Hyperbolic sine

Percentage Accurate: 54.5% → 100.0%

Time: 9.8s

Alternatives: 17

Speedup: 31.0×

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return sinh(x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.9%

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

   1. sinh-def N/A

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

   2. lower-sinh.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 68.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (exp x) (exp (- x))) 0.1) x (* x (* x x))))

C

double code(double x) {
	double tmp;
	if ((exp(x) - exp(-x)) <= 0.1) {
		tmp = x;
	} else {
		tmp = x * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((exp(x) - exp(-x)) <= 0.1d0) then
        tmp = x
    else
        tmp = x * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((Math.exp(x) - Math.exp(-x)) <= 0.1) {
		tmp = x;
	} else {
		tmp = x * (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (math.exp(x) - math.exp(-x)) <= 0.1:
		tmp = x
	else:
		tmp = x * (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) - exp(Float64(-x))) <= 0.1)
		tmp = x;
	else
		tmp = Float64(x * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((exp(x) - exp(-x)) <= 0.1)
		tmp = x;
	else
		tmp = x * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 38.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 95.0

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

   5. Simplified 95.0%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 68.8%

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

      1. *-rgt-identity 68.8

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

   10. Applied egg-rr 68.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 79.2

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

   5. Simplified 79.2%

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

   6. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 85.1

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

   8. Simplified 85.1%

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

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 74.9

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

   11. Simplified 74.9%

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

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{{x}^{3}} \]
   13. Step-by-step derivation

       1. cube-mult N/A

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

       2. unpow2 N/A

          \[\leadsto x \cdot \color{blue}{{x}^{2}} \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot {x}^{2}} \]

       4. unpow2 N/A

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

       5. lower-*.f64 68.8

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

   14. Simplified 68.8%

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

Alternative 3: 64.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (exp x) (exp (- x))) 0.1) x (* x x)))

C

double code(double x) {
	double tmp;
	if ((exp(x) - exp(-x)) <= 0.1) {
		tmp = x;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((exp(x) - exp(-x)) <= 0.1d0) then
        tmp = x
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((Math.exp(x) - Math.exp(-x)) <= 0.1) {
		tmp = x;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (math.exp(x) - math.exp(-x)) <= 0.1:
		tmp = x
	else:
		tmp = x * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) - exp(Float64(-x))) <= 0.1)
		tmp = x;
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((exp(x) - exp(-x)) <= 0.1)
		tmp = x;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 38.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 95.0

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

   5. Simplified 95.0%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 68.8%

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

      1. *-rgt-identity 68.8

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

   10. Applied egg-rr 68.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 79.2

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

   5. Simplified 79.2%

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

   6. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 85.1

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

   8. Simplified 85.1%

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

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 74.9

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

   11. Simplified 74.9%

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

   12. Taylor expanded in x around -inf

       \[\leadsto \color{blue}{{x}^{2}} \]
   13. Step-by-step derivation

       1. unpow2 N/A

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

       2. lower-*.f64 58.4

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

   14. Simplified 58.4%

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

Alternative 4: 94.4% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.9%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-*.f64 92.6

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

5. Simplified 92.6%

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

6. Taylor expanded in x around inf

   \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{5040} \cdot {x}^{5}}, \frac{1}{6}\right), 1\right) \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{5040} \cdot {x}^{5}}, \frac{1}{6}\right), 1\right) \]

   2. metadata-eval N/A

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

   3. pow-plus N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \frac{1}{5040} \cdot \color{blue}{\left({x}^{4} \cdot x\right)}, \frac{1}{6}\right), 1\right) \]

   4. lower-*.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \frac{1}{5040} \cdot \color{blue}{\left({x}^{4} \cdot x\right)}, \frac{1}{6}\right), 1\right) \]

   5. metadata-eval N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \frac{1}{5040} \cdot \left({x}^{\color{blue}{\left(3 + 1\right)}} \cdot x\right), \frac{1}{6}\right), 1\right) \]

   6. pow-plus N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \frac{1}{5040} \cdot \left(\color{blue}{\left({x}^{3} \cdot x\right)} \cdot x\right), \frac{1}{6}\right), 1\right) \]

   7. *-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \frac{1}{5040} \cdot \left(\color{blue}{\left(x \cdot {x}^{3}\right)} \cdot x\right), \frac{1}{6}\right), 1\right) \]

   8. lower-*.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \frac{1}{5040} \cdot \left(\color{blue}{\left(x \cdot {x}^{3}\right)} \cdot x\right), \frac{1}{6}\right), 1\right) \]

   9. cube-mult N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 94.2

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

8. Simplified 94.2%

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

9. Final simplification 94.2%

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

Alternative 5: 72.3% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.9%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-*.f64 92.6

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

5. Simplified 92.6%

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

6. Taylor expanded in x around 0

   \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{3}\right)}, \frac{1}{6}\right), 1\right) \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{3}\right)}, \frac{1}{6}\right), 1\right) \]

   2. +-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{5040} \cdot {x}^{3} + \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{5040}, {x}^{3}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

   4. cube-mult N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 72.4

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

8. Simplified 72.4%

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

Alternative 6: 72.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.9%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-*.f64 92.6

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

5. Simplified 92.6%

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

6. Taylor expanded in x around inf

   \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{5040} \cdot {x}^{5}}, \frac{1}{6}\right), 1\right) \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{5040} \cdot {x}^{5}}, \frac{1}{6}\right), 1\right) \]

   2. metadata-eval N/A

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

   3. pow-plus N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \frac{1}{5040} \cdot \color{blue}{\left({x}^{4} \cdot x\right)}, \frac{1}{6}\right), 1\right) \]

   4. lower-*.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \frac{1}{5040} \cdot \color{blue}{\left({x}^{4} \cdot x\right)}, \frac{1}{6}\right), 1\right) \]

   5. metadata-eval N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \frac{1}{5040} \cdot \left({x}^{\color{blue}{\left(3 + 1\right)}} \cdot x\right), \frac{1}{6}\right), 1\right) \]

   6. pow-plus N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \frac{1}{5040} \cdot \left(\color{blue}{\left({x}^{3} \cdot x\right)} \cdot x\right), \frac{1}{6}\right), 1\right) \]

   7. *-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \frac{1}{5040} \cdot \left(\color{blue}{\left(x \cdot {x}^{3}\right)} \cdot x\right), \frac{1}{6}\right), 1\right) \]

   8. lower-*.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \frac{1}{5040} \cdot \left(\color{blue}{\left(x \cdot {x}^{3}\right)} \cdot x\right), \frac{1}{6}\right), 1\right) \]

   9. cube-mult N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 94.2

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

8. Simplified 94.2%

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

9. Taylor expanded in x around inf

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

    1. unpow2 N/A

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

    2. lower-*.f64 72.3

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

11. Simplified 72.3%

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

12. Final simplification 72.3%

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

Alternative 7: 92.9% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.9%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-*.f64 92.6

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

5. Simplified 92.6%

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

Alternative 8: 92.8% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. unpow2 N/A

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

   13. associate-*l* N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 89.5

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

5. Simplified 89.5%

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

6. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

   2. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\frac{1}{120} \cdot {x}^{\color{blue}{\left(3 + 1\right)}} + \left(\frac{1}{6} \cdot \frac{1}{{x}^{4}}\right) \cdot {x}^{4}\right), x\right) \]

   3. pow-plus N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\frac{1}{120} \cdot \color{blue}{\left({x}^{3} \cdot x\right)} + \left(\frac{1}{6} \cdot \frac{1}{{x}^{4}}\right) \cdot {x}^{4}\right), x\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{\left(\frac{1}{120} \cdot {x}^{3}\right) \cdot x} + \left(\frac{1}{6} \cdot \frac{1}{{x}^{4}}\right) \cdot {x}^{4}\right), x\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\color{blue}{x \cdot \left(\frac{1}{120} \cdot {x}^{3}\right)} + \left(\frac{1}{6} \cdot \frac{1}{{x}^{4}}\right) \cdot {x}^{4}\right), x\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(\frac{1}{120} \cdot {x}^{3}\right) + \color{blue}{\frac{1}{6} \cdot \left(\frac{1}{{x}^{4}} \cdot {x}^{4}\right)}\right), x\right) \]

   7. lft-mult-inverse N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. cube-mult N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 92.4

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

8. Simplified 92.4%

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

9. Final simplification 92.4%

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

Alternative 9: 92.6% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. unpow2 N/A

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

   13. associate-*l* N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 89.5

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

5. Simplified 89.5%

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

6. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. pow-plus N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. cube-mult N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 92.0

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

8. Simplified 92.0%

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

9. Final simplification 92.0%

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

Alternative 10: 71.1% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.9%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-*.f64 92.6

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

5. Simplified 92.6%

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

6. Taylor expanded in x around inf

   \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{5040} \cdot {x}^{5}}, \frac{1}{6}\right), 1\right) \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{5040} \cdot {x}^{5}}, \frac{1}{6}\right), 1\right) \]

   2. metadata-eval N/A

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

   3. pow-plus N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \frac{1}{5040} \cdot \color{blue}{\left({x}^{4} \cdot x\right)}, \frac{1}{6}\right), 1\right) \]

   4. lower-*.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \frac{1}{5040} \cdot \color{blue}{\left({x}^{4} \cdot x\right)}, \frac{1}{6}\right), 1\right) \]

   5. metadata-eval N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \frac{1}{5040} \cdot \left({x}^{\color{blue}{\left(3 + 1\right)}} \cdot x\right), \frac{1}{6}\right), 1\right) \]

   6. pow-plus N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \frac{1}{5040} \cdot \left(\color{blue}{\left({x}^{3} \cdot x\right)} \cdot x\right), \frac{1}{6}\right), 1\right) \]

   7. *-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \frac{1}{5040} \cdot \left(\color{blue}{\left(x \cdot {x}^{3}\right)} \cdot x\right), \frac{1}{6}\right), 1\right) \]

   8. lower-*.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \frac{1}{5040} \cdot \left(\color{blue}{\left(x \cdot {x}^{3}\right)} \cdot x\right), \frac{1}{6}\right), 1\right) \]

   9. cube-mult N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 94.2

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

8. Simplified 94.2%

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

9. Taylor expanded in x around inf

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

    1. unpow2 N/A

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

    2. lower-*.f64 71.2

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

11. Simplified 71.2%

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

Alternative 11: 70.9% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. unpow2 N/A

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

   13. associate-*l* N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 89.5

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

5. Simplified 89.5%

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

6. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. pow-plus N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. cube-mult N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 92.0

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

8. Simplified 92.0%

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

9. Taylor expanded in x around inf

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

    1. unpow2 N/A

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

    2. lower-*.f64 70.8

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

11. Simplified 70.8%

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

Alternative 12: 90.0% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. unpow2 N/A

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

   13. associate-*l* N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 89.5

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

5. Simplified 89.5%

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

6. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. pow-plus N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. cube-mult N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 92.0

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

8. Simplified 92.0%

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

9. Taylor expanded in x around inf

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

    1. unpow2 N/A

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

    2. lower-*.f64 89.1

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

11. Simplified 89.1%

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

Alternative 13: 69.0% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. unpow2 N/A

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

   13. associate-*l* N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 89.5

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

5. Simplified 89.5%

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

6. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. pow-plus N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. cube-mult N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 92.0

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

8. Simplified 92.0%

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

9. Taylor expanded in x around inf

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

    1. unpow2 N/A

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

    2. lower-*.f64 69.0

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

11. Simplified 69.0%

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

Alternative 14: 84.2% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.9%

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

   1. sinh-def N/A

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

   2. lower-sinh.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 85.0

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

7. Simplified 85.0%

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

8. Final simplification 85.0%

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

Alternative 15: 83.9% accurate, 18.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. unpow2 N/A

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

   13. associate-*l* N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 89.5

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

5. Simplified 89.5%

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

6. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. pow-plus N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. cube-mult N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 92.0

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

8. Simplified 92.0%

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

9. Taylor expanded in x around inf

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

    1. unpow2 N/A

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

    2. lower-*.f64 69.0

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

11. Simplified 69.0%

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

12. Taylor expanded in x around inf

    \[\leadsto \color{blue}{{x}^{3} \cdot \left(1 + \frac{1}{{x}^{2}}\right)} \]
13. Step-by-step derivation

    1. cube-mult N/A

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

    2. unpow2 N/A

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

    3. associate-*l* N/A

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

    4. distribute-lft-in N/A

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

    5. *-rgt-identity N/A

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

    6. rgt-mult-inverse N/A

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

    7. distribute-lft-out N/A

       \[\leadsto \color{blue}{x \cdot {x}^{2} + x \cdot 1} \]

    8. *-rgt-identity N/A

       \[\leadsto x \cdot {x}^{2} + \color{blue}{x} \]

    9. lower-fma.f64 N/A

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

    10. unpow2 N/A

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

    11. lower-*.f64 84.9

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

14. Simplified 84.9%

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

Alternative 16: 62.6% accurate, 31.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return fma(x, x, x)
end

Wolfram

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

Derivation

1. Initial program 53.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. unpow2 N/A

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

   13. associate-*l* N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 89.5

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

5. Simplified 89.5%

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

6. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. pow-plus N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. cube-mult N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 92.0

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

8. Simplified 92.0%

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

9. Taylor expanded in x around inf

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

    1. unpow2 N/A

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

    2. lower-*.f64 69.0

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

11. Simplified 69.0%

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

12. Taylor expanded in x around -inf

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

    1. unpow2 N/A

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

    2. associate-*l* N/A

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

    3. distribute-lft-in N/A

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

    4. *-rgt-identity N/A

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

    5. rgt-mult-inverse N/A

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

    6. distribute-lft-out N/A

       \[\leadsto \color{blue}{x \cdot x + x \cdot 1} \]

    7. *-rgt-identity N/A

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

    8. lower-fma.f64 64.0

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

14. Simplified 64.0%

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

Alternative 17: 51.8% accurate, 217.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 53.9%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-*.f64 92.6

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

5. Simplified 92.6%

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

6. Taylor expanded in x around 0

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

8. Simplified 53.0%

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

   1. *-rgt-identity 53.0

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

10. Applied egg-rr 53.0%

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

Reproduce

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