval N/A

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

   2. pow-sqr N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. unpow2 N/A

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

   6. unpow3 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. unpow3 N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 92.0

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

8. Applied rewrites 92.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. distribute-lft-in N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. pow3 N/A

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

   11. associate-*r* N/A

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

   12. *-rgt-identity N/A

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

   13. lower-fma.f64 N/A

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

10. Applied rewrites 92.0%

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

Alternative 11: 92.5% accurate, 5.9× speedup

Math

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

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(x * x), Float64(x * Float64(x * Float64(Float64(x * x) * 0.0001984126984126984))), 1.0))
end

Wolfram

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

Derivation

1. Initial program 53.9%

   \[\frac{e^{x} - e^{-x}}{2} \]
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. lower-*.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. lower-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. lower-*.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. lower-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. lower-*.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. lower-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. lower-*.f64 92.6

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

5. Applied rewrites 92.6%

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

6. Taylor expanded in x around inf

   \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{5040} \cdot {x}^{4}}, 1\right) \]
7. Step-by-step derivation

   1. metadata-eval N/A

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

   2. pow-sqr N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. unpow2 N/A

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

   6. unpow3 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. unpow3 N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 92.0

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

8. Applied rewrites 92.0%

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

Alternative 12: 90.2% accurate, 7.8× speedup

Math

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

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(x, Float64(x * 0.008333333333333333), 0.16666666666666666)), x)
end

Wolfram

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

Derivation

1. Initial program 53.9%

   \[\frac{e^{x} - e^{-x}}{2} \]
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. lower-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. lower-*.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. lower-*.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. lower-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. lower-*.f64 89.5

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

5. Applied rewrites 89.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 89.5

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

7. Applied rewrites 89.5%

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

8. Final simplification 89.5%

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

Alternative 13: 89.9% accurate, 8.0× speedup

Math

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

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(x, Float64(x * Float64(x * Float64(x * Float64(x * 0.008333333333333333)))), x)
end

Wolfram

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

Derivation

1. Initial program 53.9%

   \[\frac{e^{x} - e^{-x}}{2} \]
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. lower-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. lower-*.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. lower-*.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. lower-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. lower-*.f64 89.5

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

5. Applied rewrites 89.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 89.5

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

7. Applied rewrites 89.5%

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

8. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 89.0

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

10. Applied rewrites 89.0%

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

    1. associate-*r* N/A

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

    2. lift-*.f64 N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

    5. lift-*.f64 N/A

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

    6. associate-*r* N/A

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

    7. lift-*.f64 N/A

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

    8. lift-*.f64 N/A

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

    9. lower-*.f64 N/A

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

    10. lift-*.f64 N/A

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

    11. lift-*.f64 N/A

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

    12. associate-*r* N/A

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

    13. lift-*.f64 N/A

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

    14. lower-*.f64 N/A

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

    15. lower-*.f64 89.0

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

12. Applied rewrites 89.0%

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

13. Final simplification 89.0%

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

Alternative 14: 68.5% accurate, 9.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 2.5) {
		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 <= 2.5d0) 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 <= 2.5) {
		tmp = x;
	} else {
		tmp = x * (x * (x * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.5:
		tmp = x
	else:
		tmp = x * (x * (x * 0.16666666666666666))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.5)
		tmp = x;
	else
		tmp = x * (x * (x * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.5

   1. Initial program 38.6%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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. lower-*.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. lower-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. lower-*.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. lower-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. lower-*.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. lower-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. lower-*.f64 95.0

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 68.8%

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

      1. *-rgt-identity 68.8

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

   10. Applied rewrites 68.8%

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

   if 2.5 < 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. lower-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. lower-*.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. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 68.8

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

   8. Applied rewrites 68.8%

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

Alternative 15: 84.2% accurate, 12.8× speedup

Math

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

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(Float64(x * Float64(x * x)), 0.16666666666666666, x)
end

Wolfram

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

Derivation

1. Initial program 53.9%

   \[\frac{e^{x} - e^{-x}}{2} \]
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. lower-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. lower-*.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. lower-*.f64 85.0

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

5. Applied rewrites 85.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. pow3 N/A

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

   7. lower-fma.f64 N/A

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

   8. cube-mult N/A

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

   9. lift-*.f64 N/A

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

   10. lower-*.f64 85.3

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

7. Applied rewrites 85.3%

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

Alternative 16: 84.2% accurate, 12.8× speedup

Math

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

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(Float64(x * x) * 0.16666666666666666), x)
end

Wolfram

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

Derivation

1. Initial program 53.9%

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

   1. sinh-def N/A

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

   2. lower-sinh.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

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

7. Applied rewrites 92.6%

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

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. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 85.0

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

10. Applied rewrites 85.0%

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

11. Final simplification 85.0%

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

Alternative 17: 84.1% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.9%

   \[\frac{e^{x} - e^{-x}}{2} \]
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. lower-*.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. lower-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. lower-*.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. lower-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. lower-*.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. lower-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. lower-*.f64 92.6

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

5. Applied rewrites 92.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 85.0%

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

Alternative 18: 51.8% 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 53.9%

   \[\frac{e^{x} - e^{-x}}{2} \]
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. lower-*.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. lower-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. lower-*.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. lower-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. lower-*.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. lower-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. lower-*.f64 92.6

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

5. Applied rewrites 92.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 53.0%

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

   1. *-rgt-identity 53.0

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

10. Applied rewrites 53.0%

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

Reproduce

herbie shell --seed 2024214 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))