exp2 (problem 3.3.7)

Percentage Accurate: 54.0% → 99.2%

Time: 10.7s

Alternatives: 6

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.0% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  (* 4.96031746031746e-5 (pow x 8.0))
  (+
   (* 0.002777777777777778 (pow x 6.0))
   (fma x x (* 0.08333333333333333 (pow x 4.0))))))

C

double code(double x) {
	return (4.96031746031746e-5 * pow(x, 8.0)) + ((0.002777777777777778 * pow(x, 6.0)) + fma(x, x, (0.08333333333333333 * pow(x, 4.0))));
}

Julia

function code(x)
	return Float64(Float64(4.96031746031746e-5 * (x ^ 8.0)) + Float64(Float64(0.002777777777777778 * (x ^ 6.0)) + fma(x, x, Float64(0.08333333333333333 * (x ^ 4.0)))))
end

Wolfram

code[x_] := N[(N[(4.96031746031746e-5 * N[Power[x, 8.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(x * x + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.7%

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

   1. associate-+l- 50.7%

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

   2. sub-neg 50.7%

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

   3. sub-neg 50.7%

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

   4. distribute-neg-in 50.7%

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

   5. remove-double-neg 50.7%

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

   6. +-commutative 50.7%

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

   7. metadata-eval 50.7%

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

3. Simplified 50.7%

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

5. Taylor expanded in x around 0 99.3%

   \[\leadsto \color{blue}{4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \left(0.08333333333333333 \cdot {x}^{4} + {x}^{2}\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative 99.3%

      \[\leadsto 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \color{blue}{\left({x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)}\right) \]

   2. unpow2 99.3%

      \[\leadsto 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \left(\color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4}\right)\right) \]

   3. fma-define 99.3%

      \[\leadsto 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \color{blue}{\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)}\right) \]

7. Applied egg-rr 99.3%

   \[\leadsto 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \color{blue}{\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)}\right) \]

8. Final simplification 99.3%

   \[\leadsto 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\right) \]
9. Add Preprocessing

Alternative 2: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \left(0.002777777777777778 \cdot {x}^{6} + \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\right) + {x}^{8} \cdot 3.8580246913580246 \cdot 10^{-5} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  (+
   (* 0.002777777777777778 (pow x 6.0))
   (fma x x (* 0.08333333333333333 (pow x 4.0))))
  (* (pow x 8.0) 3.8580246913580246e-5)))

C

double code(double x) {
	return ((0.002777777777777778 * pow(x, 6.0)) + fma(x, x, (0.08333333333333333 * pow(x, 4.0)))) + (pow(x, 8.0) * 3.8580246913580246e-5);
}

Julia

function code(x)
	return Float64(Float64(Float64(0.002777777777777778 * (x ^ 6.0)) + fma(x, x, Float64(0.08333333333333333 * (x ^ 4.0)))) + Float64((x ^ 8.0) * 3.8580246913580246e-5))
end

Wolfram

code[x_] := N[(N[(N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(x * x + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, 8.0], $MachinePrecision] * 3.8580246913580246e-5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.7%

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

   1. associate-+l- 50.7%

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

   2. sub-neg 50.7%

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

   3. sub-neg 50.7%

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

   4. distribute-neg-in 50.7%

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

   5. remove-double-neg 50.7%

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

   6. +-commutative 50.7%

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

   7. metadata-eval 50.7%

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

3. Simplified 50.7%

   \[\leadsto \color{blue}{e^{x} + \left(e^{-x} + -2\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative 50.7%

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

   2. associate-+r+ 50.7%

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

   3. metadata-eval 50.7%

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

   4. sub-neg 50.7%

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

   5. add-exp-log 50.7%

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

   6. +-commutative 50.7%

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

   7. sub-neg 50.7%

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

   8. metadata-eval 50.7%

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

   9. associate-+r+ 50.7%

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

   10. +-commutative 50.7%

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

   11. +-commutative 50.7%

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

   12. cosh-undef 50.7%

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

6. Applied egg-rr 50.7%

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

7. Taylor expanded in x around 0 46.4%

   \[\leadsto e^{\color{blue}{-0.0006944444444444445 \cdot {x}^{4} + \left(0.08333333333333333 \cdot {x}^{2} + 2 \cdot \log x\right)}} \]

8. Taylor expanded in x around 0 99.2%

   \[\leadsto \color{blue}{3.8580246913580246 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \left(0.08333333333333333 \cdot {x}^{4} + {x}^{2}\right)\right)} \]
9. Step-by-step derivation

   1. +-commutative 99.3%

      \[\leadsto 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \color{blue}{\left({x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)}\right) \]

   2. unpow2 99.3%

      \[\leadsto 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \left(\color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4}\right)\right) \]

   3. fma-define 99.3%

      \[\leadsto 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \color{blue}{\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)}\right) \]

10. Applied egg-rr 99.2%

    \[\leadsto 3.8580246913580246 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \color{blue}{\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)}\right) \]

11. Final simplification 99.2%

    \[\leadsto \left(0.002777777777777778 \cdot {x}^{6} + \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\right) + {x}^{8} \cdot 3.8580246913580246 \cdot 10^{-5} \]
12. Add Preprocessing

Alternative 3: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ 0.002777777777777778 \cdot {x}^{6} + \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  (* 0.002777777777777778 (pow x 6.0))
  (fma x x (* 0.08333333333333333 (pow x 4.0)))))

C

double code(double x) {
	return (0.002777777777777778 * pow(x, 6.0)) + fma(x, x, (0.08333333333333333 * pow(x, 4.0)));
}

Julia

function code(x)
	return Float64(Float64(0.002777777777777778 * (x ^ 6.0)) + fma(x, x, Float64(0.08333333333333333 * (x ^ 4.0))))
end

Wolfram

code[x_] := N[(N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(x * x + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.7%

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

   1. associate-+l- 50.7%

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

   2. sub-neg 50.7%

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

   3. sub-neg 50.7%

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

   4. distribute-neg-in 50.7%

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

   5. remove-double-neg 50.7%

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

   6. +-commutative 50.7%

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

   7. metadata-eval 50.7%

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

3. Simplified 50.7%

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

5. Taylor expanded in x around 0 99.2%

   \[\leadsto \color{blue}{0.002777777777777778 \cdot {x}^{6} + \left(0.08333333333333333 \cdot {x}^{4} + {x}^{2}\right)} \]
6. Step-by-step derivation

   1. +-commutative 99.3%

      \[\leadsto 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \color{blue}{\left({x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)}\right) \]

   2. unpow2 99.3%

      \[\leadsto 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \left(\color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4}\right)\right) \]

   3. fma-define 99.3%

      \[\leadsto 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \color{blue}{\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)}\right) \]

7. Applied egg-rr 99.2%

   \[\leadsto 0.002777777777777778 \cdot {x}^{6} + \color{blue}{\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)} \]

8. Final simplification 99.2%

   \[\leadsto 0.002777777777777778 \cdot {x}^{6} + \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right) \]
9. Add Preprocessing

Alternative 4: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot {\left(e^{0.08333333333333333}\right)}^{\left({x}^{2}\right)}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (* x (pow (exp 0.08333333333333333) (pow x 2.0)))))

C

double code(double x) {
	return x * (x * pow(exp(0.08333333333333333), pow(x, 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * (exp(0.08333333333333333d0) ** (x ** 2.0d0)))
end function

Java

public static double code(double x) {
	return x * (x * Math.pow(Math.exp(0.08333333333333333), Math.pow(x, 2.0)));
}

Python

def code(x):
	return x * (x * math.pow(math.exp(0.08333333333333333), math.pow(x, 2.0)))

Julia

function code(x)
	return Float64(x * Float64(x * (exp(0.08333333333333333) ^ (x ^ 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * (exp(0.08333333333333333) ^ (x ^ 2.0)));
end

Wolfram

code[x_] := N[(x * N[(x * N[Power[N[Exp[0.08333333333333333], $MachinePrecision], N[Power[x, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.7%

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

   1. associate-+l- 50.7%

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

   2. sub-neg 50.7%

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

   3. sub-neg 50.7%

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

   4. distribute-neg-in 50.7%

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

   5. remove-double-neg 50.7%

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

   6. +-commutative 50.7%

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

   7. metadata-eval 50.7%

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

3. Simplified 50.7%

   \[\leadsto \color{blue}{e^{x} + \left(e^{-x} + -2\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative 50.7%

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

   2. associate-+r+ 50.7%

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

   3. metadata-eval 50.7%

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

   4. sub-neg 50.7%

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

   5. add-exp-log 50.7%

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

   6. +-commutative 50.7%

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

   7. sub-neg 50.7%

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

   8. metadata-eval 50.7%

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

   9. associate-+r+ 50.7%

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

   10. +-commutative 50.7%

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

   11. +-commutative 50.7%

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

   12. cosh-undef 50.7%

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

6. Applied egg-rr 50.7%

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

7. Taylor expanded in x around 0 46.3%

   \[\leadsto e^{\color{blue}{0.08333333333333333 \cdot {x}^{2} + 2 \cdot \log x}} \]
8. Step-by-step derivation

   1. exp-sum 46.3%

      \[\leadsto \color{blue}{e^{0.08333333333333333 \cdot {x}^{2}} \cdot e^{2 \cdot \log x}} \]

   2. *-commutative 46.3%

      \[\leadsto e^{0.08333333333333333 \cdot {x}^{2}} \cdot e^{\color{blue}{\log x \cdot 2}} \]

   3. pow-to-exp 98.9%

      \[\leadsto e^{0.08333333333333333 \cdot {x}^{2}} \cdot \color{blue}{{x}^{2}} \]

   4. unpow2 98.9%

      \[\leadsto e^{0.08333333333333333 \cdot {x}^{2}} \cdot \color{blue}{\left(x \cdot x\right)} \]

   5. associate-*r* 98.9%

      \[\leadsto \color{blue}{\left(e^{0.08333333333333333 \cdot {x}^{2}} \cdot x\right) \cdot x} \]

   6. exp-prod 98.9%

      \[\leadsto \left(\color{blue}{{\left(e^{0.08333333333333333}\right)}^{\left({x}^{2}\right)}} \cdot x\right) \cdot x \]

9. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{\left({\left(e^{0.08333333333333333}\right)}^{\left({x}^{2}\right)} \cdot x\right) \cdot x} \]

10. Final simplification 98.9%

    \[\leadsto x \cdot \left(x \cdot {\left(e^{0.08333333333333333}\right)}^{\left({x}^{2}\right)}\right) \]
11. Add Preprocessing

Alternative 5: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ {x}^{2} \cdot e^{0.08333333333333333 \cdot {x}^{2}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (pow x 2.0) (exp (* 0.08333333333333333 (pow x 2.0)))))

C

double code(double x) {
	return pow(x, 2.0) * exp((0.08333333333333333 * pow(x, 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** 2.0d0) * exp((0.08333333333333333d0 * (x ** 2.0d0)))
end function

Java

public static double code(double x) {
	return Math.pow(x, 2.0) * Math.exp((0.08333333333333333 * Math.pow(x, 2.0)));
}

Python

def code(x):
	return math.pow(x, 2.0) * math.exp((0.08333333333333333 * math.pow(x, 2.0)))

Julia

function code(x)
	return Float64((x ^ 2.0) * exp(Float64(0.08333333333333333 * (x ^ 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = (x ^ 2.0) * exp((0.08333333333333333 * (x ^ 2.0)));
end

Wolfram

code[x_] := N[(N[Power[x, 2.0], $MachinePrecision] * N[Exp[N[(0.08333333333333333 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.7%

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

   1. associate-+l- 50.7%

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

   2. sub-neg 50.7%

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

   3. sub-neg 50.7%

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

   4. distribute-neg-in 50.7%

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

   5. remove-double-neg 50.7%

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

   6. +-commutative 50.7%

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

   7. metadata-eval 50.7%

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

3. Simplified 50.7%

   \[\leadsto \color{blue}{e^{x} + \left(e^{-x} + -2\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative 50.7%

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

   2. associate-+r+ 50.7%

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

   3. metadata-eval 50.7%

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

   4. sub-neg 50.7%

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

   5. add-exp-log 50.7%

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

   6. +-commutative 50.7%

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

   7. sub-neg 50.7%

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

   8. metadata-eval 50.7%

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

   9. associate-+r+ 50.7%

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

   10. +-commutative 50.7%

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

   11. +-commutative 50.7%

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

   12. cosh-undef 50.7%

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

6. Applied egg-rr 50.7%

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

7. Taylor expanded in x around 0 46.3%

   \[\leadsto e^{\color{blue}{0.08333333333333333 \cdot {x}^{2} + 2 \cdot \log x}} \]
8. Step-by-step derivation

   1. exp-sum 46.3%

      \[\leadsto \color{blue}{e^{0.08333333333333333 \cdot {x}^{2}} \cdot e^{2 \cdot \log x}} \]

   2. *-commutative 46.3%

      \[\leadsto e^{0.08333333333333333 \cdot {x}^{2}} \cdot e^{\color{blue}{\log x \cdot 2}} \]

   3. pow-to-exp 98.9%

      \[\leadsto e^{0.08333333333333333 \cdot {x}^{2}} \cdot \color{blue}{{x}^{2}} \]

   4. unpow2 98.9%

      \[\leadsto e^{0.08333333333333333 \cdot {x}^{2}} \cdot \color{blue}{\left(x \cdot x\right)} \]

   5. associate-*r* 98.9%

      \[\leadsto \color{blue}{\left(e^{0.08333333333333333 \cdot {x}^{2}} \cdot x\right) \cdot x} \]

   6. exp-prod 98.9%

      \[\leadsto \left(\color{blue}{{\left(e^{0.08333333333333333}\right)}^{\left({x}^{2}\right)}} \cdot x\right) \cdot x \]

9. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{\left({\left(e^{0.08333333333333333}\right)}^{\left({x}^{2}\right)} \cdot x\right) \cdot x} \]

10. Taylor expanded in x around inf 98.9%

    \[\leadsto \color{blue}{{x}^{2} \cdot e^{0.08333333333333333 \cdot {x}^{2}}} \]

11. Final simplification 98.9%

    \[\leadsto {x}^{2} \cdot e^{0.08333333333333333 \cdot {x}^{2}} \]
12. Add Preprocessing

Alternative 6: 98.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {x}^{2} \end{array} \]

FPCore

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

C

double code(double x) {
	return pow(x, 2.0);
}

Fortran

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

Java

public static double code(double x) {
	return Math.pow(x, 2.0);
}

Python

def code(x):
	return math.pow(x, 2.0)

Julia

function code(x)
	return x ^ 2.0
end

MATLAB

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

Wolfram

code[x_] := N[Power[x, 2.0], $MachinePrecision]

Derivation

1. Initial program 50.7%

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

   1. associate-+l- 50.7%

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

   2. sub-neg 50.7%

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

   3. sub-neg 50.7%

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

   4. distribute-neg-in 50.7%

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

   5. remove-double-neg 50.7%

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

   6. +-commutative 50.7%

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

   7. metadata-eval 50.7%

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

3. Simplified 50.7%

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

5. Taylor expanded in x around 0 98.6%

   \[\leadsto \color{blue}{{x}^{2}} \]

6. Final simplification 98.6%

   \[\leadsto {x}^{2} \]
7. Add Preprocessing

Developer target: 99.9% accurate, 1.0× 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 2024044 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64
  :pre (<= (fabs x) 710.0)

  :herbie-target
  (* 4.0 (* (sinh (/ x 2.0)) (sinh (/ x 2.0))))

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