exp2 (problem 3.3.7)

Percentage Accurate: 53.1% → 99.2%

Time: 9.9s

Alternatives: 8

Speedup: 68.7×

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: 53.1% 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 57.6%

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

   1. associate-+l- 57.6%

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

   2. sub-neg 57.6%

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

   3. sub-neg 57.6%

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

   4. distribute-neg-in 57.6%

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

   5. remove-double-neg 57.6%

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

   6. +-commutative 57.6%

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

   7. metadata-eval 57.6%

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

3. Simplified 57.6%

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

4. Taylor expanded in x around 0 99.3%

   \[\leadsto \color{blue}{4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \left(0.08333333333333333 \cdot {x}^{4} + {x}^{2}\right)\right)} \]
5. 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-def 99.3%

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

6. Applied egg-rr 99.3%

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

7. Final simplification 99.3%

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

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 57.6%

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

   1. associate-+l- 57.6%

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

   2. sub-neg 57.6%

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

   3. sub-neg 57.6%

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

   4. distribute-neg-in 57.6%

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

   5. remove-double-neg 57.6%

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

   6. +-commutative 57.6%

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

   7. metadata-eval 57.6%

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

3. Simplified 57.6%

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

4. Taylor expanded in x around 0 99.1%

   \[\leadsto \color{blue}{0.002777777777777778 \cdot {x}^{6} + \left(0.08333333333333333 \cdot {x}^{4} + {x}^{2}\right)} \]
5. 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-def 99.3%

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

6. Applied egg-rr 99.1%

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

7. Final simplification 99.1%

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

Alternative 3: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \left(1 + \left({x}^{4} \cdot 0.003472222222222222 + 0.08333333333333333 \cdot {x}^{2}\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (*
   x
   (+
    1.0
    (+
     (* (pow x 4.0) 0.003472222222222222)
     (* 0.08333333333333333 (pow x 2.0)))))))

C

double code(double x) {
	return x * (x * (1.0 + ((pow(x, 4.0) * 0.003472222222222222) + (0.08333333333333333 * pow(x, 2.0)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * (1.0d0 + (((x ** 4.0d0) * 0.003472222222222222d0) + (0.08333333333333333d0 * (x ** 2.0d0)))))
end function

Java

public static double code(double x) {
	return x * (x * (1.0 + ((Math.pow(x, 4.0) * 0.003472222222222222) + (0.08333333333333333 * Math.pow(x, 2.0)))));
}

Python

def code(x):
	return x * (x * (1.0 + ((math.pow(x, 4.0) * 0.003472222222222222) + (0.08333333333333333 * math.pow(x, 2.0)))))

Julia

function code(x)
	return Float64(x * Float64(x * Float64(1.0 + Float64(Float64((x ^ 4.0) * 0.003472222222222222) + Float64(0.08333333333333333 * (x ^ 2.0))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * (1.0 + (((x ^ 4.0) * 0.003472222222222222) + (0.08333333333333333 * (x ^ 2.0)))));
end

Wolfram

code[x_] := N[(x * N[(x * N[(1.0 + N[(N[(N[Power[x, 4.0], $MachinePrecision] * 0.003472222222222222), $MachinePrecision] + N[(0.08333333333333333 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.6%

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

   1. associate-+l- 57.6%

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

   2. sub-neg 57.6%

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

   3. sub-neg 57.6%

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

   4. distribute-neg-in 57.6%

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

   5. remove-double-neg 57.6%

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

   6. +-commutative 57.6%

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

   7. metadata-eval 57.6%

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

3. Simplified 57.6%

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

   1. +-commutative 57.6%

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

   2. associate-+r+ 57.6%

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

   3. metadata-eval 57.6%

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

   4. sub-neg 57.6%

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

   5. add-exp-log 57.5%

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

   6. sub-neg 57.5%

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

   7. metadata-eval 57.5%

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

   8. associate-+r+ 57.5%

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

   9. +-commutative 57.5%

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

   10. associate-+r+ 57.5%

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

   11. +-commutative 57.5%

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

   12. cosh-undef 57.5%

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

5. Applied egg-rr 57.5%

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

6. Taylor expanded in x around 0 49.8%

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

   1. exp-sum 49.8%

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

   2. *-commutative 49.8%

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

   3. pow-to-exp 98.8%

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

   4. unpow2 98.8%

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

   5. associate-*r* 98.8%

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

   6. *-commutative 98.8%

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

   7. exp-prod 98.8%

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

8. Applied egg-rr 98.8%

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

9. Taylor expanded in x around 0 98.8%

   \[\leadsto \left(\color{blue}{\left(1 + \left(0.003472222222222222 \cdot {x}^{4} + 0.08333333333333333 \cdot {x}^{2}\right)\right)} \cdot x\right) \cdot x \]

10. Final simplification 98.8%

    \[\leadsto x \cdot \left(x \cdot \left(1 + \left({x}^{4} \cdot 0.003472222222222222 + 0.08333333333333333 \cdot {x}^{2}\right)\right)\right) \]

Alternative 4: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x + \left(0.003472222222222222 \cdot {x}^{5} + 0.08333333333333333 \cdot {x}^{3}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (+
   x
   (+
    (* 0.003472222222222222 (pow x 5.0))
    (* 0.08333333333333333 (pow x 3.0))))))

C

double code(double x) {
	return x * (x + ((0.003472222222222222 * pow(x, 5.0)) + (0.08333333333333333 * pow(x, 3.0))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x + ((0.003472222222222222d0 * (x ** 5.0d0)) + (0.08333333333333333d0 * (x ** 3.0d0))))
end function

Java

public static double code(double x) {
	return x * (x + ((0.003472222222222222 * Math.pow(x, 5.0)) + (0.08333333333333333 * Math.pow(x, 3.0))));
}

Python

def code(x):
	return x * (x + ((0.003472222222222222 * math.pow(x, 5.0)) + (0.08333333333333333 * math.pow(x, 3.0))))

Julia

function code(x)
	return Float64(x * Float64(x + Float64(Float64(0.003472222222222222 * (x ^ 5.0)) + Float64(0.08333333333333333 * (x ^ 3.0)))))
end

MATLAB

function tmp = code(x)
	tmp = x * (x + ((0.003472222222222222 * (x ^ 5.0)) + (0.08333333333333333 * (x ^ 3.0))));
end

Wolfram

code[x_] := N[(x * N[(x + N[(N[(0.003472222222222222 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.08333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.6%

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

   1. associate-+l- 57.6%

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

   2. sub-neg 57.6%

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

   3. sub-neg 57.6%

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

   4. distribute-neg-in 57.6%

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

   5. remove-double-neg 57.6%

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

   6. +-commutative 57.6%

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

   7. metadata-eval 57.6%

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

3. Simplified 57.6%

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

   1. +-commutative 57.6%

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

   2. associate-+r+ 57.6%

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

   3. metadata-eval 57.6%

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

   4. sub-neg 57.6%

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

   5. add-exp-log 57.5%

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

   6. sub-neg 57.5%

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

   7. metadata-eval 57.5%

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

   8. associate-+r+ 57.5%

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

   9. +-commutative 57.5%

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

   10. associate-+r+ 57.5%

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

   11. +-commutative 57.5%

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

   12. cosh-undef 57.5%

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

5. Applied egg-rr 57.5%

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

6. Taylor expanded in x around 0 49.8%

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

   1. exp-sum 49.8%

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

   2. *-commutative 49.8%

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

   3. pow-to-exp 98.8%

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

   4. unpow2 98.8%

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

   5. associate-*r* 98.8%

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

   6. *-commutative 98.8%

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

   7. exp-prod 98.8%

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

8. Applied egg-rr 98.8%

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

9. Taylor expanded in x around 0 98.8%

   \[\leadsto \color{blue}{\left(x + \left(0.003472222222222222 \cdot {x}^{5} + 0.08333333333333333 \cdot {x}^{3}\right)\right)} \cdot x \]

10. Final simplification 98.8%

    \[\leadsto x \cdot \left(x + \left(0.003472222222222222 \cdot {x}^{5} + 0.08333333333333333 \cdot {x}^{3}\right)\right) \]

Alternative 5: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return Float64(x * Float64(x * exp(Float64(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[Exp[N[(0.08333333333333333 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.6%

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

   1. associate-+l- 57.6%

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

   2. sub-neg 57.6%

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

   3. sub-neg 57.6%

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

   4. distribute-neg-in 57.6%

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

   5. remove-double-neg 57.6%

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

   6. +-commutative 57.6%

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

   7. metadata-eval 57.6%

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

3. Simplified 57.6%

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

   1. +-commutative 57.6%

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

   2. associate-+r+ 57.6%

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

   3. metadata-eval 57.6%

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

   4. sub-neg 57.6%

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

   5. add-exp-log 57.5%

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

   6. sub-neg 57.5%

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

   7. metadata-eval 57.5%

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

   8. associate-+r+ 57.5%

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

   9. +-commutative 57.5%

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

   10. associate-+r+ 57.5%

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

   11. +-commutative 57.5%

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

   12. cosh-undef 57.5%

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

5. Applied egg-rr 57.5%

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

6. Taylor expanded in x around 0 49.8%

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

   1. exp-sum 49.8%

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

   2. *-commutative 49.8%

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

   3. pow-to-exp 98.8%

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

   4. unpow2 98.8%

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

   5. associate-*r* 98.8%

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

   6. *-commutative 98.8%

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

   7. exp-prod 98.8%

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

8. Applied egg-rr 98.8%

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

9. Taylor expanded in x around inf 98.8%

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

10. Final simplification 98.8%

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

Alternative 6: 98.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x + 0.08333333333333333 \cdot {x}^{3}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (+ x (* 0.08333333333333333 (pow x 3.0)))))

C

double code(double x) {
	return x * (x + (0.08333333333333333 * pow(x, 3.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x + (0.08333333333333333d0 * (x ** 3.0d0)))
end function

Java

public static double code(double x) {
	return x * (x + (0.08333333333333333 * Math.pow(x, 3.0)));
}

Python

def code(x):
	return x * (x + (0.08333333333333333 * math.pow(x, 3.0)))

Julia

function code(x)
	return Float64(x * Float64(x + Float64(0.08333333333333333 * (x ^ 3.0))))
end

MATLAB

function tmp = code(x)
	tmp = x * (x + (0.08333333333333333 * (x ^ 3.0)));
end

Wolfram

code[x_] := N[(x * N[(x + N[(0.08333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.6%

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

   1. associate-+l- 57.6%

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

   2. sub-neg 57.6%

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

   3. sub-neg 57.6%

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

   4. distribute-neg-in 57.6%

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

   5. remove-double-neg 57.6%

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

   6. +-commutative 57.6%

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

   7. metadata-eval 57.6%

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

3. Simplified 57.6%

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

   1. +-commutative 57.6%

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

   2. associate-+r+ 57.6%

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

   3. metadata-eval 57.6%

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

   4. sub-neg 57.6%

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

   5. add-exp-log 57.5%

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

   6. sub-neg 57.5%

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

   7. metadata-eval 57.5%

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

   8. associate-+r+ 57.5%

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

   9. +-commutative 57.5%

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

   10. associate-+r+ 57.5%

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

   11. +-commutative 57.5%

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

   12. cosh-undef 57.5%

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

5. Applied egg-rr 57.5%

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

6. Taylor expanded in x around 0 49.8%

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

   1. exp-sum 49.8%

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

   2. *-commutative 49.8%

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

   3. pow-to-exp 98.8%

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

   4. unpow2 98.8%

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

   5. associate-*r* 98.8%

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

   6. *-commutative 98.8%

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

   7. exp-prod 98.8%

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

8. Applied egg-rr 98.8%

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

9. Taylor expanded in x around 0 98.8%

   \[\leadsto \color{blue}{\left(x + 0.08333333333333333 \cdot {x}^{3}\right)} \cdot x \]

10. Final simplification 98.8%

    \[\leadsto x \cdot \left(x + 0.08333333333333333 \cdot {x}^{3}\right) \]

Alternative 7: 98.4% accurate, 68.7× 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 57.6%

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

   1. associate-+l- 57.6%

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

   2. sub-neg 57.6%

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

   3. sub-neg 57.6%

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

   4. distribute-neg-in 57.6%

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

   5. remove-double-neg 57.6%

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

   6. +-commutative 57.6%

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

   7. metadata-eval 57.6%

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

3. Simplified 57.6%

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

   1. +-commutative 57.6%

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

   2. associate-+r+ 57.6%

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

   3. metadata-eval 57.6%

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

   4. sub-neg 57.6%

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

   5. add-exp-log 57.5%

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

   6. sub-neg 57.5%

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

   7. metadata-eval 57.5%

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

   8. associate-+r+ 57.5%

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

   9. +-commutative 57.5%

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

   10. associate-+r+ 57.5%

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

   11. +-commutative 57.5%

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

   12. cosh-undef 57.5%

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

5. Applied egg-rr 57.5%

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

6. Taylor expanded in x around 0 49.8%

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

   1. exp-sum 49.8%

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

   2. *-commutative 49.8%

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

   3. pow-to-exp 98.8%

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

   4. unpow2 98.8%

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

   5. associate-*r* 98.8%

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

   6. *-commutative 98.8%

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

   7. exp-prod 98.8%

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

8. Applied egg-rr 98.8%

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

9. Taylor expanded in x around 0 98.0%

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

10. Final simplification 98.0%

    \[\leadsto x \cdot x \]

Alternative 8: 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 57.6%

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

   1. associate-+l- 57.6%

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

   2. sub-neg 57.6%

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

   3. sub-neg 57.6%

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

   4. distribute-neg-in 57.6%

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

   5. remove-double-neg 57.6%

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

   6. +-commutative 57.6%

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

   7. metadata-eval 57.6%

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

3. Simplified 57.6%

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

4. Taylor expanded in x around 0 55.3%

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

5. Taylor expanded in x around 0 6.1%

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

6. Final simplification 6.1%

   \[\leadsto x \]

Developer target: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 4 \cdot {\sinh \left(\frac{x}{2}\right)}^{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 4.0 (pow (sinh (/ x 2.0)) 2.0)))

C

double code(double x) {
	return 4.0 * pow(sinh((x / 2.0)), 2.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 4.0d0 * (sinh((x / 2.0d0)) ** 2.0d0)
end function

Java

public static double code(double x) {
	return 4.0 * Math.pow(Math.sinh((x / 2.0)), 2.0);
}

Python

def code(x):
	return 4.0 * math.pow(math.sinh((x / 2.0)), 2.0)

Julia

function code(x)
	return Float64(4.0 * (sinh(Float64(x / 2.0)) ^ 2.0))
end

MATLAB

function tmp = code(x)
	tmp = 4.0 * (sinh((x / 2.0)) ^ 2.0);
end

Wolfram

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

Reproduce

herbie shell --seed 2023339 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64
  :pre (<= (fabs x) 710.0)

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

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