Hyperbolic sine

Percentage Accurate: 54.5% → 100.0%

Time: 12.0s

Alternatives: 16

Speedup: 12.8×

• 49.8% 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 52.4%

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

   1. sinh-def N/A

      \[\leadsto \sinh x \]

   2. sinh-lowering-sinh.f64 100.0%

      \[\leadsto \mathsf{sinh.f64}\left(x\right) \]

4. Applied egg-rr 100.0%

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

Alternative 2: 75.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right)\\ t_1 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq 2 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(t\_0, t\_0 \cdot t\_1, -0.027777777777777776\right)}{\mathsf{fma}\left(x, x \cdot t\_0, -0.16666666666666666\right)}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.008333333333333333 \cdot \left(x \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x (* x 0.0001984126984126984) 0.008333333333333333))
        (t_1 (* x (* x (* x x)))))
   (if (<= x 2e+61)
     (fma
      (/
       (* (* x x) (fma t_0 (* t_0 t_1) -0.027777777777777776))
       (fma x (* x t_0) -0.16666666666666666))
      x
      x)
     (* 0.008333333333333333 (* x t_1)))))

C

double code(double x) {
	double t_0 = fma(x, (x * 0.0001984126984126984), 0.008333333333333333);
	double t_1 = x * (x * (x * x));
	double tmp;
	if (x <= 2e+61) {
		tmp = fma((((x * x) * fma(t_0, (t_0 * t_1), -0.027777777777777776)) / fma(x, (x * t_0), -0.16666666666666666)), x, x);
	} else {
		tmp = 0.008333333333333333 * (x * t_1);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(x, Float64(x * 0.0001984126984126984), 0.008333333333333333)
	t_1 = Float64(x * Float64(x * Float64(x * x)))
	tmp = 0.0
	if (x <= 2e+61)
		tmp = fma(Float64(Float64(Float64(x * x) * fma(t_0, Float64(t_0 * t_1), -0.027777777777777776)) / fma(x, Float64(x * t_0), -0.16666666666666666)), x, x);
	else
		tmp = Float64(0.008333333333333333 * Float64(x * t_1));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2e+61], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(t$95$0 * N[(t$95$0 * t$95$1), $MachinePrecision] + -0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x * t$95$0), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision], N[(0.008333333333333333 * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9999999999999999e61

   1. Initial program 39.7%

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

   3. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right) + \color{blue}{0} \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)}, 0\right) \]

   5. Simplified 91.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right), \color{blue}{x}, x\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(x \cdot x\right), \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right), x, x\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right), x, x\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right) + \frac{1}{6}\right)\right), x, x\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(\left(x \cdot x\right), \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right), \frac{1}{6}\right)\right), x, x\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right), \frac{1}{6}\right)\right), x, x\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \left(x \cdot \frac{1}{5040}\right), \frac{1}{120}\right), \frac{1}{6}\right)\right), x, x\right) \]

      11. *-lowering-*.f64 91.6%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{5040}\right), \frac{1}{120}\right), \frac{1}{6}\right)\right), x, x\right) \]

   7. Applied egg-rr 91.6%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right) + \frac{1}{6}\right) \cdot \left(x \cdot x\right)\right), x, x\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) - \frac{1}{6} \cdot \frac{1}{6}}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right) - \frac{1}{6}} \cdot \left(x \cdot x\right)\right), x, x\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\frac{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) - \frac{1}{6} \cdot \frac{1}{6}\right) \cdot \left(x \cdot x\right)}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right) - \frac{1}{6}}\right), x, x\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) - \frac{1}{6} \cdot \frac{1}{6}\right) \cdot \left(x \cdot x\right)\right), \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right) - \frac{1}{6}\right)\right), x, x\right) \]

   9. Applied egg-rr 72.1%

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

   if 1.9999999999999999e61 < 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. +-rgt-identity N/A

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + \color{blue}{0} \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}, 0\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + \color{blue}{1}\right), 0\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + 1\right), 0\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + 1\right), 0\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(x \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x\right) + 1\right), 0\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x\right)}, 1\right), 0\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right), 1\right), 0\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right), 1\right), 0\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\frac{1}{6}}\right)\right), 1\right), 0\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{120} + \frac{1}{6}\right)\right), 1\right), 0\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{120} + \frac{1}{6}\right)\right), 1\right), 0\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right), 1\right), 0\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{120}\right)}, \frac{1}{6}\right)\right), 1\right), 0\right) \]

      15. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{120}}\right), \frac{1}{6}\right)\right), 1\right), 0\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right), 0\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left({x}^{4} \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right)}, 0\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{\left(2 \cdot 2\right)} \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right), 0\right) \]

      2. pow-sqr N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(\color{blue}{\frac{1}{120}} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right), 0\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right)}\right), 0\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{{x}^{2}} + \color{blue}{\frac{1}{120}}\right)\right)\right), 0\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2} + \color{blue}{\frac{1}{120} \cdot {x}^{2}}\right)\right), 0\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \left(\frac{1}{{x}^{2}} \cdot {x}^{2}\right) + \color{blue}{\frac{1}{120}} \cdot {x}^{2}\right)\right), 0\right) \]

      7. lft-mult-inverse N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{6} \cdot 1 + \frac{1}{120} \cdot {x}^{2}\right)\right), 0\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{120}} \cdot {x}^{2}\right)\right), 0\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {x}^{2}\right)\right), 0\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}\right), 0\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}\right), 0\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right)\right), 0\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\frac{1}{6}}\right)\right)\right), 0\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{120} + \frac{1}{6}\right)\right)\right), 0\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{120} + \frac{1}{6}\right)\right)\right), 0\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right)\right), 0\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{120} \cdot x\right) + \frac{1}{6}\right)\right)\right), 0\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{120} \cdot x\right)}, \frac{1}{6}\right)\right)\right), 0\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{120}}\right), \frac{1}{6}\right)\right)\right), 0\right) \]

      20. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{120}}\right), \frac{1}{6}\right)\right)\right), 0\right) \]

   8. Simplified 100.0%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right)\right)}, 0\right) \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right), \color{blue}{\left(x \cdot x\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right), \left(\color{blue}{x} \cdot x\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{120} + \frac{1}{6}\right)\right), \left(x \cdot x\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot x\right), \frac{1}{120}, \frac{1}{6}\right)\right), \left(x \cdot x\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{120}, \frac{1}{6}\right)\right), \left(x \cdot x\right)\right) \]

      9. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{120}, \frac{1}{6}\right)\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   10. Applied egg-rr 100.0%

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

   11. Taylor expanded in x around inf

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

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \color{blue}{\left({x}^{5}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \left({x}^{\left(4 + \color{blue}{1}\right)}\right)\right) \]

       3. pow-plus N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \left({x}^{4} \cdot \color{blue}{x}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \left(x \cdot \color{blue}{{x}^{4}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{4}\right)}\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right)\right) \]

       7. pow-plus N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \left({x}^{3} \cdot \color{blue}{x}\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{{x}^{3}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right)\right)\right) \]

       10. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   13. Simplified 100.0%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), -0.027777777777777776\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right)}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.008333333333333333 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right)\\ \mathbf{if}\;x \leq 2 \cdot 10^{+61}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(x \cdot x, t\_0 \cdot \left(\left(x \cdot x\right) \cdot t\_0\right), -1\right)}{\mathsf{fma}\left(t\_0, x \cdot x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.008333333333333333 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x (* x 0.008333333333333333) 0.16666666666666666)))
   (if (<= x 2e+61)
     (/
      (* x (fma (* x x) (* t_0 (* (* x x) t_0)) -1.0))
      (fma t_0 (* x x) -1.0))
     (* 0.008333333333333333 (* x (* x (* x (* x x))))))))

C

double code(double x) {
	double t_0 = fma(x, (x * 0.008333333333333333), 0.16666666666666666);
	double tmp;
	if (x <= 2e+61) {
		tmp = (x * fma((x * x), (t_0 * ((x * x) * t_0)), -1.0)) / fma(t_0, (x * x), -1.0);
	} else {
		tmp = 0.008333333333333333 * (x * (x * (x * (x * x))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(x, Float64(x * 0.008333333333333333), 0.16666666666666666)
	tmp = 0.0
	if (x <= 2e+61)
		tmp = Float64(Float64(x * fma(Float64(x * x), Float64(t_0 * Float64(Float64(x * x) * t_0)), -1.0)) / fma(t_0, Float64(x * x), -1.0));
	else
		tmp = Float64(0.008333333333333333 * Float64(x * Float64(x * Float64(x * Float64(x * x)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]}, If[LessEqual[x, 2e+61], N[(N[(x * N[(N[(x * x), $MachinePrecision] * N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(0.008333333333333333 * N[(x * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9999999999999999e61

   1. Initial program 39.7%

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

   3. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + \color{blue}{0} \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}, 0\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + \color{blue}{1}\right), 0\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + 1\right), 0\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + 1\right), 0\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(x \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x\right) + 1\right), 0\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x\right)}, 1\right), 0\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right), 1\right), 0\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right), 1\right), 0\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\frac{1}{6}}\right)\right), 1\right), 0\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{120} + \frac{1}{6}\right)\right), 1\right), 0\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{120} + \frac{1}{6}\right)\right), 1\right), 0\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right), 1\right), 0\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{120}\right)}, \frac{1}{6}\right)\right), 1\right), 0\right) \]

      15. *-lowering-*.f64 89.2%

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{120}}\right), \frac{1}{6}\right)\right), 1\right), 0\right) \]

   5. Simplified 89.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. flip-+ N/A

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

      4. associate-*l/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right)\right) - 1 \cdot 1\right) \cdot x\right), \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right) - 1\right)}\right) \]

   7. Applied egg-rr 71.5%

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

   if 1.9999999999999999e61 < 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. +-rgt-identity N/A

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + \color{blue}{0} \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}, 0\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + \color{blue}{1}\right), 0\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + 1\right), 0\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + 1\right), 0\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(x \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x\right) + 1\right), 0\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x\right)}, 1\right), 0\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right), 1\right), 0\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right), 1\right), 0\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\frac{1}{6}}\right)\right), 1\right), 0\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{120} + \frac{1}{6}\right)\right), 1\right), 0\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{120} + \frac{1}{6}\right)\right), 1\right), 0\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right), 1\right), 0\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{120}\right)}, \frac{1}{6}\right)\right), 1\right), 0\right) \]

      15. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{120}}\right), \frac{1}{6}\right)\right), 1\right), 0\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right), 0\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left({x}^{4} \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right)}, 0\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{\left(2 \cdot 2\right)} \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right), 0\right) \]

      2. pow-sqr N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(\color{blue}{\frac{1}{120}} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right), 0\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right)}\right), 0\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{{x}^{2}} + \color{blue}{\frac{1}{120}}\right)\right)\right), 0\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2} + \color{blue}{\frac{1}{120} \cdot {x}^{2}}\right)\right), 0\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \left(\frac{1}{{x}^{2}} \cdot {x}^{2}\right) + \color{blue}{\frac{1}{120}} \cdot {x}^{2}\right)\right), 0\right) \]

      7. lft-mult-inverse N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{6} \cdot 1 + \frac{1}{120} \cdot {x}^{2}\right)\right), 0\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{120}} \cdot {x}^{2}\right)\right), 0\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {x}^{2}\right)\right), 0\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}\right), 0\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}\right), 0\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right)\right), 0\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\frac{1}{6}}\right)\right)\right), 0\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{120} + \frac{1}{6}\right)\right)\right), 0\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{120} + \frac{1}{6}\right)\right)\right), 0\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right)\right), 0\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{120} \cdot x\right) + \frac{1}{6}\right)\right)\right), 0\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{120} \cdot x\right)}, \frac{1}{6}\right)\right)\right), 0\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{120}}\right), \frac{1}{6}\right)\right)\right), 0\right) \]

      20. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{120}}\right), \frac{1}{6}\right)\right)\right), 0\right) \]

   8. Simplified 100.0%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right)\right)}, 0\right) \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right), \color{blue}{\left(x \cdot x\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right), \left(\color{blue}{x} \cdot x\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{120} + \frac{1}{6}\right)\right), \left(x \cdot x\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot x\right), \frac{1}{120}, \frac{1}{6}\right)\right), \left(x \cdot x\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{120}, \frac{1}{6}\right)\right), \left(x \cdot x\right)\right) \]

      9. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{120}, \frac{1}{6}\right)\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   10. Applied egg-rr 100.0%

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

   11. Taylor expanded in x around inf

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

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \color{blue}{\left({x}^{5}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \left({x}^{\left(4 + \color{blue}{1}\right)}\right)\right) \]

       3. pow-plus N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \left({x}^{4} \cdot \color{blue}{x}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \left(x \cdot \color{blue}{{x}^{4}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{4}\right)}\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right)\right) \]

       7. pow-plus N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \left({x}^{3} \cdot \color{blue}{x}\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{{x}^{3}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right)\right)\right) \]

       10. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   13. Simplified 100.0%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+61}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right) \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right)\right), -1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), x \cdot x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.008333333333333333 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x \cdot 0.008333333333333333, x \cdot \left(x \cdot x\right), \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.6)
   (*
    x
    (fma
     (* x 0.008333333333333333)
     (* x (* x x))
     (fma x (* x 0.16666666666666666) 1.0)))
   (*
    x
    (*
     x
     (*
      x
      (* x (* x (fma x (* x 0.0001984126984126984) 0.008333333333333333))))))))

C

double code(double x) {
	double tmp;
	if (x <= 5.6) {
		tmp = x * fma((x * 0.008333333333333333), (x * (x * x)), fma(x, (x * 0.16666666666666666), 1.0));
	} else {
		tmp = x * (x * (x * (x * (x * fma(x, (x * 0.0001984126984126984), 0.008333333333333333)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.6)
		tmp = Float64(x * fma(Float64(x * 0.008333333333333333), Float64(x * Float64(x * x)), fma(x, Float64(x * 0.16666666666666666), 1.0)));
	else
		tmp = Float64(x * Float64(x * Float64(x * Float64(x * Float64(x * fma(x, Float64(x * 0.0001984126984126984), 0.008333333333333333))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5.6], N[(x * N[(N[(x * 0.008333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999996

   1. Initial program 35.2%

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

      1. sinh-def N/A

         \[\leadsto \sinh x \]

      2. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{sinh.f64}\left(x\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + \color{blue}{1}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)}, 1\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right), 1\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right), 1\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \color{blue}{\frac{1}{6}}\right), 1\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}\right), 1\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + \frac{1}{6}\right), 1\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)}, \frac{1}{6}\right), 1\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)}\right), \frac{1}{6}\right), 1\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{5040} \cdot {x}^{2} + \color{blue}{\frac{1}{120}}\right)\right), \frac{1}{6}\right), 1\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{5040} + \frac{1}{120}\right)\right), \frac{1}{6}\right), 1\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{5040}}, \frac{1}{120}\right)\right), \frac{1}{6}\right), 1\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot x\right), \frac{1}{5040}, \frac{1}{120}\right)\right), \frac{1}{6}\right), 1\right)\right) \]

      15. *-lowering-*.f64 96.5%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{5040}, \frac{1}{120}\right)\right), \frac{1}{6}\right), 1\right)\right) \]

   7. Simplified 96.5%

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

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right)\right) \cdot \left(x \cdot x\right) + \frac{1}{6} \cdot \left(x \cdot x\right)\right) + 1\right)\right) \]

      2. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right)\right) \cdot \left(x \cdot x\right) + \color{blue}{\left(\frac{1}{6} \cdot \left(x \cdot x\right) + 1\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) \cdot x\right) \cdot \left(x \cdot x\right) + \left(\color{blue}{\frac{1}{6}} \cdot \left(x \cdot x\right) + 1\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(\color{blue}{\frac{1}{6} \cdot \left(x \cdot x\right)} + 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right), \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}, \left(\frac{1}{6} \cdot \left(x \cdot x\right) + 1\right)\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left(\left(\left(x \cdot x\right) \cdot \frac{1}{5040}\right) \cdot x + \frac{1}{120} \cdot x\right), \left(\color{blue}{x} \cdot \left(x \cdot x\right)\right), \left(\frac{1}{6} \cdot \left(x \cdot x\right) + 1\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left(\left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) \cdot x + \frac{1}{120} \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right), \left(\frac{1}{6} \cdot \left(x \cdot x\right) + 1\right)\right)\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right), \left(\color{blue}{x} \cdot \left(x \cdot x\right)\right), \left(\frac{1}{6} \cdot \left(x \cdot x\right) + 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right), \left(\color{blue}{x} \cdot \left(x \cdot x\right)\right), \left(\frac{1}{6} \cdot \left(x \cdot x\right) + 1\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \frac{1}{5040}\right), \frac{1}{120}\right)\right), \left(x \cdot \left(x \cdot x\right)\right), \left(\frac{1}{6} \cdot \left(x \cdot x\right) + 1\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{5040}\right), \frac{1}{120}\right)\right), \left(x \cdot \left(x \cdot x\right)\right), \left(\frac{1}{6} \cdot \left(x \cdot x\right) + 1\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{5040}\right), \frac{1}{120}\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot x\right)}\right), \left(\frac{1}{6} \cdot \left(x \cdot x\right) + 1\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{5040}\right), \frac{1}{120}\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right), \left(\frac{1}{6} \cdot \left(x \cdot x\right) + 1\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{5040}\right), \frac{1}{120}\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(\left(x \cdot x\right) \cdot \frac{1}{6} + 1\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{5040}\right), \frac{1}{120}\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(x \cdot \left(x \cdot \frac{1}{6}\right) + 1\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{5040}\right), \frac{1}{120}\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{fma.f64}\left(x, \left(x \cdot \frac{1}{6}\right), 1\right)\right)\right) \]

      17. *-lowering-*.f64 96.5%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{5040}\right), \frac{1}{120}\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{6}\right), 1\right)\right)\right) \]

   9. Applied egg-rr 96.5%

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

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\color{blue}{\left(\frac{1}{120} \cdot x\right)}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{6}\right), 1\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot \frac{1}{120}\right), \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{6}\right), 1\right)\right)\right) \]

       2. *-lowering-*.f64 95.4%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{120}\right), \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{6}\right), 1\right)\right)\right) \]

   12. Simplified 95.4%

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

   if 5.5999999999999996 < x

   1. Initial program 100.0%

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

      1. sinh-def N/A

         \[\leadsto \sinh x \]

      2. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{sinh.f64}\left(x\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + \color{blue}{1}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)}, 1\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right), 1\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right), 1\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \color{blue}{\frac{1}{6}}\right), 1\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}\right), 1\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + \frac{1}{6}\right), 1\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)}, \frac{1}{6}\right), 1\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)}\right), \frac{1}{6}\right), 1\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{5040} \cdot {x}^{2} + \color{blue}{\frac{1}{120}}\right)\right), \frac{1}{6}\right), 1\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{5040} + \frac{1}{120}\right)\right), \frac{1}{6}\right), 1\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{5040}}, \frac{1}{120}\right)\right), \frac{1}{6}\right), 1\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot x\right), \frac{1}{5040}, \frac{1}{120}\right)\right), \frac{1}{6}\right), 1\right)\right) \]

      15. *-lowering-*.f64 84.7%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{5040}, \frac{1}{120}\right)\right), \frac{1}{6}\right), 1\right)\right) \]

   7. Simplified 84.7%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{6} \cdot \left(\frac{1}{5040} + \frac{1}{120} \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]

   10. Simplified 84.7%

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

Alternative 5: 91.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.6)
   (fma (* x (* x x)) (fma x (* x 0.008333333333333333) 0.16666666666666666) x)
   (*
    x
    (*
     x
     (*
      x
      (* x (* x (fma x (* x 0.0001984126984126984) 0.008333333333333333))))))))

C

double code(double x) {
	double tmp;
	if (x <= 5.6) {
		tmp = fma((x * (x * x)), fma(x, (x * 0.008333333333333333), 0.16666666666666666), x);
	} else {
		tmp = x * (x * (x * (x * (x * fma(x, (x * 0.0001984126984126984), 0.008333333333333333)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.6)
		tmp = fma(Float64(x * Float64(x * x)), fma(x, Float64(x * 0.008333333333333333), 0.16666666666666666), x);
	else
		tmp = Float64(x * Float64(x * Float64(x * Float64(x * Float64(x * fma(x, Float64(x * 0.0001984126984126984), 0.008333333333333333))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5.6], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + x), $MachinePrecision], N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999996

   1. Initial program 35.2%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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. +-rgt-identity N/A

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + \color{blue}{0} \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}, 0\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + \color{blue}{1}\right), 0\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + 1\right), 0\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + 1\right), 0\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(x \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x\right) + 1\right), 0\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x\right)}, 1\right), 0\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right), 1\right), 0\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right), 1\right), 0\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\frac{1}{6}}\right)\right), 1\right), 0\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{120} + \frac{1}{6}\right)\right), 1\right), 0\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{120} + \frac{1}{6}\right)\right), 1\right), 0\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right), 1\right), 0\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{120}\right)}, \frac{1}{6}\right)\right), 1\right), 0\right) \]

      15. *-lowering-*.f64 95.4%

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{120}}\right), \frac{1}{6}\right)\right), 1\right), 0\right) \]

   5. Simplified 95.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. cube-unmult N/A

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

      6. *-rgt-identity N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(\left({x}^{3}\right), \color{blue}{\left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)}, x\right) \]

      8. cube-unmult N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{120}\right)} + \frac{1}{6}\right), x\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{120}\right)} + \frac{1}{6}\right), x\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{120}\right)} + \frac{1}{6}\right), x\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{120}\right)}, \frac{1}{6}\right), x\right) \]

      12. *-lowering-*.f64 95.4%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{120}}\right), \frac{1}{6}\right), x\right) \]

   7. Applied egg-rr 95.4%

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

   if 5.5999999999999996 < x

   1. Initial program 100.0%

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

      1. sinh-def N/A

         \[\leadsto \sinh x \]

      2. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{sinh.f64}\left(x\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + \color{blue}{1}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)}, 1\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right), 1\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right), 1\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \color{blue}{\frac{1}{6}}\right), 1\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}\right), 1\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + \frac{1}{6}\right), 1\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)}, \frac{1}{6}\right), 1\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)}\right), \frac{1}{6}\right), 1\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{5040} \cdot {x}^{2} + \color{blue}{\frac{1}{120}}\right)\right), \frac{1}{6}\right), 1\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{5040} + \frac{1}{120}\right)\right), \frac{1}{6}\right), 1\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{5040}}, \frac{1}{120}\right)\right), \frac{1}{6}\right), 1\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot x\right), \frac{1}{5040}, \frac{1}{120}\right)\right), \frac{1}{6}\right), 1\right)\right) \]

      15. *-lowering-*.f64 84.7%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{5040}, \frac{1}{120}\right)\right), \frac{1}{6}\right), 1\right)\right) \]

   7. Simplified 84.7%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{6} \cdot \left(\frac{1}{5040} + \frac{1}{120} \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]

   10. Simplified 84.7%

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

Alternative 6: 91.5% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq 7.6:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot t\_0\right) \cdot \left(0.0001984126984126984 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= x 7.6)
     (fma t_0 (fma x (* x 0.008333333333333333) 0.16666666666666666) x)
     (* (* x t_0) (* 0.0001984126984126984 t_0)))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= 7.6) {
		tmp = fma(t_0, fma(x, (x * 0.008333333333333333), 0.16666666666666666), x);
	} else {
		tmp = (x * t_0) * (0.0001984126984126984 * t_0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= 7.6)
		tmp = fma(t_0, fma(x, Float64(x * 0.008333333333333333), 0.16666666666666666), x);
	else
		tmp = Float64(Float64(x * t_0) * Float64(0.0001984126984126984 * t_0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 7.6], N[(t$95$0 * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + x), $MachinePrecision], N[(N[(x * t$95$0), $MachinePrecision] * N[(0.0001984126984126984 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 7.5999999999999996

   1. Initial program 35.2%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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. +-rgt-identity N/A

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + \color{blue}{0} \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}, 0\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + \color{blue}{1}\right), 0\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + 1\right), 0\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + 1\right), 0\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(x \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x\right) + 1\right), 0\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x\right)}, 1\right), 0\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right), 1\right), 0\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right), 1\right), 0\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\frac{1}{6}}\right)\right), 1\right), 0\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{120} + \frac{1}{6}\right)\right), 1\right), 0\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{120} + \frac{1}{6}\right)\right), 1\right), 0\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right), 1\right), 0\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{120}\right)}, \frac{1}{6}\right)\right), 1\right), 0\right) \]

      15. *-lowering-*.f64 95.4%

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{120}}\right), \frac{1}{6}\right)\right), 1\right), 0\right) \]

   5. Simplified 95.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. cube-unmult N/A

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

      6. *-rgt-identity N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(\left({x}^{3}\right), \color{blue}{\left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)}, x\right) \]

      8. cube-unmult N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{120}\right)} + \frac{1}{6}\right), x\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{120}\right)} + \frac{1}{6}\right), x\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{120}\right)} + \frac{1}{6}\right), x\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{120}\right)}, \frac{1}{6}\right), x\right) \]

      12. *-lowering-*.f64 95.4%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{120}}\right), \frac{1}{6}\right), x\right) \]

   7. Applied egg-rr 95.4%

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

   if 7.5999999999999996 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right) + \color{blue}{0} \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)}, 0\right) \]

   5. Simplified 84.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right), 0\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{5040} \cdot {x}^{6}\right)}, 0\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\frac{1}{5040} \cdot {x}^{\left(2 \cdot \color{blue}{3}\right)}\right), 0\right) \]

      2. pow-sqr N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\frac{1}{5040} \cdot \left({x}^{3} \cdot \color{blue}{{x}^{3}}\right)\right), 0\right) \]

      3. cube-prod N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\frac{1}{5040} \cdot {\left(x \cdot x\right)}^{\color{blue}{3}}\right), 0\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\frac{1}{5040} \cdot {\left({x}^{2}\right)}^{3}\right), 0\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\frac{1}{5040} \cdot \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}}\right)\right), 0\right) \]

      6. pow-sqr N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\frac{1}{5040} \cdot \left({x}^{\left(2 \cdot 2\right)} \cdot {\color{blue}{x}}^{2}\right)\right), 0\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\frac{1}{5040} \cdot \left({x}^{4} \cdot {x}^{2}\right)\right), 0\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\left(\frac{1}{5040} \cdot {x}^{4}\right) \cdot \color{blue}{{x}^{2}}\right), 0\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\left({x}^{4} \cdot \frac{1}{5040}\right) \cdot {\color{blue}{x}}^{2}\right), 0\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{4} \cdot \color{blue}{\left(\frac{1}{5040} \cdot {x}^{2}\right)}\right), 0\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{\left(2 \cdot 2\right)} \cdot \left(\frac{1}{5040} \cdot {x}^{2}\right)\right), 0\right) \]

      12. pow-sqr N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(\color{blue}{\frac{1}{5040}} \cdot {x}^{2}\right)\right), 0\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{5040} \cdot {x}^{2}\right)\right)}\right), 0\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2}} \cdot \left(\frac{1}{5040} \cdot {x}^{2}\right)\right)\right), 0\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{5040} \cdot {x}^{2}\right)\right)\right)}\right), 0\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{5040} \cdot {x}^{2}\right)\right)\right)}\right), 0\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{5040} \cdot {x}^{2}\right)\right)}\right)\right), 0\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{5040} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}}\right)\right)\right), 0\right) \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{5040} \cdot \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)}\right)\right)\right), 0\right) \]

      20. pow-sqr N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{5040} \cdot {x}^{\color{blue}{\left(2 \cdot 2\right)}}\right)\right)\right), 0\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{5040} \cdot {x}^{4}\right)\right)\right), 0\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{5040}, \color{blue}{\left({x}^{4}\right)}\right)\right)\right), 0\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{5040}, \left({x}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right)\right), 0\right) \]

      24. pow-sqr N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{5040}, \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right)\right)\right)\right), 0\right) \]

      25. unpow2 N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{5040}, \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right)\right)\right)\right), 0\right) \]

      26. associate-*l* N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{5040}, \left(x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right)\right)\right)\right), 0\right) \]

   8. Simplified 84.7%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \left(0.0001984126984126984 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}, 0\right) \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{5040}\right)\right), \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right)\right), \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) \cdot x\right), \left(\color{blue}{x} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(x \cdot x\right) \cdot \frac{1}{5040}\right) \cdot x\right), \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{5040} \cdot \left(x \cdot x\right)\right) \cdot x\right), \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{5040} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \left(\color{blue}{x} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{5040} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{5040}, \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\color{blue}{x} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{5040}, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right), \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

      19. *-lowering-*.f64 84.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   10. Applied egg-rr 84.7%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.6:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.0001984126984126984 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.8% accurate, 5.6× speedup

Math

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

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(Float64(x * 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[(N[(x * x), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.4%

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

   1. sinh-def N/A

      \[\leadsto \sinh x \]

   2. sinh-lowering-sinh.f64 100.0%

      \[\leadsto \mathsf{sinh.f64}\left(x\right) \]

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + \color{blue}{1}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)}, 1\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right), 1\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right), 1\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \color{blue}{\frac{1}{6}}\right), 1\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}\right), 1\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + \frac{1}{6}\right), 1\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)}, \frac{1}{6}\right), 1\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)}\right), \frac{1}{6}\right), 1\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{5040} \cdot {x}^{2} + \color{blue}{\frac{1}{120}}\right)\right), \frac{1}{6}\right), 1\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{5040} + \frac{1}{120}\right)\right), \frac{1}{6}\right), 1\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{5040}}, \frac{1}{120}\right)\right), \frac{1}{6}\right), 1\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot x\right), \frac{1}{5040}, \frac{1}{120}\right)\right), \frac{1}{6}\right), 1\right)\right) \]

   15. *-lowering-*.f64 93.4%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{5040}, \frac{1}{120}\right)\right), \frac{1}{6}\right), 1\right)\right) \]

7. Simplified 93.4%

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

Alternative 8: 92.4% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.4%

   \[\frac{e^{x} - e^{-x}}{2} \]
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. +-rgt-identity N/A

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right) + \color{blue}{0} \]

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

      \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)}, 0\right) \]

5. Simplified 93.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right), 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

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

      \[\leadsto \mathsf{fma.f64}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right), \color{blue}{x}, x\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(x \cdot x\right), \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right), x, x\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right), x, x\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right) + \frac{1}{6}\right)\right), x, x\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(\left(x \cdot x\right), \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right), \frac{1}{6}\right)\right), x, x\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(x \cdot \frac{1}{5040}\right) + \frac{1}{120}\right), \frac{1}{6}\right)\right), x, x\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \left(x \cdot \frac{1}{5040}\right), \frac{1}{120}\right), \frac{1}{6}\right)\right), x, x\right) \]

   11. *-lowering-*.f64 93.4%

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{5040}\right), \frac{1}{120}\right), \frac{1}{6}\right)\right), x, x\right) \]

7. Applied egg-rr 93.4%

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

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \color{blue}{\left({x}^{4} \cdot \left(\frac{1}{5040} + \frac{1}{120} \cdot \frac{1}{{x}^{2}}\right)\right)}\right), x, x\right) \]
9. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{4} + \left(\frac{1}{120} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{4}\right)\right), x, x\right) \]

   2. associate-*r/ N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{4} + \frac{\frac{1}{120} \cdot 1}{{x}^{2}} \cdot {x}^{4}\right)\right), x, x\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{4} + \frac{\frac{1}{120}}{{x}^{2}} \cdot {x}^{4}\right)\right), x, x\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{4} + \frac{\frac{1}{120} \cdot {x}^{4}}{{x}^{2}}\right)\right), x, x\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{4} + \frac{1}{120} \cdot \frac{{x}^{4}}{{x}^{2}}\right)\right), x, x\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{4} + \frac{1}{120} \cdot \frac{{x}^{\left(2 \cdot 2\right)}}{{x}^{2}}\right)\right), x, x\right) \]

   7. pow-sqr N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{4} + \frac{1}{120} \cdot \frac{{x}^{2} \cdot {x}^{2}}{{x}^{2}}\right)\right), x, x\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{4} + \frac{1}{120} \cdot \left({x}^{2} \cdot \frac{{x}^{2}}{{x}^{2}}\right)\right)\right), x, x\right) \]

   9. *-rgt-identity N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{4} + \frac{1}{120} \cdot \left({x}^{2} \cdot \frac{{x}^{2} \cdot 1}{{x}^{2}}\right)\right)\right), x, x\right) \]

   10. associate-*r/ N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{4} + \frac{1}{120} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right), x, x\right) \]

   11. rgt-mult-inverse N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{4} + \frac{1}{120} \cdot \left({x}^{2} \cdot 1\right)\right)\right), x, x\right) \]

   12. *-rgt-identity N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{4} + \frac{1}{120} \cdot {x}^{2}\right)\right), x, x\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{4} + \frac{1}{120} \cdot \left(x \cdot x\right)\right)\right), x, x\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{4} + \frac{1}{120} \cdot {x}^{2}\right)\right), x, x\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{\left(3 + 1\right)} + \frac{1}{120} \cdot {x}^{2}\right)\right), x, x\right) \]

   16. pow-plus N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot \left({x}^{3} \cdot x\right) + \frac{1}{120} \cdot {x}^{2}\right)\right), x, x\right) \]

   17. associate-*l* N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(\frac{1}{5040} \cdot {x}^{3}\right) \cdot x + \frac{1}{120} \cdot {x}^{2}\right)\right), x, x\right) \]

10. Simplified 92.8%

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

Alternative 9: 92.4% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(x * fma(Float64(x * x), Float64(x * Float64(x * fma(x, Float64(x * 0.0001984126984126984), 0.008333333333333333))), 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]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.4%

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

   1. sinh-def N/A

      \[\leadsto \sinh x \]

   2. sinh-lowering-sinh.f64 100.0%

      \[\leadsto \mathsf{sinh.f64}\left(x\right) \]

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + \color{blue}{1}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)}, 1\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right), 1\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right), 1\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \color{blue}{\frac{1}{6}}\right), 1\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}\right), 1\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + \frac{1}{6}\right), 1\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)}, \frac{1}{6}\right), 1\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)}\right), \frac{1}{6}\right), 1\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{5040} \cdot {x}^{2} + \color{blue}{\frac{1}{120}}\right)\right), \frac{1}{6}\right), 1\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{5040} + \frac{1}{120}\right)\right), \frac{1}{6}\right), 1\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{5040}}, \frac{1}{120}\right)\right), \frac{1}{6}\right), 1\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot x\right), \frac{1}{5040}, \frac{1}{120}\right)\right), \frac{1}{6}\right), 1\right)\right) \]

   15. *-lowering-*.f64 93.4%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{5040}, \frac{1}{120}\right)\right), \frac{1}{6}\right), 1\right)\right) \]

7. Simplified 93.4%

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

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \color{blue}{\left({x}^{4} \cdot \left(\frac{1}{5040} + \frac{1}{120} \cdot \frac{1}{{x}^{2}}\right)\right)}, 1\right)\right) \]
9. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{4} + \color{blue}{\left(\frac{1}{120} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{4}}\right), 1\right)\right) \]

   2. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{4} + \frac{\frac{1}{120} \cdot 1}{{x}^{2}} \cdot {\color{blue}{x}}^{4}\right), 1\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{4} + \frac{\frac{1}{120}}{{x}^{2}} \cdot {x}^{4}\right), 1\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{4} + \frac{\frac{1}{120} \cdot {x}^{4}}{\color{blue}{{x}^{2}}}\right), 1\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{4} + \frac{1}{120} \cdot \color{blue}{\frac{{x}^{4}}{{x}^{2}}}\right), 1\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{4} + \frac{1}{120} \cdot \frac{{x}^{\left(2 \cdot 2\right)}}{{x}^{2}}\right), 1\right)\right) \]

   7. pow-sqr N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{4} + \frac{1}{120} \cdot \frac{{x}^{2} \cdot {x}^{2}}{{\color{blue}{x}}^{2}}\right), 1\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{4} + \frac{1}{120} \cdot \left({x}^{2} \cdot \color{blue}{\frac{{x}^{2}}{{x}^{2}}}\right)\right), 1\right)\right) \]

   9. *-rgt-identity N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{4} + \frac{1}{120} \cdot \left({x}^{2} \cdot \frac{{x}^{2} \cdot 1}{{\color{blue}{x}}^{2}}\right)\right), 1\right)\right) \]

   10. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{4} + \frac{1}{120} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \color{blue}{\frac{1}{{x}^{2}}}\right)\right)\right), 1\right)\right) \]

   11. rgt-mult-inverse N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{4} + \frac{1}{120} \cdot \left({x}^{2} \cdot 1\right)\right), 1\right)\right) \]

   12. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{4} + \frac{1}{120} \cdot {x}^{\color{blue}{2}}\right), 1\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot {x}^{\left(3 + 1\right)} + \frac{1}{120} \cdot {x}^{2}\right), 1\right)\right) \]

   14. pow-plus N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5040} \cdot \left({x}^{3} \cdot x\right) + \frac{1}{120} \cdot {x}^{2}\right), 1\right)\right) \]

   15. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(\frac{1}{5040} \cdot {x}^{3}\right) \cdot x + \color{blue}{\frac{1}{120}} \cdot {x}^{2}\right), 1\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(\frac{1}{5040} \cdot {x}^{3}\right) \cdot x + \frac{1}{120} \cdot \left(x \cdot \color{blue}{x}\right)\right), 1\right)\right) \]

   17. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(\frac{1}{5040} \cdot {x}^{3}\right) \cdot x + \left(\frac{1}{120} \cdot x\right) \cdot \color{blue}{x}\right), 1\right)\right) \]

   18. distribute-rgt-in N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \color{blue}{\left(\frac{1}{5040} \cdot {x}^{3} + \frac{1}{120} \cdot x\right)}\right), 1\right)\right) \]

10. Simplified 92.8%

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

Alternative 10: 86.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.3:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, x \cdot 0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.3)
   (fma (* x x) (* x 0.16666666666666666) x)
   (* x (* x (* x (fma x (* x 0.008333333333333333) 0.16666666666666666))))))

C

double code(double x) {
	double tmp;
	if (x <= 3.3) {
		tmp = fma((x * x), (x * 0.16666666666666666), x);
	} else {
		tmp = x * (x * (x * fma(x, (x * 0.008333333333333333), 0.16666666666666666)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.3)
		tmp = fma(Float64(x * x), Float64(x * 0.16666666666666666), x);
	else
		tmp = Float64(x * Float64(x * Float64(x * fma(x, Float64(x * 0.008333333333333333), 0.16666666666666666))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3.3], N[(N[(x * x), $MachinePrecision] * N[(x * 0.16666666666666666), $MachinePrecision] + x), $MachinePrecision], N[(x * N[(x * N[(x * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.2999999999999998

   1. Initial program 35.2%

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

   3. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

         \[\leadsto x \cdot \left(1 + \frac{1}{6} \cdot {x}^{2}\right) + \color{blue}{0} \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left(1 + \frac{1}{6} \cdot {x}^{2}\right)}, 0\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\frac{1}{6} \cdot {x}^{2} + \color{blue}{1}\right), 0\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\frac{1}{6} \cdot \left(x \cdot x\right) + 1\right), 0\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\left(\frac{1}{6} \cdot x\right) \cdot x + 1\right), 0\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{6} \cdot x\right) + 1\right), 0\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot x\right)}, 1\right), 0\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{6}}\right), 1\right), 0\right) \]

      9. *-lowering-*.f64 90.6%

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{6}}\right), 1\right), 0\right) \]

   5. Simplified 90.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 1\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. associate-*r* N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(\left(x \cdot x\right), \color{blue}{\left(x \cdot \frac{1}{6}\right)}, x\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{x} \cdot \frac{1}{6}\right), x\right) \]

      7. *-lowering-*.f64 90.6%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{6}}\right), x\right) \]

   7. Applied egg-rr 90.6%

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

   if 3.2999999999999998 < 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. +-rgt-identity N/A

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + \color{blue}{0} \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}, 0\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + \color{blue}{1}\right), 0\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + 1\right), 0\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + 1\right), 0\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(x \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x\right) + 1\right), 0\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x\right)}, 1\right), 0\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right), 1\right), 0\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right), 1\right), 0\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\frac{1}{6}}\right)\right), 1\right), 0\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{120} + \frac{1}{6}\right)\right), 1\right), 0\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{120} + \frac{1}{6}\right)\right), 1\right), 0\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right), 1\right), 0\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{120}\right)}, \frac{1}{6}\right)\right), 1\right), 0\right) \]

      15. *-lowering-*.f64 80.6%

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{120}}\right), \frac{1}{6}\right)\right), 1\right), 0\right) \]

   5. Simplified 80.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right), 0\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left({x}^{4} \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right)}, 0\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{\left(2 \cdot 2\right)} \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right), 0\right) \]

      2. pow-sqr N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(\color{blue}{\frac{1}{120}} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right), 0\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right)}\right), 0\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{{x}^{2}} + \color{blue}{\frac{1}{120}}\right)\right)\right), 0\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2} + \color{blue}{\frac{1}{120} \cdot {x}^{2}}\right)\right), 0\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \left(\frac{1}{{x}^{2}} \cdot {x}^{2}\right) + \color{blue}{\frac{1}{120}} \cdot {x}^{2}\right)\right), 0\right) \]

      7. lft-mult-inverse N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{6} \cdot 1 + \frac{1}{120} \cdot {x}^{2}\right)\right), 0\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{120}} \cdot {x}^{2}\right)\right), 0\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {x}^{2}\right)\right), 0\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}\right), 0\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}\right), 0\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right)\right), 0\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\frac{1}{6}}\right)\right)\right), 0\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{120} + \frac{1}{6}\right)\right)\right), 0\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{120} + \frac{1}{6}\right)\right)\right), 0\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right)\right), 0\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{120} \cdot x\right) + \frac{1}{6}\right)\right)\right), 0\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{120} \cdot x\right)}, \frac{1}{6}\right)\right)\right), 0\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{120}}\right), \frac{1}{6}\right)\right)\right), 0\right) \]

      20. *-lowering-*.f64 80.6%

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{120}}\right), \frac{1}{6}\right)\right)\right), 0\right) \]

   8. Simplified 80.6%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right)\right)}, 0\right) \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right), \color{blue}{\left(x \cdot x\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right), \left(\color{blue}{x} \cdot x\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{120} + \frac{1}{6}\right)\right), \left(x \cdot x\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot x\right), \frac{1}{120}, \frac{1}{6}\right)\right), \left(x \cdot x\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{120}, \frac{1}{6}\right)\right), \left(x \cdot x\right)\right) \]

      9. *-lowering-*.f64 80.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{120}, \frac{1}{6}\right)\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   10. Applied egg-rr 80.6%

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(x \cdot x\right) \cdot \frac{1}{120} + \frac{1}{6}\right) \cdot \left(x \cdot x\right)\right), \color{blue}{x}\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(\left(x \cdot x\right) \cdot \frac{1}{120} + \frac{1}{6}\right) \cdot x\right) \cdot x\right), x\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{120} + \frac{1}{6}\right)\right) \cdot x\right), x\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{120} + \frac{1}{6}\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{120} + \frac{1}{6}\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{120} + \frac{1}{6}\right)\right)\right), x\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right)\right), x\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \frac{1}{120}\right), \frac{1}{6}\right)\right)\right), x\right) \]

       11. *-lowering-*.f64 80.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{120}\right), \frac{1}{6}\right)\right)\right), x\right) \]

   12. Applied egg-rr 80.6%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.3:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, x \cdot 0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.8% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 5

   1. Initial program 35.2%

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

   3. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

         \[\leadsto x \cdot \left(1 + \frac{1}{6} \cdot {x}^{2}\right) + \color{blue}{0} \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left(1 + \frac{1}{6} \cdot {x}^{2}\right)}, 0\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\frac{1}{6} \cdot {x}^{2} + \color{blue}{1}\right), 0\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\frac{1}{6} \cdot \left(x \cdot x\right) + 1\right), 0\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\left(\frac{1}{6} \cdot x\right) \cdot x + 1\right), 0\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{6} \cdot x\right) + 1\right), 0\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot x\right)}, 1\right), 0\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{6}}\right), 1\right), 0\right) \]

      9. *-lowering-*.f64 90.6%

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{6}}\right), 1\right), 0\right) \]

   5. Simplified 90.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 1\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. associate-*r* N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(\left(x \cdot x\right), \color{blue}{\left(x \cdot \frac{1}{6}\right)}, x\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{x} \cdot \frac{1}{6}\right), x\right) \]

      7. *-lowering-*.f64 90.6%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{6}}\right), x\right) \]

   7. Applied egg-rr 90.6%

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

   if 5 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + \color{blue}{0} \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}, 0\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + \color{blue}{1}\right), 0\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + 1\right), 0\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + 1\right), 0\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(x \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x\right) + 1\right), 0\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x\right)}, 1\right), 0\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right), 1\right), 0\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right), 1\right), 0\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\frac{1}{6}}\right)\right), 1\right), 0\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{120} + \frac{1}{6}\right)\right), 1\right), 0\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{120} + \frac{1}{6}\right)\right), 1\right), 0\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right), 1\right), 0\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{120}\right)}, \frac{1}{6}\right)\right), 1\right), 0\right) \]

      15. *-lowering-*.f64 80.6%

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{120}}\right), \frac{1}{6}\right)\right), 1\right), 0\right) \]

   5. Simplified 80.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right), 0\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left({x}^{4} \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right)}, 0\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{\left(2 \cdot 2\right)} \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right), 0\right) \]

      2. pow-sqr N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(\color{blue}{\frac{1}{120}} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right), 0\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right)}\right), 0\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{{x}^{2}} + \color{blue}{\frac{1}{120}}\right)\right)\right), 0\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2} + \color{blue}{\frac{1}{120} \cdot {x}^{2}}\right)\right), 0\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \left(\frac{1}{{x}^{2}} \cdot {x}^{2}\right) + \color{blue}{\frac{1}{120}} \cdot {x}^{2}\right)\right), 0\right) \]

      7. lft-mult-inverse N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{6} \cdot 1 + \frac{1}{120} \cdot {x}^{2}\right)\right), 0\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{120}} \cdot {x}^{2}\right)\right), 0\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {x}^{2}\right)\right), 0\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}\right), 0\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}\right), 0\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right)\right), 0\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\frac{1}{6}}\right)\right)\right), 0\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{120} + \frac{1}{6}\right)\right)\right), 0\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{120} + \frac{1}{6}\right)\right)\right), 0\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right)\right), 0\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{120} \cdot x\right) + \frac{1}{6}\right)\right)\right), 0\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{120} \cdot x\right)}, \frac{1}{6}\right)\right)\right), 0\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{120}}\right), \frac{1}{6}\right)\right)\right), 0\right) \]

      20. *-lowering-*.f64 80.6%

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{120}}\right), \frac{1}{6}\right)\right)\right), 0\right) \]

   8. Simplified 80.6%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right)\right)}, 0\right) \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right), \color{blue}{\left(x \cdot x\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right), \left(\color{blue}{x} \cdot x\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{120} + \frac{1}{6}\right)\right), \left(x \cdot x\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot x\right), \frac{1}{120}, \frac{1}{6}\right)\right), \left(x \cdot x\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{120}, \frac{1}{6}\right)\right), \left(x \cdot x\right)\right) \]

      9. *-lowering-*.f64 80.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{120}, \frac{1}{6}\right)\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   10. Applied egg-rr 80.6%

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

   11. Taylor expanded in x around inf

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

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \color{blue}{\left({x}^{5}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \left({x}^{\left(4 + \color{blue}{1}\right)}\right)\right) \]

       3. pow-plus N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \left({x}^{4} \cdot \color{blue}{x}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \left(x \cdot \color{blue}{{x}^{4}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{4}\right)}\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right)\right) \]

       7. pow-plus N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \left({x}^{3} \cdot \color{blue}{x}\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{{x}^{3}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right)\right)\right) \]

       10. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 80.6%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   13. Simplified 80.6%

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

Alternative 12: 90.0% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.4%

   \[\frac{e^{x} - e^{-x}}{2} \]
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. +-rgt-identity N/A

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + \color{blue}{0} \]

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

      \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}, 0\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + \color{blue}{1}\right), 0\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{fma.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + 1\right), 0\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{fma.f64}\left(x, \left(x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + 1\right), 0\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma.f64}\left(x, \left(x \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x\right) + 1\right), 0\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x\right)}, 1\right), 0\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right), 1\right), 0\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right), 1\right), 0\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\frac{1}{6}}\right)\right), 1\right), 0\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{120} + \frac{1}{6}\right)\right), 1\right), 0\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{120} + \frac{1}{6}\right)\right), 1\right), 0\right) \]

   13. associate-*l* N/A

       \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right), 1\right), 0\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{120}\right)}, \frac{1}{6}\right)\right), 1\right), 0\right) \]

   15. *-lowering-*.f64 91.4%

       \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{120}}\right), \frac{1}{6}\right)\right), 1\right), 0\right) \]

5. Simplified 91.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right), 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. cube-unmult N/A

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

   6. *-rgt-identity N/A

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

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

      \[\leadsto \mathsf{fma.f64}\left(\left({x}^{3}\right), \color{blue}{\left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)}, x\right) \]

   8. cube-unmult N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{120}\right)} + \frac{1}{6}\right), x\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{120}\right)} + \frac{1}{6}\right), x\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{120}\right)} + \frac{1}{6}\right), x\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{120}\right)}, \frac{1}{6}\right), x\right) \]

   12. *-lowering-*.f64 91.4%

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{120}}\right), \frac{1}{6}\right), x\right) \]

7. Applied egg-rr 91.4%

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

Alternative 13: 89.7% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.008333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right), 1\right), 0\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma x (fma x (* 0.008333333333333333 (* x (* x x))) 1.0) 0.0))

C

double code(double x) {
	return fma(x, fma(x, (0.008333333333333333 * (x * (x * x))), 1.0), 0.0);
}

Julia

function code(x)
	return fma(x, fma(x, Float64(0.008333333333333333 * Float64(x * Float64(x * x))), 1.0), 0.0)
end

Wolfram

code[x_] := N[(x * N[(x * N[(0.008333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 0.0), $MachinePrecision]

Derivation

1. Initial program 52.4%

   \[\frac{e^{x} - e^{-x}}{2} \]
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. +-rgt-identity N/A

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + \color{blue}{0} \]

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

      \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}, 0\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + \color{blue}{1}\right), 0\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{fma.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + 1\right), 0\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{fma.f64}\left(x, \left(x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + 1\right), 0\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma.f64}\left(x, \left(x \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x\right) + 1\right), 0\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot x\right)}, 1\right), 0\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right), 1\right), 0\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right), 1\right), 0\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\frac{1}{6}}\right)\right), 1\right), 0\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{120} + \frac{1}{6}\right)\right), 1\right), 0\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{120} + \frac{1}{6}\right)\right), 1\right), 0\right) \]

   13. associate-*l* N/A

       \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \frac{1}{6}\right)\right), 1\right), 0\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{120}\right)}, \frac{1}{6}\right)\right), 1\right), 0\right) \]

   15. *-lowering-*.f64 91.4%

       \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{120}}\right), \frac{1}{6}\right)\right), 1\right), 0\right) \]

5. Simplified 91.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right), 0\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{120} \cdot {x}^{3}\right)}, 1\right), 0\right) \]
7. Step-by-step derivation

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

      \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{120}, \color{blue}{\left({x}^{3}\right)}\right), 1\right), 0\right) \]

   2. cube-mult N/A

      \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{120}, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right), 1\right), 0\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{120}, \left(x \cdot {x}^{\color{blue}{2}}\right)\right), 1\right), 0\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right), 1\right), 0\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right), 1\right), 0\right) \]

   6. *-lowering-*.f64 90.9%

      \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right), 1\right), 0\right) \]

8. Simplified 90.9%

   \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{0.008333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)}, 1\right), 0\right) \]
9. Add Preprocessing

Alternative 14: 67.2% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 35.2%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 71.1%

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

   if 2.39999999999999991 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

         \[\leadsto x \cdot \left(1 + \frac{1}{6} \cdot {x}^{2}\right) + \color{blue}{0} \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left(1 + \frac{1}{6} \cdot {x}^{2}\right)}, 0\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\frac{1}{6} \cdot {x}^{2} + \color{blue}{1}\right), 0\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\frac{1}{6} \cdot \left(x \cdot x\right) + 1\right), 0\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(\left(\frac{1}{6} \cdot x\right) \cdot x + 1\right), 0\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{6} \cdot x\right) + 1\right), 0\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot x\right)}, 1\right), 0\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{6}}\right), 1\right), 0\right) \]

      9. *-lowering-*.f64 68.1%

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{6}}\right), 1\right), 0\right) \]

   5. Simplified 68.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 1\right), 0\right)} \]

   6. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot {x}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      8. *-lowering-*.f64 68.1%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   8. Simplified 68.1%

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

4. Final simplification 70.3%

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

Alternative 15: 83.7% accurate, 12.8× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 52.4%

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

3. Taylor expanded in x around 0

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

   1. +-rgt-identity N/A

      \[\leadsto x \cdot \left(1 + \frac{1}{6} \cdot {x}^{2}\right) + \color{blue}{0} \]

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

      \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left(1 + \frac{1}{6} \cdot {x}^{2}\right)}, 0\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{fma.f64}\left(x, \left(\frac{1}{6} \cdot {x}^{2} + \color{blue}{1}\right), 0\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{fma.f64}\left(x, \left(\frac{1}{6} \cdot \left(x \cdot x\right) + 1\right), 0\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{fma.f64}\left(x, \left(\left(\frac{1}{6} \cdot x\right) \cdot x + 1\right), 0\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{6} \cdot x\right) + 1\right), 0\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot x\right)}, 1\right), 0\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{6}}\right), 1\right), 0\right) \]

   9. *-lowering-*.f64 84.6%

      \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{6}}\right), 1\right), 0\right) \]

5. Simplified 84.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 1\right), 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. associate-*r* N/A

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

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

      \[\leadsto \mathsf{fma.f64}\left(\left(x \cdot x\right), \color{blue}{\left(x \cdot \frac{1}{6}\right)}, x\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{x} \cdot \frac{1}{6}\right), x\right) \]

   7. *-lowering-*.f64 84.6%

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{6}}\right), x\right) \]

7. Applied egg-rr 84.6%

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

Alternative 16: 51.9% accurate, 217.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 52.4%

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

3. Taylor expanded in x around 0

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

5. Simplified 53.6%

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

Reproduce

herbie shell --seed 2024193 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))