exp2 (problem 3.3.7)

Percentage Accurate: 53.2% → 99.2%

Time: 11.1s

Alternatives: 6

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: 53.2% 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: 99.2% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.4%

   \[\left(e^{x} - 2\right) + e^{-x} \]
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. lower-*.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. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. *-rgt-identity N/A

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

   9. lower-fma.f64 N/A

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

5. Simplified 98.9%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. distribute-rgt-in N/A

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

   8. lift-*.f64 N/A

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

7. Applied egg-rr 98.9%

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

8. Final simplification 98.9%

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

Alternative 2: 99.2% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.4%

   \[\left(e^{x} - 2\right) + e^{-x} \]
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. lower-*.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. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. *-rgt-identity N/A

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

   9. lower-fma.f64 N/A

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

5. Simplified 98.9%

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

Alternative 3: 98.8% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.4%

   \[\left(e^{x} - 2\right) + e^{-x} \]
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. lower-*.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. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. *-rgt-identity N/A

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

   9. lower-fma.f64 N/A

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

5. Simplified 98.9%

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

6. Taylor expanded in x around inf

   \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, \color{blue}{\frac{1}{20160} \cdot {x}^{3}}, \frac{1}{12}\right), x \cdot \left(x \cdot x\right), x\right) \]
7. Step-by-step derivation

   1. unpow3 N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 98.9

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

8. Simplified 98.9%

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

Alternative 4: 98.8% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.4%

   \[\left(e^{x} - 2\right) + e^{-x} \]
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. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. *-rgt-identity N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 98.9

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

5. Simplified 98.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. lift-*.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. lift-*.f64 N/A

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

   9. lower-*.f64 98.9

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

7. Applied egg-rr 98.9%

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

8. Final simplification 98.9%

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

Alternative 5: 98.8% accurate, 9.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.4%

   \[\left(e^{x} - 2\right) + e^{-x} \]
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. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. *-rgt-identity N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 98.9

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

5. Simplified 98.9%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 98.9

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

7. Applied egg-rr 98.9%

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

8. Final simplification 98.9%

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

Alternative 6: 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 53.4%

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

3. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 98.7

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

5. Simplified 98.7%

   \[\leadsto \color{blue}{x \cdot x} \]
6. 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 2024215 
(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))))