exp2 (problem 3.3.7)

Percentage Accurate: 53.4% → 99.2%

Time: 10.0s

Alternatives: 7

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: 99.2% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, x \cdot x, 0.002777777777777778\right), x \cdot x, 0.08333333333333333\right), x\right) \cdot x} \]
6. Step-by-step derivation

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.5%

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

Alternative 2: 99.2% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, x \cdot x, 0.002777777777777778\right), x \cdot x, 0.08333333333333333\right), x\right) \cdot x} \]
6. Step-by-step derivation

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.5%

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

Alternative 3: 99.0% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.8%

   \[\left(e^{x} - 2\right) + e^{-x} \]
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. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. associate-*r* N/A

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

   8. *-rgt-identity N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. cube-unmult N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{1}{12} + \frac{1}{360} \cdot {x}^{2}, x\right) \cdot x \]

   12. metadata-eval N/A

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

   13. lower-pow.f64 N/A

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

   14. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, \frac{1}{12} + \frac{1}{360} \cdot {x}^{2}, x\right) \cdot x \]

   15. +-commutative N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{1}{360} \cdot {x}^{2} + \frac{1}{12}}, x\right) \cdot x \]

   16. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{1}{360}, {x}^{2}, \frac{1}{12}\right)}, x\right) \cdot x \]

   17. unpow2 N/A

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

   18. lower-*.f64 99.5

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.002777777777777778, \color{blue}{x \cdot x}, 0.08333333333333333\right), x\right) \cdot x \]

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x\right) \cdot x} \]
6. Step-by-step derivation

7. Applied rewrites 99.5%

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

Alternative 4: 98.9% accurate, 9.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.8%

   \[\left(e^{x} - 2\right) + e^{-x} \]
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. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. associate-*r* N/A

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

   8. *-rgt-identity N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. cube-unmult N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{1}{12} + \frac{1}{360} \cdot {x}^{2}, x\right) \cdot x \]

   12. metadata-eval N/A

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

   13. lower-pow.f64 N/A

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

   14. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, \frac{1}{12} + \frac{1}{360} \cdot {x}^{2}, x\right) \cdot x \]

   15. +-commutative N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{1}{360} \cdot {x}^{2} + \frac{1}{12}}, x\right) \cdot x \]

   16. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{1}{360}, {x}^{2}, \frac{1}{12}\right)}, x\right) \cdot x \]

   17. unpow2 N/A

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

   18. lower-*.f64 99.5

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.002777777777777778, \color{blue}{x \cdot x}, 0.08333333333333333\right), x\right) \cdot x \]

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x\right) \cdot x} \]
6. Step-by-step derivation

7. Applied rewrites 99.5%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.2%

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

Alternative 5: 98.9% accurate, 9.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, x \cdot x, 0.002777777777777778\right), x \cdot x, 0.08333333333333333\right), x\right) \cdot x} \]
6. Step-by-step derivation

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.5%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 99.2%

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

Alternative 6: 98.4% 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 54.8%

   \[\left(e^{x} - 2\right) + e^{-x} \]
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 98.6

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

5. Applied rewrites 98.6%

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

Alternative 7: 51.1% accurate, 52.3× speedup

Math

\[\begin{array}{l} \\ 2 - 2 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return 2.0 - 2.0;
}

Python

def code(x):
	return 2.0 - 2.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. associate-+r- N/A

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

   5. +-commutative N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. lower--.f64 N/A

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

   10. cosh-undef N/A

       \[\leadsto \color{blue}{2 \cdot \cosh x} - 2 \]

   11. lower-*.f64 N/A

       \[\leadsto \color{blue}{2 \cdot \cosh x} - 2 \]

   12. lower-cosh.f64 54.7

       \[\leadsto 2 \cdot \color{blue}{\cosh x} - 2 \]

4. Applied rewrites 54.7%

   \[\leadsto \color{blue}{2 \cdot \cosh x - 2} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{2} - 2 \]
6. Step-by-step derivation

7. Applied rewrites 52.6%

   \[\leadsto \color{blue}{2} - 2 \]
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 2024326 
(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))))