exp2 (problem 3.3.7)

Percentage Accurate: 53.4% → 99.2%

Time: 9.2s

Alternatives: 4

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, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({x}^{6}, \mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, x \cdot x, 0.002777777777777778\right), \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right), 0.08333333333333333, x \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (pow x 6.0)
  (fma 4.96031746031746e-5 (* x x) 0.002777777777777778)
  (fma (* (* x x) (* x x)) 0.08333333333333333 (* x x))))

C

double code(double x) {
	return fma(pow(x, 6.0), fma(4.96031746031746e-5, (x * x), 0.002777777777777778), fma(((x * x) * (x * x)), 0.08333333333333333, (x * x)));
}

Julia

function code(x)
	return fma((x ^ 6.0), fma(4.96031746031746e-5, Float64(x * x), 0.002777777777777778), fma(Float64(Float64(x * x) * Float64(x * x)), 0.08333333333333333, Float64(x * x)))
end

Wolfram

code[x_] := N[(N[Power[x, 6.0], $MachinePrecision] * N[(4.96031746031746e-5 * N[(x * x), $MachinePrecision] + 0.002777777777777778), $MachinePrecision] + N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.08333333333333333 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.8%

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

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

   2. *-commutative N/A

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

   3. associate-+r+ N/A

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

   4. distribute-lft-in N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

5. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{6}, \mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, x \cdot x, 0.002777777777777778\right), \mathsf{fma}\left({x}^{4}, 0.08333333333333333, x \cdot x\right)\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.7%

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

Alternative 2: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, x \cdot x, 0.002777777777777778\right), x \cdot x, 0.08333333333333333\right), x\right) \cdot x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma
   (pow x 3.0)
   (fma
    (fma 4.96031746031746e-5 (* x x) 0.002777777777777778)
    (* x x)
    0.08333333333333333)
   x)
  x))

C

double code(double x) {
	return fma(pow(x, 3.0), fma(fma(4.96031746031746e-5, (x * x), 0.002777777777777778), (x * x), 0.08333333333333333), x) * x;
}

Julia

function code(x)
	return Float64(fma((x ^ 3.0), fma(fma(4.96031746031746e-5, Float64(x * x), 0.002777777777777778), Float64(x * x), 0.08333333333333333), x) * x)
end

Wolfram

code[x_] := N[(N[(N[Power[x, 3.0], $MachinePrecision] * N[(N[(4.96031746031746e-5 * N[(x * x), $MachinePrecision] + 0.002777777777777778), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] + x), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 48.8%

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

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

   5. associate-+r+ N/A

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

   6. +-commutative N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. cosh-undef N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-cosh.f64 N/A

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

   13. metadata-eval 48.7

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

4. Applied rewrites 48.7%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(2, \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}, -2\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(2, \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1}, -2\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(2, \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1, -2\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(2, \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), {x}^{2}, 1\right)}, -2\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(2, \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, {x}^{2}, 1\right), -2\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(2, \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{2}, {x}^{2}, 1\right), -2\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(2, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right), -2\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(2, \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right), -2\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(2, \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{1}{24}\right)}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right), -2\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(2, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right), -2\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(2, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right), -2\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(2, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right), -2\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(2, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right), -2\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(2, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right), -2\right) \]

   14. lower-*.f64 48.4

       \[\leadsto \mathsf{fma}\left(2, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right), -2\right) \]

7. Applied rewrites 48.4%

   \[\leadsto \mathsf{fma}\left(2, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}, -2\right) \]

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(2, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right), -2\right) \]
9. Step-by-step derivation

10. Applied rewrites 48.2%

    \[\leadsto \mathsf{fma}\left(2, \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right), x \cdot x, 0.5\right), x \cdot x, 1\right), -2\right) \]

11. Taylor expanded in x around 0

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

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

13. Applied rewrites 99.7%

    \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, x \cdot x, 0.002777777777777778\right), x \cdot x, 0.08333333333333333\right), x\right) \cdot x} \]
14. Add Preprocessing

Alternative 3: 98.9% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.8%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. pow-sqr N/A

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

   8. lower-pow.f64 N/A

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

   9. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{4}}, \frac{1}{12}, {x}^{2}\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{4}, \frac{1}{12}, \color{blue}{x \cdot x}\right) \]

   11. lower-*.f64 99.5

       \[\leadsto \mathsf{fma}\left({x}^{4}, 0.08333333333333333, \color{blue}{x \cdot x}\right) \]

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, 0.08333333333333333, x \cdot x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.5%

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

Alternative 4: 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 48.8%

   \[\left(e^{x} - 2\right) + e^{-x} \]
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 99.3

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

5. Applied rewrites 99.3%

   \[\leadsto \color{blue}{x \cdot x} \]
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 2024306 
(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))))