exp2 (problem 3.3.7)

Percentage Accurate: 54.9% → 100.0%

Time: 18.3s

Alternatives: 11

Speedup: 34.8×

Specification

\[\left|x\right| \leq 710\]

Math

\[\begin{array}{l} \\ \left(e^{x} - 2\right) + e^{-x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(e^{x} - 2\right) + e^{-x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{0 - x} \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(x, \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right)\right) \cdot \left(x \cdot x\right), x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \cosh x, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (- (exp x) 2.0) (exp (- 0.0 x))) 0.005)
   (fma
    x
    (*
     (*
      x
      (fma
       x
       (* x (fma x (* x 4.96031746031746e-5) 0.002777777777777778))
       0.08333333333333333))
     (* x x))
    (* x x))
   (fma 2.0 (cosh x) -2.0)))

C

double code(double x) {
	double tmp;
	if (((exp(x) - 2.0) + exp((0.0 - x))) <= 0.005) {
		tmp = fma(x, ((x * fma(x, (x * fma(x, (x * 4.96031746031746e-5), 0.002777777777777778)), 0.08333333333333333)) * (x * x)), (x * x));
	} else {
		tmp = fma(2.0, cosh(x), -2.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) - 2.0) + exp(Float64(0.0 - x))) <= 0.005)
		tmp = fma(x, Float64(Float64(x * fma(x, Float64(x * fma(x, Float64(x * 4.96031746031746e-5), 0.002777777777777778)), 0.08333333333333333)) * Float64(x * x)), Float64(x * x));
	else
		tmp = fma(2.0, cosh(x), -2.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[N[(0.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.005], N[(x * N[(N[(x * N[(x * N[(x * N[(x * N[(x * 4.96031746031746e-5), $MachinePrecision] + 0.002777777777777778), $MachinePrecision]), $MachinePrecision] + 0.08333333333333333), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Cosh[x], $MachinePrecision] + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x))) < 0.0050000000000000001

   1. Initial program 53.0%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{e^{\mathsf{neg}\left(x\right)} + \left(e^{x} - 2\right)} \]

      2. sub-neg N/A

         \[\leadsto e^{\mathsf{neg}\left(x\right)} + \color{blue}{\left(e^{x} + \left(\mathsf{neg}\left(2\right)\right)\right)} \]

      3. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + e^{x}\right) + \left(\mathsf{neg}\left(2\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(e^{x} + e^{\mathsf{neg}\left(x\right)}\right)} + \left(\mathsf{neg}\left(2\right)\right) \]

      5. cosh-undef N/A

         \[\leadsto \color{blue}{2 \cdot \cosh x} + \left(\mathsf{neg}\left(2\right)\right) \]

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

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

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

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

      8. metadata-eval 53.0

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

   4. Applied egg-rr 53.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right) \]

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

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right) + 1\right)} \]

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

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right), 1\right)} \]

      6. unpow2 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right), 1\right) \]

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

         \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right), 1\right) \]

      8. +-commutative N/A

         \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right) + \frac{1}{12}}, 1\right) \]

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

         \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}, \frac{1}{12}\right)}, 1\right) \]

      10. unpow2 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}, \frac{1}{12}\right), 1\right) \]

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

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}, \frac{1}{12}\right), 1\right) \]

      12. +-commutative N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{20160} \cdot {x}^{2} + \frac{1}{360}}, \frac{1}{12}\right), 1\right) \]

      13. *-commutative N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{20160}} + \frac{1}{360}, \frac{1}{12}\right), 1\right) \]

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

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{20160}, \frac{1}{360}\right)}, \frac{1}{12}\right), 1\right) \]

      15. unpow2 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{20160}, \frac{1}{360}\right), \frac{1}{12}\right), 1\right) \]

      16. *-lowering-*.f64 100.0

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right), 1\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right), 1\right)} \]
   8. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{20160} + \frac{1}{360}\right) + \frac{1}{12}\right)\right) \cdot \left(x \cdot x\right) + 1 \cdot \left(x \cdot x\right)} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{20160} + \frac{1}{360}\right) + \frac{1}{12}\right)\right)\right)} \cdot \left(x \cdot x\right) + 1 \cdot \left(x \cdot x\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{20160} + \frac{1}{360}\right) + \frac{1}{12}\right)\right) \cdot \left(x \cdot x\right)\right)} + 1 \cdot \left(x \cdot x\right) \]

      4. *-lft-identity N/A

         \[\leadsto x \cdot \left(\left(x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{20160} + \frac{1}{360}\right) + \frac{1}{12}\right)\right) \cdot \left(x \cdot x\right)\right) + \color{blue}{x \cdot x} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{20160} + \frac{1}{360}\right) + \frac{1}{12}\right)\right) \cdot \left(x \cdot x\right), x \cdot x\right)} \]

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right)\right) \cdot \left(x \cdot x\right), x \cdot x\right)} \]

   if 0.0050000000000000001 < (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x)))

   1. Initial program 99.8%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{e^{\mathsf{neg}\left(x\right)} + \left(e^{x} - 2\right)} \]

      2. sub-neg N/A

         \[\leadsto e^{\mathsf{neg}\left(x\right)} + \color{blue}{\left(e^{x} + \left(\mathsf{neg}\left(2\right)\right)\right)} \]

      3. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + e^{x}\right) + \left(\mathsf{neg}\left(2\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(e^{x} + e^{\mathsf{neg}\left(x\right)}\right)} + \left(\mathsf{neg}\left(2\right)\right) \]

      5. cosh-undef N/A

         \[\leadsto \color{blue}{2 \cdot \cosh x} + \left(\mathsf{neg}\left(2\right)\right) \]

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

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

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

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

      8. metadata-eval 99.8

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{0 - x} \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(x, \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right)\right) \cdot \left(x \cdot x\right), x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \cosh x, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right)\right)\right), x, \mathsf{fma}\left(x, x \cdot 0.08333333333333333, 1\right)\right), 0\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (fma
   x
   (fma
    (* x (* x (* x (fma x (* x 4.96031746031746e-5) 0.002777777777777778))))
    x
    (fma x (* x 0.08333333333333333) 1.0))
   0.0)))

C

double code(double x) {
	return x * fma(x, fma((x * (x * (x * fma(x, (x * 4.96031746031746e-5), 0.002777777777777778)))), x, fma(x, (x * 0.08333333333333333), 1.0)), 0.0);
}

Julia

function code(x)
	return Float64(x * fma(x, fma(Float64(x * Float64(x * Float64(x * fma(x, Float64(x * 4.96031746031746e-5), 0.002777777777777778)))), x, fma(x, Float64(x * 0.08333333333333333), 1.0)), 0.0))
end

Wolfram

code[x_] := N[(x * N[(x * N[(N[(x * N[(x * N[(x * N[(x * N[(x * 4.96031746031746e-5), $MachinePrecision] + 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + N[(x * N[(x * 0.08333333333333333), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)} \]

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

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)} \]

   4. +-rgt-identity N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right) + 0\right)} \]

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

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, 1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right), 0\right)} \]

5. Simplified 97.3%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right), 1\right), 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right) + \frac{1}{12}\right) + 1, 0\right) \]

   2. distribute-lft-in N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right)\right) + \left(x \cdot x\right) \cdot \frac{1}{12}\right)} + 1, 0\right) \]

   3. associate-*r* N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right)\right) + \color{blue}{x \cdot \left(x \cdot \frac{1}{12}\right)}\right) + 1, 0\right) \]

   4. associate-+l+ N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right)\right) + \left(x \cdot \left(x \cdot \frac{1}{12}\right) + 1\right)}, 0\right) \]

   5. *-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right) \cdot x\right)} + \left(x \cdot \left(x \cdot \frac{1}{12}\right) + 1\right), 0\right) \]

   6. associate-*r* N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right)\right) \cdot x} + \left(x \cdot \left(x \cdot \frac{1}{12}\right) + 1\right), 0\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right), x, x \cdot \left(x \cdot \frac{1}{12}\right) + 1\right)}, 0\right) \]

7. Applied egg-rr 97.3%

   \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right) \cdot \left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right)\right), x, \mathsf{fma}\left(x, x \cdot 0.08333333333333333, 1\right)\right)}, 0\right) \]
8. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right), x, \mathsf{fma}\left(x, x \cdot \frac{1}{12}, 1\right)\right), 0\right) \]

   2. *-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot x\right) \cdot \color{blue}{\left(\left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right) \cdot x\right)}, x, \mathsf{fma}\left(x, x \cdot \frac{1}{12}, 1\right)\right), 0\right) \]

   3. +-rgt-identity N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{20160} + \frac{1}{360}\right) \cdot x\right), x, \mathsf{fma}\left(x, x \cdot \frac{1}{12}, 1\right)\right), 0\right) \]

   4. associate-*r* N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right) \cdot x}, x, \mathsf{fma}\left(x, x \cdot \frac{1}{12}, 1\right)\right), 0\right) \]

   5. associate-*r* N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right)\right)} \cdot x, x, \mathsf{fma}\left(x, x \cdot \frac{1}{12}, 1\right)\right), 0\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right)\right) \cdot x}, x, \mathsf{fma}\left(x, x \cdot \frac{1}{12}, 1\right)\right), 0\right) \]

9. Applied egg-rr 97.3%

   \[\leadsto x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right)\right)\right) \cdot x}, x, \mathsf{fma}\left(x, x \cdot 0.08333333333333333, 1\right)\right), 0\right) \]

10. Final simplification 97.3%

    \[\leadsto x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right)\right)\right), x, \mathsf{fma}\left(x, x \cdot 0.08333333333333333, 1\right)\right), 0\right) \]
11. Add Preprocessing

Alternative 3: 99.0% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right)\right) \cdot \left(x \cdot x\right), x \cdot x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  x
  (*
   (*
    x
    (fma
     x
     (* x (fma x (* x 4.96031746031746e-5) 0.002777777777777778))
     0.08333333333333333))
   (* x x))
  (* x x)))

C

double code(double x) {
	return fma(x, ((x * fma(x, (x * fma(x, (x * 4.96031746031746e-5), 0.002777777777777778)), 0.08333333333333333)) * (x * x)), (x * x));
}

Julia

function code(x)
	return fma(x, Float64(Float64(x * fma(x, Float64(x * fma(x, Float64(x * 4.96031746031746e-5), 0.002777777777777778)), 0.08333333333333333)) * Float64(x * x)), Float64(x * x))
end

Wolfram

code[x_] := N[(x * N[(N[(x * N[(x * N[(x * N[(x * N[(x * 4.96031746031746e-5), $MachinePrecision] + 0.002777777777777778), $MachinePrecision]), $MachinePrecision] + 0.08333333333333333), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(x\right)} + \left(e^{x} - 2\right)} \]

   2. sub-neg N/A

      \[\leadsto e^{\mathsf{neg}\left(x\right)} + \color{blue}{\left(e^{x} + \left(\mathsf{neg}\left(2\right)\right)\right)} \]

   3. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + e^{x}\right) + \left(\mathsf{neg}\left(2\right)\right)} \]

   4. +-commutative N/A

      \[\leadsto \color{blue}{\left(e^{x} + e^{\mathsf{neg}\left(x\right)}\right)} + \left(\mathsf{neg}\left(2\right)\right) \]

   5. cosh-undef N/A

      \[\leadsto \color{blue}{2 \cdot \cosh x} + \left(\mathsf{neg}\left(2\right)\right) \]

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

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

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

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

   8. metadata-eval 54.5

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

4. Applied egg-rr 54.5%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

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

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)} \]

   2. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right) \]

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

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right) + 1\right)} \]

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

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right), 1\right)} \]

   6. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right), 1\right) \]

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

      \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right), 1\right) \]

   8. +-commutative N/A

      \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right) + \frac{1}{12}}, 1\right) \]

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

      \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}, \frac{1}{12}\right)}, 1\right) \]

   10. unpow2 N/A

       \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}, \frac{1}{12}\right), 1\right) \]

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

       \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}, \frac{1}{12}\right), 1\right) \]

   12. +-commutative N/A

       \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{20160} \cdot {x}^{2} + \frac{1}{360}}, \frac{1}{12}\right), 1\right) \]

   13. *-commutative N/A

       \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{20160}} + \frac{1}{360}, \frac{1}{12}\right), 1\right) \]

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

       \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{20160}, \frac{1}{360}\right)}, \frac{1}{12}\right), 1\right) \]

   15. unpow2 N/A

       \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{20160}, \frac{1}{360}\right), \frac{1}{12}\right), 1\right) \]

   16. *-lowering-*.f64 97.3

       \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right), 1\right) \]

7. Simplified 97.3%

   \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right), 1\right)} \]
8. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{20160} + \frac{1}{360}\right) + \frac{1}{12}\right)\right) \cdot \left(x \cdot x\right) + 1 \cdot \left(x \cdot x\right)} \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{20160} + \frac{1}{360}\right) + \frac{1}{12}\right)\right)\right)} \cdot \left(x \cdot x\right) + 1 \cdot \left(x \cdot x\right) \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(\left(x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{20160} + \frac{1}{360}\right) + \frac{1}{12}\right)\right) \cdot \left(x \cdot x\right)\right)} + 1 \cdot \left(x \cdot x\right) \]

   4. *-lft-identity N/A

      \[\leadsto x \cdot \left(\left(x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{20160} + \frac{1}{360}\right) + \frac{1}{12}\right)\right) \cdot \left(x \cdot x\right)\right) + \color{blue}{x \cdot x} \]

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{20160} + \frac{1}{360}\right) + \frac{1}{12}\right)\right) \cdot \left(x \cdot x\right), x \cdot x\right)} \]

9. Applied egg-rr 97.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right)\right) \cdot \left(x \cdot x\right), x \cdot x\right)} \]
10. Add Preprocessing

Alternative 4: 99.0% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right), x \cdot \mathsf{fma}\left(x, x, 0\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (fma
   (fma
    x
    (* x (fma x (* x 4.96031746031746e-5) 0.002777777777777778))
    0.08333333333333333)
   (* x (fma x x 0.0))
   x)))

C

double code(double x) {
	return x * fma(fma(x, (x * fma(x, (x * 4.96031746031746e-5), 0.002777777777777778)), 0.08333333333333333), (x * fma(x, x, 0.0)), x);
}

Julia

function code(x)
	return Float64(x * fma(fma(x, Float64(x * fma(x, Float64(x * 4.96031746031746e-5), 0.002777777777777778)), 0.08333333333333333), Float64(x * fma(x, x, 0.0)), x))
end

Wolfram

code[x_] := N[(x * N[(N[(x * N[(x * N[(x * N[(x * 4.96031746031746e-5), $MachinePrecision] + 0.002777777777777778), $MachinePrecision]), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(x * N[(x * x + 0.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)} \]

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

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)} \]

   4. +-rgt-identity N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right) + 0\right)} \]

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

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, 1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right), 0\right)} \]

5. Simplified 97.3%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right), 1\right), 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right) + \frac{1}{12}\right) + 1, 0\right) \]

   2. distribute-lft-in N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right)\right) + \left(x \cdot x\right) \cdot \frac{1}{12}\right)} + 1, 0\right) \]

   3. associate-*r* N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right)\right) + \color{blue}{x \cdot \left(x \cdot \frac{1}{12}\right)}\right) + 1, 0\right) \]

   4. associate-+l+ N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right)\right) + \left(x \cdot \left(x \cdot \frac{1}{12}\right) + 1\right)}, 0\right) \]

   5. *-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right) \cdot x\right)} + \left(x \cdot \left(x \cdot \frac{1}{12}\right) + 1\right), 0\right) \]

   6. associate-*r* N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right)\right) \cdot x} + \left(x \cdot \left(x \cdot \frac{1}{12}\right) + 1\right), 0\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right), x, x \cdot \left(x \cdot \frac{1}{12}\right) + 1\right)}, 0\right) \]

7. Applied egg-rr 97.3%

   \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right) \cdot \left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right)\right), x, \mathsf{fma}\left(x, x \cdot 0.08333333333333333, 1\right)\right)}, 0\right) \]
8. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right)\right) \cdot x + \left(x \cdot \left(x \cdot \frac{1}{12}\right) + 1\right)\right)\right)} \]

   2. associate-+r+ N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\left(\left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right)\right) \cdot x + x \cdot \left(x \cdot \frac{1}{12}\right)\right) + 1\right)}\right) \]

9. Applied egg-rr 97.3%

   \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right), x \cdot \mathsf{fma}\left(x, x, 0\right), x\right)} \]
10. Add Preprocessing

Alternative 5: 99.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right), 1\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (*
   x
   (fma
    (* x x)
    (fma
     x
     (* x (fma x (* x 4.96031746031746e-5) 0.002777777777777778))
     0.08333333333333333)
    1.0))))

C

double code(double x) {
	return x * (x * fma((x * x), fma(x, (x * fma(x, (x * 4.96031746031746e-5), 0.002777777777777778)), 0.08333333333333333), 1.0));
}

Julia

function code(x)
	return Float64(x * Float64(x * fma(Float64(x * x), fma(x, Float64(x * fma(x, Float64(x * 4.96031746031746e-5), 0.002777777777777778)), 0.08333333333333333), 1.0)))
end

Wolfram

code[x_] := N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 4.96031746031746e-5), $MachinePrecision] + 0.002777777777777778), $MachinePrecision]), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)} \]

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

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)} \]

   4. +-rgt-identity N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right) + 0\right)} \]

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

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, 1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right), 0\right)} \]

5. Simplified 97.3%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right), 1\right), 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right) + \frac{1}{12}\right) + 1, 0\right) \]

   2. distribute-lft-in N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right)\right) + \left(x \cdot x\right) \cdot \frac{1}{12}\right)} + 1, 0\right) \]

   3. associate-*r* N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right)\right) + \color{blue}{x \cdot \left(x \cdot \frac{1}{12}\right)}\right) + 1, 0\right) \]

   4. associate-+l+ N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right)\right) + \left(x \cdot \left(x \cdot \frac{1}{12}\right) + 1\right)}, 0\right) \]

   5. *-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right) \cdot x\right)} + \left(x \cdot \left(x \cdot \frac{1}{12}\right) + 1\right), 0\right) \]

   6. associate-*r* N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right)\right) \cdot x} + \left(x \cdot \left(x \cdot \frac{1}{12}\right) + 1\right), 0\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right), x, x \cdot \left(x \cdot \frac{1}{12}\right) + 1\right)}, 0\right) \]

7. Applied egg-rr 97.3%

   \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right) \cdot \left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right)\right), x, \mathsf{fma}\left(x, x \cdot 0.08333333333333333, 1\right)\right)}, 0\right) \]

8. Taylor expanded in x around 0

   \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)} \]
9. Step-by-step derivation

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

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right) + 1\right)}\right) \]

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

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right), 1\right)}\right) \]

   4. unpow2 N/A

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right), 1\right)\right) \]

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

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right), 1\right)\right) \]

   6. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right) + \frac{1}{12}}, 1\right)\right) \]

   7. unpow2 N/A

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right) + \frac{1}{12}, 1\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)} + \frac{1}{12}, 1\right)\right) \]

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

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right), \frac{1}{12}\right)}, 1\right)\right) \]

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

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)}, \frac{1}{12}\right), 1\right)\right) \]

   11. +-commutative N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{20160} \cdot {x}^{2} + \frac{1}{360}\right)}, \frac{1}{12}\right), 1\right)\right) \]

   12. *-commutative N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{20160}} + \frac{1}{360}\right), \frac{1}{12}\right), 1\right)\right) \]

   13. unpow2 N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{20160} + \frac{1}{360}\right), \frac{1}{12}\right), 1\right)\right) \]

   14. associate-*l* N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{20160}\right)} + \frac{1}{360}\right), \frac{1}{12}\right), 1\right)\right) \]

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

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{20160}, \frac{1}{360}\right)}, \frac{1}{12}\right), 1\right)\right) \]

   16. *-lowering-*.f64 97.3

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 4.96031746031746 \cdot 10^{-5}}, 0.002777777777777778\right), 0.08333333333333333\right), 1\right)\right) \]

10. Simplified 97.3%

    \[\leadsto x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right), 1\right)\right)} \]
11. Add Preprocessing

Alternative 6: 99.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, x, x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.002777777777777778, 0.08333333333333333\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  x
  x
  (*
   x
   (* x (* x (* x (fma x (* x 0.002777777777777778) 0.08333333333333333)))))))

C

double code(double x) {
	return fma(x, x, (x * (x * (x * (x * fma(x, (x * 0.002777777777777778), 0.08333333333333333))))));
}

Julia

function code(x)
	return fma(x, x, Float64(x * Float64(x * Float64(x * Float64(x * fma(x, Float64(x * 0.002777777777777778), 0.08333333333333333))))))
end

Wolfram

code[x_] := N[(x * x + N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * 0.002777777777777778), $MachinePrecision] + 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)} \]
4. Step-by-step derivation

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

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)} \]

   2. remove-double-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \]

   3. +-rgt-identity N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left({x}^{2}\right)\right) + 0\right)}\right)\right) \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \]

   4. distribute-neg-in N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right) + \left(\mathsf{neg}\left(0\right)\right)\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \]

   5. remove-double-neg N/A

      \[\leadsto \left(\color{blue}{{x}^{2}} + \left(\mathsf{neg}\left(0\right)\right)\right) \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \left(\color{blue}{x \cdot x} + \left(\mathsf{neg}\left(0\right)\right)\right) \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \left(x \cdot x + \color{blue}{0}\right) \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \]

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, x, 0\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + 1\right)} \]

   10. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x, x, 0\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + 1\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(x, x, 0\right) \cdot \left(\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)} + 1\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, x, 0\right) \cdot \left(x \cdot \color{blue}{\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot x\right)} + 1\right) \]

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

       \[\leadsto \mathsf{fma}\left(x, x, 0\right) \cdot \color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot x, 1\right)} \]

5. Simplified 97.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.002777777777777778, 0.08333333333333333\right), 1\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{360} + \frac{1}{12}\right)\right) + 1\right) \]

   2. +-commutative N/A

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{360} + \frac{1}{12}\right)\right)\right)} \]

   3. distribute-rgt-in N/A

      \[\leadsto \color{blue}{1 \cdot \left(x \cdot x\right) + \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{360} + \frac{1}{12}\right)\right)\right) \cdot \left(x \cdot x\right)} \]

   4. *-lft-identity N/A

      \[\leadsto \color{blue}{x \cdot x} + \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{360} + \frac{1}{12}\right)\right)\right) \cdot \left(x \cdot x\right) \]

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{360} + \frac{1}{12}\right)\right)\right) \cdot \left(x \cdot x\right)\right)} \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{360} + \frac{1}{12}\right)\right)\right)}\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{360} + \frac{1}{12}\right)\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{360} + \frac{1}{12}\right)\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{fma}\left(x, x, x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{360} + \frac{1}{12}\right)\right)\right)\right)}\right) \]

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

       \[\leadsto \mathsf{fma}\left(x, x, x \cdot \left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{360} + \frac{1}{12}\right)\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{fma}\left(x, x, x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{360} + \frac{1}{12}\right)\right)}\right)\right)\right) \]

   12. +-rgt-identity N/A

       \[\leadsto \mathsf{fma}\left(x, x, x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{360} + \frac{1}{12}\right)\right)\right)\right)\right) \]

   13. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(x, x, x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{360}\right)} + \frac{1}{12}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{fma}\left(x, x, x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{360}, \frac{1}{12}\right)}\right)\right)\right)\right) \]

   15. *-lowering-*.f64 97.2

       \[\leadsto \mathsf{fma}\left(x, x, x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.002777777777777778}, 0.08333333333333333\right)\right)\right)\right)\right) \]

7. Applied egg-rr 97.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.002777777777777778, 0.08333333333333333\right)\right)\right)\right)\right)} \]
8. Add Preprocessing

Alternative 7: 98.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.002777777777777778, 0.08333333333333333\right), 1\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (*
   x
   (fma (* x x) (fma x (* x 0.002777777777777778) 0.08333333333333333) 1.0))))

C

double code(double x) {
	return x * (x * fma((x * x), fma(x, (x * 0.002777777777777778), 0.08333333333333333), 1.0));
}

Julia

function code(x)
	return Float64(x * Float64(x * fma(Float64(x * x), fma(x, Float64(x * 0.002777777777777778), 0.08333333333333333), 1.0)))
end

Wolfram

code[x_] := N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.002777777777777778), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)} \]

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

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)} \]

   4. +-rgt-identity N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right) + 0\right)} \]

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

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, 1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right), 0\right)} \]

5. Simplified 97.3%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right), 1\right), 0\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)} \]
7. Step-by-step derivation

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

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + 1\right)}\right) \]

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

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{12} + \frac{1}{360} \cdot {x}^{2}, 1\right)}\right) \]

   4. unpow2 N/A

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{12} + \frac{1}{360} \cdot {x}^{2}, 1\right)\right) \]

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

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{12} + \frac{1}{360} \cdot {x}^{2}, 1\right)\right) \]

   6. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{360} \cdot {x}^{2} + \frac{1}{12}}, 1\right)\right) \]

   7. *-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{360}} + \frac{1}{12}, 1\right)\right) \]

   8. unpow2 N/A

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{360} + \frac{1}{12}, 1\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{360}\right)} + \frac{1}{12}, 1\right)\right) \]

   10. *-commutative N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\frac{1}{360} \cdot x\right)} + \frac{1}{12}, 1\right)\right) \]

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

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{360} \cdot x, \frac{1}{12}\right)}, 1\right)\right) \]

   12. *-commutative N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{360}}, \frac{1}{12}\right), 1\right)\right) \]

   13. *-lowering-*.f64 97.2

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.002777777777777778}, 0.08333333333333333\right), 1\right)\right) \]

8. Simplified 97.2%

   \[\leadsto x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.002777777777777778, 0.08333333333333333\right), 1\right)\right)} \]
9. Add Preprocessing

Alternative 8: 98.8% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, x, x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.08333333333333333\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma x x (* x (* x (* x (* x 0.08333333333333333))))))

C

double code(double x) {
	return fma(x, x, (x * (x * (x * (x * 0.08333333333333333)))));
}

Julia

function code(x)
	return fma(x, x, Float64(x * Float64(x * Float64(x * Float64(x * 0.08333333333333333)))))
end

Wolfram

code[x_] := N[(x * x + N[(x * N[(x * N[(x * N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)} \]
4. Step-by-step derivation

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

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)} \]

   2. remove-double-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \]

   3. +-rgt-identity N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left({x}^{2}\right)\right) + 0\right)}\right)\right) \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \]

   4. distribute-neg-in N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right) + \left(\mathsf{neg}\left(0\right)\right)\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \]

   5. remove-double-neg N/A

      \[\leadsto \left(\color{blue}{{x}^{2}} + \left(\mathsf{neg}\left(0\right)\right)\right) \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \left(\color{blue}{x \cdot x} + \left(\mathsf{neg}\left(0\right)\right)\right) \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \left(x \cdot x + \color{blue}{0}\right) \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \]

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, x, 0\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + 1\right)} \]

   10. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x, x, 0\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + 1\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(x, x, 0\right) \cdot \left(\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)} + 1\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, x, 0\right) \cdot \left(x \cdot \color{blue}{\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot x\right)} + 1\right) \]

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

       \[\leadsto \mathsf{fma}\left(x, x, 0\right) \cdot \color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot x, 1\right)} \]

5. Simplified 97.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.002777777777777778, 0.08333333333333333\right), 1\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{360} + \frac{1}{12}\right)\right) + 1\right) \]

   2. +-commutative N/A

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{360} + \frac{1}{12}\right)\right)\right)} \]

   3. distribute-rgt-in N/A

      \[\leadsto \color{blue}{1 \cdot \left(x \cdot x\right) + \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{360} + \frac{1}{12}\right)\right)\right) \cdot \left(x \cdot x\right)} \]

   4. *-lft-identity N/A

      \[\leadsto \color{blue}{x \cdot x} + \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{360} + \frac{1}{12}\right)\right)\right) \cdot \left(x \cdot x\right) \]

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{360} + \frac{1}{12}\right)\right)\right) \cdot \left(x \cdot x\right)\right)} \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{360} + \frac{1}{12}\right)\right)\right)}\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{360} + \frac{1}{12}\right)\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{360} + \frac{1}{12}\right)\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{fma}\left(x, x, x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{360} + \frac{1}{12}\right)\right)\right)\right)}\right) \]

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

       \[\leadsto \mathsf{fma}\left(x, x, x \cdot \left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{360} + \frac{1}{12}\right)\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{fma}\left(x, x, x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{360} + \frac{1}{12}\right)\right)}\right)\right)\right) \]

   12. +-rgt-identity N/A

       \[\leadsto \mathsf{fma}\left(x, x, x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{360} + \frac{1}{12}\right)\right)\right)\right)\right) \]

   13. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(x, x, x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{360}\right)} + \frac{1}{12}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{fma}\left(x, x, x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{360}, \frac{1}{12}\right)}\right)\right)\right)\right) \]

   15. *-lowering-*.f64 97.2

       \[\leadsto \mathsf{fma}\left(x, x, x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.002777777777777778}, 0.08333333333333333\right)\right)\right)\right)\right) \]

7. Applied egg-rr 97.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.002777777777777778, 0.08333333333333333\right)\right)\right)\right)\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(x, x, x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\frac{1}{12}}\right)\right)\right)\right) \]
9. Step-by-step derivation

10. Simplified 97.1%

    \[\leadsto \mathsf{fma}\left(x, x, x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{0.08333333333333333}\right)\right)\right)\right) \]
11. Add Preprocessing

Alternative 9: 98.8% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, x, 0\right), 0.08333333333333333, x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (fma (* x (fma x x 0.0)) 0.08333333333333333 x)))

C

double code(double x) {
	return x * fma((x * fma(x, x, 0.0)), 0.08333333333333333, x);
}

Julia

function code(x)
	return Float64(x * fma(Float64(x * fma(x, x, 0.0)), 0.08333333333333333, x))
end

Wolfram

code[x_] := N[(x * N[(N[(x * N[(x * x + 0.0), $MachinePrecision]), $MachinePrecision] * 0.08333333333333333 + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)\right)} \]

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

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)\right)} \]

   4. +-rgt-identity N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right) + 0\right)} \]

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

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, 1 + \frac{1}{12} \cdot {x}^{2}, 0\right)} \]

   6. +-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\frac{1}{12} \cdot {x}^{2} + 1}, 0\right) \]

   7. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \frac{1}{12} \cdot \color{blue}{\left(x \cdot x\right)} + 1, 0\right) \]

   8. associate-*r* N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\left(\frac{1}{12} \cdot x\right) \cdot x} + 1, 0\right) \]

   9. *-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{12} \cdot x\right)} + 1, 0\right) \]

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

       \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{12} \cdot x, 1\right)}, 0\right) \]

   11. *-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{12}}, 1\right), 0\right) \]

   12. *-lowering-*.f64 97.1

       \[\leadsto x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.08333333333333333}, 1\right), 0\right) \]

5. Simplified 97.1%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot 0.08333333333333333, 1\right), 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{12}\right) + 1\right)\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{12}\right)\right) + x \cdot 1\right)} \]

   3. associate-*r* N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)} + x \cdot 1\right) \]

   4. associate-*r* N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12}} + x \cdot 1\right) \]

   5. cube-unmult N/A

      \[\leadsto x \cdot \left(\color{blue}{{x}^{3}} \cdot \frac{1}{12} + x \cdot 1\right) \]

   6. *-rgt-identity N/A

      \[\leadsto x \cdot \left({x}^{3} \cdot \frac{1}{12} + \color{blue}{x}\right) \]

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

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{12}, x\right)} \]

   8. cube-unmult N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot x\right)}, \frac{1}{12}, x\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot x\right)}, \frac{1}{12}, x\right) \]

   10. +-rgt-identity N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot \color{blue}{\left(x \cdot x + 0\right)}, \frac{1}{12}, x\right) \]

   11. accelerator-lowering-fma.f64 97.1

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot \color{blue}{\mathsf{fma}\left(x, x, 0\right)}, 0.08333333333333333, x\right) \]

7. Applied egg-rr 97.1%

   \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, x, 0\right), 0.08333333333333333, x\right)} \]
8. Add Preprocessing

Alternative 10: 98.8% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (* x (fma 0.08333333333333333 (* x x) 1.0))))

C

double code(double x) {
	return x * (x * fma(0.08333333333333333, (x * x), 1.0));
}

Julia

function code(x)
	return Float64(x * Float64(x * fma(0.08333333333333333, Float64(x * x), 1.0)))
end

Wolfram

code[x_] := N[(x * N[(x * N[(0.08333333333333333 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)} \]

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

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)} \]

   4. +-rgt-identity N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right) + 0\right)} \]

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

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, 1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right), 0\right)} \]

5. Simplified 97.3%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right), 1\right), 0\right)} \]

6. Taylor expanded in x around 0

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

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

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\frac{1}{12} \cdot {x}^{2} + 1\right)}\right) \]

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

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{12}, {x}^{2}, 1\right)}\right) \]

   4. unpow2 N/A

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\frac{1}{12}, \color{blue}{x \cdot x}, 1\right)\right) \]

   5. *-lowering-*.f64 97.1

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(0.08333333333333333, \color{blue}{x \cdot x}, 1\right)\right) \]

8. Simplified 97.1%

   \[\leadsto x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right)\right)} \]
9. Add Preprocessing

Alternative 11: 98.3% accurate, 34.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.5%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. remove-double-neg N/A

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

   2. +-rgt-identity N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left({x}^{2}\right)\right) + 0\right)}\right) \]

   3. distribute-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right) + \left(\mathsf{neg}\left(0\right)\right)} \]

   4. remove-double-neg N/A

      \[\leadsto \color{blue}{{x}^{2}} + \left(\mathsf{neg}\left(0\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \color{blue}{x \cdot x} + \left(\mathsf{neg}\left(0\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto x \cdot x + \color{blue}{0} \]

   7. accelerator-lowering-fma.f64 96.9

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

5. Simplified 96.9%

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

   1. +-rgt-identity N/A

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

   2. *-lowering-*.f64 96.9

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

7. Applied egg-rr 96.9%

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

Developer Target 1: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sinh \left(\frac{x}{2}\right)\\ 4 \cdot \left(t\_0 \cdot t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sinh (/ x 2.0)))) (* 4.0 (* t_0 t_0))))

C

double code(double x) {
	double t_0 = sinh((x / 2.0));
	return 4.0 * (t_0 * t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sinh((x / 2.0d0))
    code = 4.0d0 * (t_0 * t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.sinh((x / 2.0));
	return 4.0 * (t_0 * t_0);
}

Python

def code(x):
	t_0 = math.sinh((x / 2.0))
	return 4.0 * (t_0 * t_0)

Julia

function code(x)
	t_0 = sinh(Float64(x / 2.0))
	return Float64(4.0 * Float64(t_0 * t_0))
end

MATLAB

function tmp = code(x)
	t_0 = sinh((x / 2.0));
	tmp = 4.0 * (t_0 * t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sinh[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024197 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64
  :pre (<= (fabs x) 710.0)

  :alt
  (! :herbie-platform default (* 4 (* (sinh (/ x 2)) (sinh (/ x 2)))))

  (+ (- (exp x) 2.0) (exp (- x))))