exp2 (problem 3.3.7)

Percentage Accurate: 53.7% → 99.1%

Time: 11.1s

Alternatives: 7

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: 53.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

   1. associate-+l- 51.6%

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

   2. sub-neg 51.6%

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

   3. sub-neg 51.6%

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

   4. distribute-neg-in 51.6%

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

   5. remove-double-neg 51.6%

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

   6. +-commutative 51.6%

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

   7. metadata-eval 51.6%

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

3. Simplified 51.6%

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

5. Taylor expanded in x around 0 99.4%

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

   1. +-commutative 99.4%

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

   2. unpow2 99.4%

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

   3. fma-define 99.4%

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

7. Applied egg-rr 99.4%

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

   1. +-commutative 99.4%

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

   2. fma-undefine 99.4%

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

   3. pow2 99.4%

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

   4. metadata-eval 99.4%

      \[\leadsto \left({x}^{\color{blue}{\left(\frac{4}{2}\right)}} + 0.08333333333333333 \cdot {x}^{4}\right) + 0.002777777777777778 \cdot {x}^{6} \]

   5. associate-+l+ 99.4%

      \[\leadsto \color{blue}{{x}^{\left(\frac{4}{2}\right)} + \left(0.08333333333333333 \cdot {x}^{4} + 0.002777777777777778 \cdot {x}^{6}\right)} \]

   6. metadata-eval 99.4%

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

   7. pow2 99.4%

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

   8. fma-define 99.4%

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

   9. fma-define 99.4%

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

9. Applied egg-rr 99.4%

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

    1. fma-undefine 99.4%

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

11. Applied egg-rr 99.4%

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

12. Final simplification 99.4%

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

Alternative 2: 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 51.7%

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

   1. associate-+l- 51.6%

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

   2. sub-neg 51.6%

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

   3. sub-neg 51.6%

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

   4. distribute-neg-in 51.6%

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

   5. remove-double-neg 51.6%

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

   6. +-commutative 51.6%

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

   7. metadata-eval 51.6%

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

3. Simplified 51.6%

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

5. Taylor expanded in x around 0 99.4%

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

   1. +-commutative 99.4%

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

   2. unpow2 99.4%

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

   3. fma-define 99.4%

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

7. Applied egg-rr 99.4%

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

8. Final simplification 99.4%

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

Alternative 3: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(x + 0.041666666666666664 \cdot {x}^{3}\right)}^{2} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow (+ x (* 0.041666666666666664 (pow x 3.0))) 2.0))

C

double code(double x) {
	return pow((x + (0.041666666666666664 * pow(x, 3.0))), 2.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x + (0.041666666666666664d0 * (x ** 3.0d0))) ** 2.0d0
end function

Java

public static double code(double x) {
	return Math.pow((x + (0.041666666666666664 * Math.pow(x, 3.0))), 2.0);
}

Python

def code(x):
	return math.pow((x + (0.041666666666666664 * math.pow(x, 3.0))), 2.0)

Julia

function code(x)
	return Float64(x + Float64(0.041666666666666664 * (x ^ 3.0))) ^ 2.0
end

MATLAB

function tmp = code(x)
	tmp = (x + (0.041666666666666664 * (x ^ 3.0))) ^ 2.0;
end

Wolfram

code[x_] := N[Power[N[(x + N[(0.041666666666666664 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]

Derivation

1. Initial program 51.7%

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

   1. associate-+l- 51.6%

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

   2. sub-neg 51.6%

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

   3. sub-neg 51.6%

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

   4. distribute-neg-in 51.6%

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

   5. remove-double-neg 51.6%

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

   6. +-commutative 51.6%

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

   7. metadata-eval 51.6%

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

3. Simplified 51.6%

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

   1. +-commutative 51.6%

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

   2. associate-+r+ 51.7%

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

   3. metadata-eval 51.7%

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

   4. sub-neg 51.7%

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

   5. add-sqr-sqrt 51.7%

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

   6. pow2 51.7%

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

   7. sub-neg 51.7%

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

   8. metadata-eval 51.7%

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

   9. associate-+r+ 51.6%

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

   10. +-commutative 51.6%

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

   11. associate-+r+ 51.5%

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

   12. +-commutative 51.5%

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

   13. cosh-undef 51.5%

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

6. Applied egg-rr 51.5%

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

7. Taylor expanded in x around 0 99.2%

   \[\leadsto {\color{blue}{\left(x + 0.041666666666666664 \cdot {x}^{3}\right)}}^{2} \]

8. Final simplification 99.2%

   \[\leadsto {\left(x + 0.041666666666666664 \cdot {x}^{3}\right)}^{2} \]
9. Add Preprocessing

Alternative 4: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

   1. associate-+l- 51.6%

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

   2. sub-neg 51.6%

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

   3. sub-neg 51.6%

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

   4. distribute-neg-in 51.6%

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

   5. remove-double-neg 51.6%

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

   6. +-commutative 51.6%

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

   7. metadata-eval 51.6%

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

3. Simplified 51.6%

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

5. Taylor expanded in x around 0 99.1%

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

   1. +-commutative 99.4%

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

   2. unpow2 99.4%

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

   3. fma-define 99.4%

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

7. Applied egg-rr 99.1%

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

8. Final simplification 99.1%

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

Alternative 5: 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 51.7%

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

   1. associate-+l- 51.6%

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

   2. sub-neg 51.6%

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

   3. sub-neg 51.6%

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

   4. distribute-neg-in 51.6%

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

   5. remove-double-neg 51.6%

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

   6. +-commutative 51.6%

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

   7. metadata-eval 51.6%

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

3. Simplified 51.6%

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

5. Taylor expanded in x around 0 98.5%

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

6. Final simplification 98.5%

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

Alternative 6: 6.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (expm1 x))

C

double code(double x) {
	return expm1(x);
}

Java

public static double code(double x) {
	return Math.expm1(x);
}

Python

def code(x):
	return math.expm1(x)

Julia

function code(x)
	return expm1(x)
end

Wolfram

code[x_] := N[(Exp[x] - 1), $MachinePrecision]

Derivation

1. Initial program 51.7%

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

   1. associate-+l- 51.6%

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

   2. sub-neg 51.6%

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

   3. sub-neg 51.6%

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

   4. distribute-neg-in 51.6%

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

   5. remove-double-neg 51.6%

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

   6. +-commutative 51.6%

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

   7. metadata-eval 51.6%

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

3. Simplified 51.6%

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

5. Taylor expanded in x around 0 49.8%

   \[\leadsto e^{x} + \color{blue}{-1} \]

6. Taylor expanded in x around inf 49.8%

   \[\leadsto \color{blue}{e^{x} - 1} \]
7. Step-by-step derivation

   1. expm1-define 6.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right)} \]

8. Simplified 6.3%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right)} \]

9. Final simplification 6.3%

   \[\leadsto \mathsf{expm1}\left(x\right) \]
10. Add Preprocessing

Alternative 7: 5.9% accurate, 206.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 51.7%

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

   1. associate-+l- 51.6%

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

   2. sub-neg 51.6%

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

   3. sub-neg 51.6%

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

   4. distribute-neg-in 51.6%

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

   5. remove-double-neg 51.6%

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

   6. +-commutative 51.6%

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

   7. metadata-eval 51.6%

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

3. Simplified 51.6%

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

5. Taylor expanded in x around 0 49.8%

   \[\leadsto e^{x} + \color{blue}{-1} \]

6. Taylor expanded in x around 0 6.0%

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

7. Final simplification 6.0%

   \[\leadsto x \]
8. 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 2024048 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64
  :pre (<= (fabs x) 710.0)

  :alt
  (* 4.0 (* (sinh (/ x 2.0)) (sinh (/ x 2.0))))

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