Hyperbolic sine

Percentage Accurate: 54.7% → 100.0%

Time: 10.9s

Alternatives: 14

Speedup: 12.8×

• 50.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.7% 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 55.6%

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

   1. sinh-def N/A

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

   2. sinh-lowering-sinh.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 91.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 5:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (exp x) (exp (- x))) 5.0)
   (*
    x
    (fma (* x x) (fma x (* x 0.008333333333333333) 0.16666666666666666) 1.0))
   (*
    x
    (*
     (fma (* x x) 0.0001984126984126984 0.008333333333333333)
     (* x (* x (* x x)))))))

C

double code(double x) {
	double tmp;
	if ((exp(x) - exp(-x)) <= 5.0) {
		tmp = x * fma((x * x), fma(x, (x * 0.008333333333333333), 0.16666666666666666), 1.0);
	} else {
		tmp = x * (fma((x * x), 0.0001984126984126984, 0.008333333333333333) * (x * (x * (x * x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) - exp(Float64(-x))) <= 5.0)
		tmp = Float64(x * fma(Float64(x * x), fma(x, Float64(x * 0.008333333333333333), 0.16666666666666666), 1.0));
	else
		tmp = Float64(x * Float64(fma(Float64(x * x), 0.0001984126984126984, 0.008333333333333333) * Float64(x * Float64(x * Float64(x * x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 5.0], N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(N[(x * x), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 40.5%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

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

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

      16. *-lowering-*.f64 94.9

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

   5. Simplified 94.9%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 91.8

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

   8. Simplified 91.8%

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

   if 5 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

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

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

      16. *-lowering-*.f64 89.9

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

   5. Simplified 89.9%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 89.9%

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

Alternative 3: 91.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 5:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(0.0001984126984126984 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (exp x) (exp (- x))) 5.0)
   (*
    x
    (fma (* x x) (fma x (* x 0.008333333333333333) 0.16666666666666666) 1.0))
   (* x (* (* x x) (* x (* 0.0001984126984126984 (* x (* x x))))))))

C

double code(double x) {
	double tmp;
	if ((exp(x) - exp(-x)) <= 5.0) {
		tmp = x * fma((x * x), fma(x, (x * 0.008333333333333333), 0.16666666666666666), 1.0);
	} else {
		tmp = x * ((x * x) * (x * (0.0001984126984126984 * (x * (x * x)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) - exp(Float64(-x))) <= 5.0)
		tmp = Float64(x * fma(Float64(x * x), fma(x, Float64(x * 0.008333333333333333), 0.16666666666666666), 1.0));
	else
		tmp = Float64(x * Float64(Float64(x * x) * Float64(x * Float64(0.0001984126984126984 * Float64(x * Float64(x * x))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 5.0], N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(0.0001984126984126984 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 40.5%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

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

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

      16. *-lowering-*.f64 94.9

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

   5. Simplified 94.9%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 91.8

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

   8. Simplified 91.8%

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

   if 5 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

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

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

      16. *-lowering-*.f64 89.9

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

   5. Simplified 89.9%

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

   6. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. cube-prod N/A

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

      4. unpow2 N/A

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

      5. unpow3 N/A

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

      6. pow-sqr N/A

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

      7. metadata-eval N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. pow-sqr N/A

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

      13. associate-*r* N/A

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

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

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

      15. unpow2 N/A

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

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

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. *-commutative N/A

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

      20. associate-*r* N/A

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

      21. unpow2 N/A

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

      22. unpow3 N/A

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

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

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

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

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

      25. cube-mult N/A

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

      26. unpow2 N/A

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

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

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

   8. Simplified 89.9%

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

Alternative 4: 86.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 5:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x \cdot x, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (exp x) (exp (- x))) 5.0)
   (* x (fma (* x x) 0.16666666666666666 1.0))
   (* (* x x) (* x (fma x (* x 0.008333333333333333) 0.16666666666666666)))))

C

double code(double x) {
	double tmp;
	if ((exp(x) - exp(-x)) <= 5.0) {
		tmp = x * fma((x * x), 0.16666666666666666, 1.0);
	} else {
		tmp = (x * x) * (x * fma(x, (x * 0.008333333333333333), 0.16666666666666666));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) - exp(Float64(-x))) <= 5.0)
		tmp = Float64(x * fma(Float64(x * x), 0.16666666666666666, 1.0));
	else
		tmp = Float64(Float64(x * x) * Float64(x * fma(x, Float64(x * 0.008333333333333333), 0.16666666666666666)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 5.0], N[(x * N[(N[(x * x), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 40.5%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

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

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

      16. *-lowering-*.f64 94.9

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

   5. Simplified 94.9%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 88.4%

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

   if 5 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*l* N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 81.2

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

   5. Simplified 81.2%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 81.2%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 81.2%

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

Alternative 5: 86.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 5:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x \cdot x, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.008333333333333333 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (exp x) (exp (- x))) 5.0)
   (* x (fma (* x x) 0.16666666666666666 1.0))
   (* x (* 0.008333333333333333 (* x (* x (* x x)))))))

C

double code(double x) {
	double tmp;
	if ((exp(x) - exp(-x)) <= 5.0) {
		tmp = x * fma((x * x), 0.16666666666666666, 1.0);
	} else {
		tmp = x * (0.008333333333333333 * (x * (x * (x * x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) - exp(Float64(-x))) <= 5.0)
		tmp = Float64(x * fma(Float64(x * x), 0.16666666666666666, 1.0));
	else
		tmp = Float64(x * Float64(0.008333333333333333 * Float64(x * Float64(x * Float64(x * x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 5.0], N[(x * N[(N[(x * x), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.008333333333333333 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 40.5%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

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

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

      16. *-lowering-*.f64 94.9

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

   5. Simplified 94.9%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 88.4%

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

   if 5 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*l* N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 81.2

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

   5. Simplified 81.2%

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

   6. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. pow-plus N/A

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

      6. associate-*l* N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. pow-sqr N/A

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

      18. metadata-eval N/A

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

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

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

      20. metadata-eval N/A

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

      21. pow-plus N/A

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

      22. *-commutative N/A

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

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

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

      24. cube-mult N/A

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

      25. unpow2 N/A

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

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

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

      27. unpow2 N/A

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

      28. *-lowering-*.f64 81.2

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

   8. Simplified 81.2%

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

Alternative 6: 67.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (exp x) (exp (- x))) 5.0) x (* 0.16666666666666666 (* x (* x x)))))

C

double code(double x) {
	double tmp;
	if ((exp(x) - exp(-x)) <= 5.0) {
		tmp = x;
	} else {
		tmp = 0.16666666666666666 * (x * (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((exp(x) - exp(-x)) <= 5.0d0) then
        tmp = x
    else
        tmp = 0.16666666666666666d0 * (x * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((Math.exp(x) - Math.exp(-x)) <= 5.0) {
		tmp = x;
	} else {
		tmp = 0.16666666666666666 * (x * (x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (math.exp(x) - math.exp(-x)) <= 5.0:
		tmp = x
	else:
		tmp = 0.16666666666666666 * (x * (x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) - exp(Float64(-x))) <= 5.0)
		tmp = x;
	else
		tmp = Float64(0.16666666666666666 * Float64(x * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((exp(x) - exp(-x)) <= 5.0)
		tmp = x;
	else
		tmp = 0.16666666666666666 * (x * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 40.5%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 66.0%

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

   if 5 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 65.2

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

   5. Simplified 65.2%

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

   6. Taylor expanded in x around inf

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 65.2

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

   8. Simplified 65.2%

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

Alternative 7: 92.6% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in x around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. unpow2 N/A

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

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

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

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

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

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

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

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

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

   16. *-lowering-*.f64 93.6

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

5. Simplified 93.6%

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

   1. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

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

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

   4. associate-*l* N/A

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

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

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

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

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

   7. associate-*r* N/A

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

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

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

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

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

   10. associate-*r* N/A

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

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

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

   12. *-lowering-*.f64 93.6

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

7. Applied egg-rr 93.6%

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

Alternative 8: 92.6% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in x around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. unpow2 N/A

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

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

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

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

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

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

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

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

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

   16. *-lowering-*.f64 93.6

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

5. Simplified 93.6%

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

Alternative 9: 92.2% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in x around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. unpow2 N/A

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

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

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

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

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

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

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

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

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

   16. *-lowering-*.f64 93.6

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

5. Simplified 93.6%

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

   1. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

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

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

   4. associate-*l* N/A

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

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

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

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

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

   7. associate-*r* N/A

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

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

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

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

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

   10. associate-*r* N/A

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

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

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

   12. *-lowering-*.f64 93.6

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

7. Applied egg-rr 93.6%

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

8. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

   2. +-commutative N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. associate-*l/ N/A

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

   6. associate-/l* N/A

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

   7. metadata-eval N/A

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

   8. pow-sqr N/A

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

   9. associate-/l* N/A

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

   10. unpow2 N/A

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

   11. associate-/l/ N/A

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

   12. unpow2 N/A

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

   13. associate-/l* N/A

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

   14. *-inverses N/A

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

   15. *-rgt-identity N/A

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

   16. *-inverses N/A

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

   17. *-rgt-identity N/A

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

   18. unpow2 N/A

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

   19. associate-*r* N/A

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

   20. metadata-eval N/A

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

   21. pow-sqr N/A

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

   22. associate-*l* N/A

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

10. Simplified 93.0%

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

Alternative 10: 92.2% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in x around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. unpow2 N/A

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

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

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

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

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

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

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

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

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

   16. *-lowering-*.f64 93.6

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

5. Simplified 93.6%

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

   1. distribute-lft-in N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-rgt-identity N/A

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

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

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

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

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

   7. associate-*r* N/A

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

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

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

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

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

   10. associate-*r* N/A

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

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

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

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

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

   13. *-lowering-*.f64 93.6

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

7. Applied egg-rr 93.6%

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

8. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. pow-plus N/A

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

   3. associate-*l* N/A

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

   4. metadata-eval N/A

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

   5. pow-sqr N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

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

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. pow-sqr N/A

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

   13. metadata-eval N/A

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

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

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

   15. metadata-eval N/A

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

   16. pow-plus N/A

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

   17. *-commutative N/A

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

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

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

   19. cube-mult N/A

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

   20. unpow2 N/A

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

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

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

   22. unpow2 N/A

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

   23. *-lowering-*.f64 93.0

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

10. Simplified 93.0%

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

Alternative 11: 89.7% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in x around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. unpow2 N/A

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

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

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

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

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

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

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

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

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

   16. *-lowering-*.f64 93.6

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

5. Simplified 93.6%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 89.1

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

8. Simplified 89.1%

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

Alternative 12: 89.4% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. associate-*l* N/A

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

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

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

   6. unpow2 N/A

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

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

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

   8. *-commutative N/A

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

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

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. unpow2 N/A

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

   13. associate-*l* N/A

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

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

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

   15. *-lowering-*.f64 89.1

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

5. Simplified 89.1%

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

6. Taylor expanded in x around inf

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

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

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

   2. cube-mult N/A

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

   3. unpow2 N/A

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

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

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 88.6

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

8. Simplified 88.6%

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

Alternative 13: 83.5% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in x around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. unpow2 N/A

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

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

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

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

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

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

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

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

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

   16. *-lowering-*.f64 93.6

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

5. Simplified 93.6%

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

6. Taylor expanded in x around 0

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

8. Simplified 82.5%

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

Alternative 14: 51.8% accurate, 217.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in x around 0

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

5. Simplified 50.5%

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

Reproduce

herbie shell --seed 2024199 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))