Hyperbolic sine

Percentage Accurate: 53.8% → 100.0%

Time: 10.9s

Alternatives: 13

Speedup: 12.8×

• 49.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return sinh(x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.7%

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

Alternative 2: 91.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (exp x) (exp (- x))) 0.002)
   (fma x (* (* x x) (fma x (* x 0.008333333333333333) 0.16666666666666666)) x)
   (*
    x
    (*
     (* x x)
     (* (* x x) (fma (* x x) 0.0001984126984126984 0.008333333333333333))))))

C

double code(double x) {
	double tmp;
	if ((exp(x) - exp(-x)) <= 0.002) {
		tmp = fma(x, ((x * x) * fma(x, (x * 0.008333333333333333), 0.16666666666666666)), x);
	} else {
		tmp = x * ((x * x) * ((x * x) * fma((x * x), 0.0001984126984126984, 0.008333333333333333)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) - exp(Float64(-x))) <= 0.002)
		tmp = fma(x, Float64(Float64(x * x) * fma(x, Float64(x * 0.008333333333333333), 0.16666666666666666)), x);
	else
		tmp = Float64(x * Float64(Float64(x * x) * Float64(Float64(x * x) * fma(Float64(x * x), 0.0001984126984126984, 0.008333333333333333))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 0.002], N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 2e-3

   1. Initial program 37.5%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 92.2

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

   5. Applied rewrites 92.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), x\right)} \]
   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 92.2

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

   7. Applied rewrites 92.2%

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

   if 2e-3 < (-.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 86.5

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

   5. Applied rewrites 86.5%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 86.5

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

   7. Applied rewrites 86.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
   9. 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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   10. Applied rewrites 86.5%

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

   11. Taylor expanded in x around inf

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

       1. distribute-rgt-in N/A

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

       2. +-commutative N/A

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

   13. Applied rewrites 86.5%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0001984126984126984\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (exp x) (exp (- x))) 0.002)
   (fma x (* (* x x) (fma x (* x 0.008333333333333333) 0.16666666666666666)) x)
   (* x (* x (* (* x (* x (* x (* x x)))) 0.0001984126984126984)))))

C

double code(double x) {
	double tmp;
	if ((exp(x) - exp(-x)) <= 0.002) {
		tmp = fma(x, ((x * x) * fma(x, (x * 0.008333333333333333), 0.16666666666666666)), x);
	} else {
		tmp = x * (x * ((x * (x * (x * (x * x)))) * 0.0001984126984126984));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) - exp(Float64(-x))) <= 0.002)
		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 * Float64(x * Float64(x * x)))) * 0.0001984126984126984)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 0.002], 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 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 2e-3

   1. Initial program 37.5%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 92.2

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

   5. Applied rewrites 92.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), x\right)} \]
   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 92.2

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

   7. Applied rewrites 92.2%

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

   if 2e-3 < (-.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 86.5

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

   5. Applied rewrites 86.5%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 86.5

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

   7. Applied rewrites 86.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
   9. 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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   10. Applied rewrites 86.5%

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

   11. Taylor expanded in x around inf

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

       1. metadata-eval N/A

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

       2. pow-plus N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. metadata-eval N/A

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

       6. pow-sqr N/A

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

       7. cube-prod N/A

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

       8. unpow2 N/A

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

       9. unpow3 N/A

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

       10. pow-sqr N/A

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

       11. metadata-eval N/A

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

       12. associate-*l* N/A

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

       13. *-commutative N/A

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

       14. associate-*l* N/A

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

       15. metadata-eval N/A

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

       16. pow-sqr N/A

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

       17. associate-*r* N/A

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

       18. *-commutative N/A

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

       19. *-commutative N/A

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

       20. lower-*.f64 N/A

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

   13. Applied rewrites 86.5%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0001984126984126984\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.002:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (exp x) (exp (- x))) 0.002)
   x
   (* x (* (* x x) 0.16666666666666666))))

C

double code(double x) {
	double tmp;
	if ((exp(x) - exp(-x)) <= 0.002) {
		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 ((exp(x) - exp(-x)) <= 0.002d0) 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 ((Math.exp(x) - Math.exp(-x)) <= 0.002) {
		tmp = x;
	} else {
		tmp = x * ((x * x) * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (math.exp(x) - math.exp(-x)) <= 0.002:
		tmp = x
	else:
		tmp = x * ((x * x) * 0.16666666666666666)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) - exp(Float64(-x))) <= 0.002)
		tmp = x;
	else
		tmp = Float64(x * Float64(Float64(x * x) * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((exp(x) - exp(-x)) <= 0.002)
		tmp = x;
	else
		tmp = x * ((x * x) * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 0.002], x, N[(x * N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 2e-3

   1. Initial program 37.5%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 69.2%

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

      1. *-rgt-identity 69.2

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

   10. Applied rewrites 69.2%

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

   if 2e-3 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. 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 68.2

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

   5. Applied rewrites 68.2%

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

   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. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 68.2

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

   8. Applied rewrites 68.2%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.002:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq 5 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \mathsf{fma}\left(5.787037037037037 \cdot 10^{-7}, t\_0 \cdot t\_0, 0.004629629629629629\right), \frac{1}{\mathsf{fma}\left(6.944444444444444 \cdot 10^{-5}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), 0.027777777777777776 - \left(x \cdot x\right) \cdot 0.001388888888888889\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.008333333333333333 \cdot \left(x \cdot \left(x \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= x 5e+60)
     (fma
      (* t_0 (fma 5.787037037037037e-7 (* t_0 t_0) 0.004629629629629629))
      (/
       1.0
       (fma
        6.944444444444444e-5
        (* (* x x) (* x x))
        (- 0.027777777777777776 (* (* x x) 0.001388888888888889))))
      x)
     (* 0.008333333333333333 (* x (* x t_0))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= 5e+60) {
		tmp = fma((t_0 * fma(5.787037037037037e-7, (t_0 * t_0), 0.004629629629629629)), (1.0 / fma(6.944444444444444e-5, ((x * x) * (x * x)), (0.027777777777777776 - ((x * x) * 0.001388888888888889)))), x);
	} else {
		tmp = 0.008333333333333333 * (x * (x * t_0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= 5e+60)
		tmp = fma(Float64(t_0 * fma(5.787037037037037e-7, Float64(t_0 * t_0), 0.004629629629629629)), Float64(1.0 / fma(6.944444444444444e-5, Float64(Float64(x * x) * Float64(x * x)), Float64(0.027777777777777776 - Float64(Float64(x * x) * 0.001388888888888889)))), x);
	else
		tmp = Float64(0.008333333333333333 * Float64(x * Float64(x * t_0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5e+60], N[(N[(t$95$0 * N[(5.787037037037037e-7 * N[(t$95$0 * t$95$0), $MachinePrecision] + 0.004629629629629629), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(6.944444444444444e-5 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(0.027777777777777776 - N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(0.008333333333333333 * N[(x * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999999999975e60

   1. Initial program 41.7%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 86.5

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

   5. Applied rewrites 86.5%

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

   7. Applied rewrites 73.4%

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

   if 4.99999999999999975e60 < 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. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. distribute-lft-in N/A

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

       7. *-rgt-identity N/A

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

       8. lower-fma.f64 N/A

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{120} \cdot {x}^{5}} \]
   13. 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. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. metadata-eval N/A

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

       7. pow-plus N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. cube-mult N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 100.0

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

   14. Applied rewrites 100.0%

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

Alternative 6: 93.1% accurate, 5.6× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 52.7%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 92.7

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

5. Applied rewrites 92.7%

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

   1. associate-*r* N/A

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

   2. lift-*.f64 N/A

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

   3. lower-fma.f64 92.7

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

7. Applied rewrites 92.7%

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

8. Taylor expanded in x around 0

   \[\leadsto x \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
9. 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. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-fma.f64 N/A

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

10. Applied rewrites 92.7%

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

Alternative 7: 92.7% accurate, 5.7× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 52.7%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 92.7

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

5. Applied rewrites 92.7%

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

   1. associate-*r* N/A

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

   2. lift-*.f64 N/A

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

   3. lower-fma.f64 92.7

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

7. Applied rewrites 92.7%

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

8. Taylor expanded in x around 0

   \[\leadsto x \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
9. 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. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-fma.f64 N/A

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

10. Applied rewrites 92.7%

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

11. Taylor expanded in x around inf

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

    1. distribute-rgt-in N/A

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

    2. associate-*r/ N/A

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

    3. metadata-eval N/A

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

    4. associate-*l/ N/A

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

    5. associate-/l* N/A

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

    6. metadata-eval N/A

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

    7. pow-sqr N/A

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

    8. associate-/l* N/A

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

    9. *-rgt-identity N/A

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

    10. associate-*r/ N/A

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

    11. rgt-mult-inverse N/A

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

    12. *-rgt-identity N/A

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

    13. metadata-eval N/A

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

    14. pow-sqr N/A

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

    15. associate-*l* N/A

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

    16. distribute-rgt-in N/A

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

    17. +-commutative N/A

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

    18. lower-*.f64 N/A

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

13. Applied rewrites 92.4%

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

Alternative 8: 92.7% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.7%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 92.7

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

5. Applied rewrites 92.7%

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

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. lower-*.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-sqr N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. unpow2 N/A

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

   7. cube-mult N/A

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

   8. lower-*.f64 N/A

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

   9. cube-mult N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 92.4

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

8. Applied rewrites 92.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

10. Applied rewrites 92.4%

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

Alternative 9: 87.5% accurate, 6.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.0)
   (fma x (* (* x x) 0.16666666666666666) x)
   (* 0.008333333333333333 (* x (* x (* x (* x x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 5.0) {
		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 (x <= 5.0)
		tmp = fma(x, Float64(Float64(x * x) * 0.16666666666666666), x);
	else
		tmp = Float64(0.008333333333333333 * Float64(x * Float64(x * Float64(x * Float64(x * x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5.0], N[(x * N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + x), $MachinePrecision], N[(0.008333333333333333 * N[(x * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5

   1. Initial program 37.5%

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

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{6} \cdot {x}^{2}\right)} \]
   6. 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 87.2

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

   7. Applied rewrites 87.2%

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

   if 5 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
   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 79.0

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

   5. Applied rewrites 79.0%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 79.0

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

   8. Applied rewrites 79.0%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. distribute-lft-in N/A

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

       7. *-rgt-identity N/A

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

       8. lower-fma.f64 N/A

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

   11. Applied rewrites 79.0%

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

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{120} \cdot {x}^{5}} \]
   13. 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. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. metadata-eval N/A

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

       7. pow-plus N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. cube-mult N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 79.0

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

   14. Applied rewrites 79.0%

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

4. Final simplification 85.2%

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

Alternative 10: 90.6% 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 52.7%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 89.0

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

5. Applied rewrites 89.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), x\right)} \]
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 89.0

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

7. Applied rewrites 89.0%

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

8. Final simplification 89.0%

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

Alternative 11: 90.2% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.7%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 89.0

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

5. Applied rewrites 89.0%

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

6. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. cube-mult N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 88.7

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

8. Applied rewrites 88.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-lft-out N/A

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

   9. lower-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r* N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. associate-*l* N/A

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

   16. lift-*.f64 N/A

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

   17. lower-fma.f64 N/A

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

   18. lower-*.f64 88.7

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

10. Applied rewrites 88.7%

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

Alternative 12: 84.5% 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 52.7%

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

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{6} \cdot {x}^{2}\right)} \]
6. 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 82.6

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

7. Applied rewrites 82.6%

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

8. Final simplification 82.6%

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

Alternative 13: 52.7% accurate, 217.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 52.7%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 92.7

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

5. Applied rewrites 92.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 53.8%

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

   1. *-rgt-identity 53.8

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

10. Applied rewrites 53.8%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))