exp2 (problem 3.3.7)

Percentage Accurate: 53.4% → 99.1%

Time: 16.3s

Alternatives: 10

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: 53.4% 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: 99.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right)\right) + 0.08333333333333333, x \cdot \left(\mathsf{fma}\left(x, x, 2\right) + -2\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (fma
   (+
    (* x (* x (fma (fma x x 0.0) 4.96031746031746e-5 0.002777777777777778)))
    0.08333333333333333)
   (* x (+ (fma x x 2.0) -2.0))
   x)))

C

double code(double x) {
	return x * fma(((x * (x * fma(fma(x, x, 0.0), 4.96031746031746e-5, 0.002777777777777778))) + 0.08333333333333333), (x * (fma(x, x, 2.0) + -2.0)), x);
}

Julia

function code(x)
	return Float64(x * fma(Float64(Float64(x * Float64(x * fma(fma(x, x, 0.0), 4.96031746031746e-5, 0.002777777777777778))) + 0.08333333333333333), Float64(x * Float64(fma(x, x, 2.0) + -2.0)), x))
end

Wolfram

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

Derivation

1. Initial program 54.7%

   \[\left(e^{x} - 2\right) + e^{-x} \]
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 98.7%

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

   1. +-rgt-identity N/A

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

   2. distribute-rgt-in N/A

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

   3. +-rgt-identity N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. pow3 N/A

      \[\leadsto x \cdot \left(\left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right) + \frac{1}{12}\right) \cdot \color{blue}{{x}^{3}} + 1 \cdot x\right) \]

   7. *-lft-identity N/A

      \[\leadsto x \cdot \left(\left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right) + \frac{1}{12}\right) \cdot {x}^{3} + \color{blue}{x}\right) \]

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

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right) + \frac{1}{12}, {x}^{3}, x\right)} \]

7. Applied egg-rr 98.7%

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

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

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

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

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

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

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

   4. +-rgt-identity N/A

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

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

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

   6. +-rgt-identity N/A

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

   7. accelerator-lowering-fma.f64 98.7

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

9. Applied egg-rr 98.7%

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

    1. *-rgt-identity N/A

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

    2. metadata-eval N/A

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

    3. associate-*l* N/A

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

    4. associate-*r* N/A

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

    5. metadata-eval N/A

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

    6. associate-+l+ N/A

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

    7. metadata-eval N/A

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

    8. distribute-rgt-in N/A

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

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

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

    10. distribute-rgt-in N/A

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

    11. associate-*r* N/A

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

    12. associate-*l* N/A

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

    13. metadata-eval N/A

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

    14. *-rgt-identity N/A

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

    15. metadata-eval N/A

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

    16. accelerator-lowering-fma.f64 98.7

        \[\leadsto x \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right)\right) + 0.08333333333333333, x \cdot \left(\color{blue}{\mathsf{fma}\left(x, x, 2\right)} + -2\right), x\right) \]

11. Applied egg-rr 98.7%

    \[\leadsto x \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right)\right) + 0.08333333333333333, x \cdot \color{blue}{\left(\mathsf{fma}\left(x, x, 2\right) + -2\right)}, x\right) \]
12. Add Preprocessing

Alternative 2: 99.1% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.7%

   \[\left(e^{x} - 2\right) + e^{-x} \]
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 98.7%

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

   1. +-rgt-identity N/A

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

   2. distribute-rgt-in N/A

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

   3. +-rgt-identity N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. pow3 N/A

      \[\leadsto x \cdot \left(\left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right) + \frac{1}{12}\right) \cdot \color{blue}{{x}^{3}} + 1 \cdot x\right) \]

   7. *-lft-identity N/A

      \[\leadsto x \cdot \left(\left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right) + \frac{1}{12}\right) \cdot {x}^{3} + \color{blue}{x}\right) \]

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

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right) + \frac{1}{12}, {x}^{3}, x\right)} \]

7. Applied egg-rr 98.7%

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

   1. +-rgt-identity N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

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

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

   5. *-lowering-*.f64 98.7

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

9. Applied egg-rr 98.7%

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

10. Final simplification 98.7%

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

Alternative 3: 99.1% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.7%

   \[\left(e^{x} - 2\right) + e^{-x} \]
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 98.7%

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

   1. +-rgt-identity N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

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

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

7. Applied egg-rr 98.7%

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

Alternative 4: 99.1% accurate, 4.8× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 54.7%

   \[\left(e^{x} - 2\right) + e^{-x} \]
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 98.7%

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

6. Taylor expanded in x around 0

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

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

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

   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. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

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

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

8. Simplified 98.7%

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

Alternative 5: 99.0% accurate, 5.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fma
  x
  x
  (*
   (fma x x 0.0)
   (*
    (fma x x 0.0)
    (fma (fma x x 0.0) 0.002777777777777778 0.08333333333333333)))))

C

double code(double x) {
	return fma(x, x, (fma(x, x, 0.0) * (fma(x, x, 0.0) * fma(fma(x, x, 0.0), 0.002777777777777778, 0.08333333333333333))));
}

Julia

function code(x)
	return fma(x, x, Float64(fma(x, x, 0.0) * Float64(fma(x, x, 0.0) * fma(fma(x, x, 0.0), 0.002777777777777778, 0.08333333333333333))))
end

Wolfram

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

Derivation

1. Initial program 54.7%

   \[\left(e^{x} - 2\right) + e^{-x} \]
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 98.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.002777777777777778, 0.08333333333333333\right), 1\right)} \]
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. *-lowering-*.f64 N/A

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

   8. +-rgt-identity N/A

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

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

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

   10. associate-*r* N/A

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

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

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

   12. +-rgt-identity N/A

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

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

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

   14. +-rgt-identity N/A

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

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

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

   16. +-rgt-identity N/A

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

   17. accelerator-lowering-fma.f64 98.3

       \[\leadsto \mathsf{fma}\left(x, x, \mathsf{fma}\left(x, x, 0\right) \cdot \left(\mathsf{fma}\left(x, x, 0\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, 0.002777777777777778, 0.08333333333333333\right)\right)\right) \]

7. Applied egg-rr 98.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, x, 0\right) \cdot \left(\mathsf{fma}\left(x, x, 0\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.002777777777777778, 0.08333333333333333\right)\right)\right)} \]
8. Add Preprocessing

Alternative 6: 99.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.002777777777777778, 0.08333333333333333\right), x \cdot \left(\mathsf{fma}\left(x, x, 2\right) + -2\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (fma
   (fma (* x x) 0.002777777777777778 0.08333333333333333)
   (* x (+ (fma x x 2.0) -2.0))
   x)))

C

double code(double x) {
	return x * fma(fma((x * x), 0.002777777777777778, 0.08333333333333333), (x * (fma(x, x, 2.0) + -2.0)), x);
}

Julia

function code(x)
	return Float64(x * fma(fma(Float64(x * x), 0.002777777777777778, 0.08333333333333333), Float64(x * Float64(fma(x, x, 2.0) + -2.0)), x))
end

Wolfram

code[x_] := N[(x * N[(N[(N[(x * x), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision] * N[(x * N[(N[(x * x + 2.0), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.7%

   \[\left(e^{x} - 2\right) + e^{-x} \]
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 98.7%

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

   1. +-rgt-identity N/A

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

   2. distribute-rgt-in N/A

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

   3. +-rgt-identity N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. pow3 N/A

      \[\leadsto x \cdot \left(\left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right) + \frac{1}{12}\right) \cdot \color{blue}{{x}^{3}} + 1 \cdot x\right) \]

   7. *-lft-identity N/A

      \[\leadsto x \cdot \left(\left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right) + \frac{1}{12}\right) \cdot {x}^{3} + \color{blue}{x}\right) \]

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

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right) + \frac{1}{12}, {x}^{3}, x\right)} \]

7. Applied egg-rr 98.7%

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

8. Taylor expanded in x around 0

   \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}}, x \cdot \mathsf{fma}\left(x, x, 0\right), x\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

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

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 98.3

      \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.002777777777777778, 0.08333333333333333\right), x \cdot \mathsf{fma}\left(x, x, 0\right), x\right) \]

10. Simplified 98.3%

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

    1. *-rgt-identity N/A

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

    2. metadata-eval N/A

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

    3. associate-*l* N/A

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

    4. associate-*r* N/A

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

    5. metadata-eval N/A

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

    6. associate-+l+ N/A

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

    7. metadata-eval N/A

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

    8. distribute-rgt-in N/A

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

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

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

    10. distribute-rgt-in N/A

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

    11. associate-*r* N/A

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

    12. associate-*l* N/A

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

    13. metadata-eval N/A

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

    14. *-rgt-identity N/A

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

    15. metadata-eval N/A

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

    16. accelerator-lowering-fma.f64 98.3

        \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.002777777777777778, 0.08333333333333333\right), x \cdot \left(\color{blue}{\mathsf{fma}\left(x, x, 2\right)} + -2\right), x\right) \]

12. Applied egg-rr 98.3%

    \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.002777777777777778, 0.08333333333333333\right), x \cdot \color{blue}{\left(\mathsf{fma}\left(x, x, 2\right) + -2\right)}, x\right) \]
13. Add Preprocessing

Alternative 7: 99.0% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.7%

   \[\left(e^{x} - 2\right) + e^{-x} \]
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 98.7%

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

   1. +-rgt-identity N/A

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

   2. distribute-rgt-in N/A

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

   3. +-rgt-identity N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. pow3 N/A

      \[\leadsto x \cdot \left(\left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right) + \frac{1}{12}\right) \cdot \color{blue}{{x}^{3}} + 1 \cdot x\right) \]

   7. *-lft-identity N/A

      \[\leadsto x \cdot \left(\left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right) + \frac{1}{12}\right) \cdot {x}^{3} + \color{blue}{x}\right) \]

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

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{20160} + \frac{1}{360}\right)\right) + \frac{1}{12}, {x}^{3}, x\right)} \]

7. Applied egg-rr 98.7%

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

8. Taylor expanded in x around 0

   \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}}, x \cdot \mathsf{fma}\left(x, x, 0\right), x\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

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

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 98.3

      \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.002777777777777778, 0.08333333333333333\right), x \cdot \mathsf{fma}\left(x, x, 0\right), x\right) \]

10. Simplified 98.3%

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

    1. +-rgt-identity N/A

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

    2. associate-*r* N/A

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

    3. associate-*r* N/A

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

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

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

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

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

    6. *-commutative N/A

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

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

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

    8. associate-*l* N/A

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

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

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

    10. *-lowering-*.f64 98.3

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

12. Applied egg-rr 98.3%

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

13. Final simplification 98.3%

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

Alternative 8: 99.0% accurate, 6.3× speedup

Math

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

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(x, Float64(x * fma(x, Float64(x * 0.002777777777777778), 0.08333333333333333)), 1.0)))
end

Wolfram

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

Derivation

1. Initial program 54.7%

   \[\left(e^{x} - 2\right) + e^{-x} \]
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 98.7%

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

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. unpow2 N/A

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

   4. associate-*l* N/A

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

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

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

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

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

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

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

   12. *-commutative N/A

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

   13. *-lowering-*.f64 98.3

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

8. Simplified 98.3%

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

Alternative 9: 98.8% accurate, 9.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.7%

   \[\left(e^{x} - 2\right) + e^{-x} \]
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 98.7%

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

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. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 97.8

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

8. Simplified 97.8%

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

Alternative 10: 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 54.7%

   \[\left(e^{x} - 2\right) + e^{-x} \]
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 2024194 
(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))))