exp2 (problem 3.3.7)

Percentage Accurate: 54.2% → 99.2%

Time: 13.4s

Alternatives: 9

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: 54.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot 0.002777777777777778\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (pow x 2.0)
  (+
   1.0
   (*
    (pow x 2.0)
    (+ 0.08333333333333333 (* (pow x 2.0) 0.002777777777777778))))))

C

double code(double x) {
	return pow(x, 2.0) * (1.0 + (pow(x, 2.0) * (0.08333333333333333 + (pow(x, 2.0) * 0.002777777777777778))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** 2.0d0) * (1.0d0 + ((x ** 2.0d0) * (0.08333333333333333d0 + ((x ** 2.0d0) * 0.002777777777777778d0))))
end function

Java

public static double code(double x) {
	return Math.pow(x, 2.0) * (1.0 + (Math.pow(x, 2.0) * (0.08333333333333333 + (Math.pow(x, 2.0) * 0.002777777777777778))));
}

Python

def code(x):
	return math.pow(x, 2.0) * (1.0 + (math.pow(x, 2.0) * (0.08333333333333333 + (math.pow(x, 2.0) * 0.002777777777777778))))

Julia

function code(x)
	return Float64((x ^ 2.0) * Float64(1.0 + Float64((x ^ 2.0) * Float64(0.08333333333333333 + Float64((x ^ 2.0) * 0.002777777777777778)))))
end

MATLAB

function tmp = code(x)
	tmp = (x ^ 2.0) * (1.0 + ((x ^ 2.0) * (0.08333333333333333 + ((x ^ 2.0) * 0.002777777777777778))));
end

Wolfram

code[x_] := N[(N[Power[x, 2.0], $MachinePrecision] * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.08333333333333333 + N[(N[Power[x, 2.0], $MachinePrecision] * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.7%

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

   1. associate-+l- 49.6%

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

   2. sub-neg 49.6%

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

   3. sub-neg 49.6%

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

   4. distribute-neg-in 49.6%

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

   5. remove-double-neg 49.6%

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

   6. +-commutative 49.6%

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

   7. metadata-eval 49.6%

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

3. Simplified 49.6%

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

5. Taylor expanded in x around 0 99.5%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {x}^{2}\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 99.5%

      \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + \color{blue}{{x}^{2} \cdot 0.002777777777777778}\right)\right) \]

7. Simplified 99.5%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot 0.002777777777777778\right)\right)} \]

8. Final simplification 99.5%

   \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot 0.002777777777777778\right)\right) \]
9. Add Preprocessing

Alternative 2: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.08333333333333333, {x}^{4}, {x}^{2}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma 0.08333333333333333 (pow x 4.0) (pow x 2.0)))

C

double code(double x) {
	return fma(0.08333333333333333, pow(x, 4.0), pow(x, 2.0));
}

Julia

function code(x)
	return fma(0.08333333333333333, (x ^ 4.0), (x ^ 2.0))
end

Wolfram

code[x_] := N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision] + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.7%

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

   1. associate-+l- 49.6%

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

   2. sub-neg 49.6%

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

   3. sub-neg 49.6%

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

   4. distribute-neg-in 49.6%

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

   5. remove-double-neg 49.6%

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

   6. +-commutative 49.6%

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

   7. metadata-eval 49.6%

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

3. Simplified 49.6%

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

5. Taylor expanded in x around 0 99.3%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {x}^{2}\right)} \]
6. Step-by-step derivation

   1. +-commutative 99.3%

      \[\leadsto {x}^{2} \cdot \color{blue}{\left(0.08333333333333333 \cdot {x}^{2} + 1\right)} \]

   2. distribute-rgt-in 99.4%

      \[\leadsto \color{blue}{\left(0.08333333333333333 \cdot {x}^{2}\right) \cdot {x}^{2} + 1 \cdot {x}^{2}} \]

   3. associate-*l* 99.4%

      \[\leadsto \color{blue}{0.08333333333333333 \cdot \left({x}^{2} \cdot {x}^{2}\right)} + 1 \cdot {x}^{2} \]

   4. *-lft-identity 99.4%

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

   5. fma-define 99.4%

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

   6. pow-sqr 99.4%

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

   7. metadata-eval 99.4%

      \[\leadsto \mathsf{fma}\left(0.08333333333333333, {x}^{\color{blue}{4}}, {x}^{2}\right) \]

7. Simplified 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.08333333333333333, {x}^{4}, {x}^{2}\right)} \]

8. Final simplification 99.4%

   \[\leadsto \mathsf{fma}\left(0.08333333333333333, {x}^{4}, {x}^{2}\right) \]
9. Add Preprocessing

Alternative 3: 99.0% 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 49.7%

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

   1. associate-+l- 49.6%

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

   2. sub-neg 49.6%

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

   3. sub-neg 49.6%

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

   4. distribute-neg-in 49.6%

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

   5. remove-double-neg 49.6%

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

   6. +-commutative 49.6%

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

   7. metadata-eval 49.6%

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

3. Simplified 49.6%

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

5. Taylor expanded in x around 0 99.3%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {x}^{2}\right)} \]
6. Step-by-step derivation

   1. *-commutative 99.3%

      \[\leadsto {x}^{2} \cdot \left(1 + \color{blue}{{x}^{2} \cdot 0.08333333333333333}\right) \]

7. Simplified 99.3%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot 0.08333333333333333\right)} \]
8. Step-by-step derivation

   1. distribute-rgt-in 99.4%

      \[\leadsto \color{blue}{1 \cdot {x}^{2} + \left({x}^{2} \cdot 0.08333333333333333\right) \cdot {x}^{2}} \]

   2. *-un-lft-identity 99.4%

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

   3. unpow2 99.4%

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

   4. fma-define 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-*r* 99.4%

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

   7. pow-sqr 99.4%

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

   8. metadata-eval 99.4%

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

9. Applied egg-rr 99.4%

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

10. Final simplification 99.4%

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

Alternative 4: 98.5% 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 49.7%

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

   1. associate-+l- 49.6%

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

   2. sub-neg 49.6%

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

   3. sub-neg 49.6%

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

   4. distribute-neg-in 49.6%

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

   5. remove-double-neg 49.6%

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

   6. +-commutative 49.6%

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

   7. metadata-eval 49.6%

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

3. Simplified 49.6%

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

5. Taylor expanded in x around 0 98.9%

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

6. Final simplification 98.9%

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

Alternative 5: 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 49.7%

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

   1. associate-+l- 49.6%

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

   2. sub-neg 49.6%

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

   3. sub-neg 49.6%

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

   4. distribute-neg-in 49.6%

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

   5. remove-double-neg 49.6%

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

   6. +-commutative 49.6%

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

   7. metadata-eval 49.6%

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

3. Simplified 49.6%

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

5. Taylor expanded in x around 0 48.5%

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

6. Taylor expanded in x around inf 48.5%

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

   1. expm1-define 6.2%

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

8. Simplified 6.2%

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

9. Final simplification 6.2%

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

Alternative 6: 5.9% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (+
   1.0
   (* x (+ 0.5 (* x (+ 0.16666666666666666 (* x 0.041666666666666664))))))))

C

double code(double x) {
	return x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + (x * (0.5d0 + (x * (0.16666666666666666d0 + (x * 0.041666666666666664d0))))))
end function

Java

public static double code(double x) {
	return x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))));
}

Python

def code(x):
	return x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(0.16666666666666666 + Float64(x * 0.041666666666666664)))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * N[(0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.7%

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

   1. associate-+l- 49.6%

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

   2. sub-neg 49.6%

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

   3. sub-neg 49.6%

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

   4. distribute-neg-in 49.6%

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

   5. remove-double-neg 49.6%

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

   6. +-commutative 49.6%

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

   7. metadata-eval 49.6%

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

3. Simplified 49.6%

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

5. Taylor expanded in x around 0 48.5%

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

6. Taylor expanded in x around 0 5.8%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + 0.041666666666666664 \cdot x\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative 5.8%

      \[\leadsto x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + \color{blue}{x \cdot 0.041666666666666664}\right)\right)\right) \]

8. Simplified 5.8%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)} \]

9. Final simplification 5.8%

   \[\leadsto x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right) \]
10. Add Preprocessing

Alternative 7: 5.8% accurate, 18.7× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666))))))

C

double code(double x) {
	return x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0))))
end function

Java

public static double code(double x) {
	return x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))));
}

Python

def code(x):
	return x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666)))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.7%

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

   1. associate-+l- 49.6%

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

   2. sub-neg 49.6%

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

   3. sub-neg 49.6%

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

   4. distribute-neg-in 49.6%

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

   5. remove-double-neg 49.6%

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

   6. +-commutative 49.6%

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

   7. metadata-eval 49.6%

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

3. Simplified 49.6%

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

5. Taylor expanded in x around 0 48.5%

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

6. Taylor expanded in x around 0 5.8%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(0.5 + 0.16666666666666666 \cdot x\right)\right)} \]

7. Final simplification 5.8%

   \[\leadsto x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right) \]
8. Add Preprocessing

Alternative 8: 5.9% accurate, 29.4× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + x \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (+ 1.0 (* x 0.5))))

C

double code(double x) {
	return x * (1.0 + (x * 0.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + (x * 0.5d0))
end function

Java

public static double code(double x) {
	return x * (1.0 + (x * 0.5));
}

Python

def code(x):
	return x * (1.0 + (x * 0.5))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(x * 0.5)))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + (x * 0.5));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.7%

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

   1. associate-+l- 49.6%

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

   2. sub-neg 49.6%

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

   3. sub-neg 49.6%

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

   4. distribute-neg-in 49.6%

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

   5. remove-double-neg 49.6%

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

   6. +-commutative 49.6%

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

   7. metadata-eval 49.6%

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

3. Simplified 49.6%

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

5. Taylor expanded in x around 0 48.5%

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

6. Taylor expanded in x around 0 5.8%

   \[\leadsto \color{blue}{x \cdot \left(1 + 0.5 \cdot x\right)} \]
7. Step-by-step derivation

   1. *-commutative 5.8%

      \[\leadsto x \cdot \left(1 + \color{blue}{x \cdot 0.5}\right) \]

8. Simplified 5.8%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot 0.5\right)} \]

9. Final simplification 5.8%

   \[\leadsto x \cdot \left(1 + x \cdot 0.5\right) \]
10. Add Preprocessing

Alternative 9: 5.8% 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 49.7%

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

   1. associate-+l- 49.6%

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

   2. sub-neg 49.6%

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

   3. sub-neg 49.6%

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

   4. distribute-neg-in 49.6%

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

   5. remove-double-neg 49.6%

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

   6. +-commutative 49.6%

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

   7. metadata-eval 49.6%

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

3. Simplified 49.6%

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

5. Taylor expanded in x around 0 48.5%

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

6. Taylor expanded in x around 0 5.8%

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

7. Final simplification 5.8%

   \[\leadsto x \]
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 2024053 
(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))))