exp2 (problem 3.3.7)

Percentage Accurate: 54.1% → 99.1%

Time: 10.2s

Alternatives: 5

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: 54.1% 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: 99.1% 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 55.4%

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

3. Taylor expanded in x around 0 99.0%

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

   1. *-commutative 99.0%

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

5. Simplified 99.0%

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

6. Final simplification 99.0%

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

Alternative 2: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {x}^{2} + 0.08333333333333333 \cdot {x}^{4} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.4%

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

3. Taylor expanded in x around 0 99.0%

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

   1. distribute-rgt-in 99.0%

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

   2. *-lft-identity 99.0%

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

   3. associate-*l* 99.0%

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

   4. pow-sqr 99.0%

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

   5. metadata-eval 99.0%

      \[\leadsto {x}^{2} + 0.08333333333333333 \cdot {x}^{\color{blue}{4}} \]

5. Simplified 99.0%

   \[\leadsto \color{blue}{{x}^{2} + 0.08333333333333333 \cdot {x}^{4}} \]

6. Final simplification 99.0%

   \[\leadsto {x}^{2} + 0.08333333333333333 \cdot {x}^{4} \]
7. Add Preprocessing

Alternative 3: 98.9% 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 55.4%

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

3. Taylor expanded in x around 0 99.0%

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

   1. distribute-rgt-in 99.0%

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

   2. *-lft-identity 99.0%

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

   3. associate-*l* 99.0%

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

   4. pow-sqr 99.0%

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

   5. metadata-eval 99.0%

      \[\leadsto {x}^{2} + 0.08333333333333333 \cdot {x}^{\color{blue}{4}} \]

5. Simplified 99.0%

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

   1. unpow2 99.0%

      \[\leadsto \color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4} \]

   2. fma-define 99.0%

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

7. Applied egg-rr 99.0%

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

8. Final simplification 99.0%

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

Alternative 4: 98.4% 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 55.4%

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

3. Taylor expanded in x around 0 98.8%

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

4. Final simplification 98.8%

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

Alternative 5: 51.8% accurate, 206.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 55.4%

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

   1. +-commutative 55.4%

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

   2. associate-+r- 55.3%

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

   3. add-sqr-sqrt 30.1%

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

   4. sqrt-unprod 54.2%

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

   5. sqr-neg 54.2%

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

   6. sqrt-unprod 24.1%

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

   7. add-sqr-sqrt 54.2%

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

   8. add-sqr-sqrt 24.1%

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

   9. sqrt-unprod 54.2%

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

   10. sqr-neg 54.2%

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

   11. sqrt-unprod 30.1%

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

   12. add-sqr-sqrt 55.3%

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

   13. cosh-undef 55.3%

       \[\leadsto \color{blue}{2 \cdot \cosh x} - 2 \]

4. Applied egg-rr 55.3%

   \[\leadsto \color{blue}{2 \cdot \cosh x - 2} \]
5. Step-by-step derivation

   1. add-sqr-sqrt 5.7%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \cosh x} \cdot \sqrt{2 \cdot \cosh x}} - 2 \]

   2. fma-neg 5.7%

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

   3. metadata-eval 5.7%

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

6. Applied egg-rr 5.7%

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

7. Taylor expanded in x around 0 4.4%

   \[\leadsto \color{blue}{{\left(\sqrt{2}\right)}^{2} - 2} \]
8. Step-by-step derivation

   1. unpow2 4.4%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{2}} - 2 \]

   2. rem-square-sqrt 53.9%

      \[\leadsto \color{blue}{2} - 2 \]

   3. metadata-eval 53.9%

      \[\leadsto \color{blue}{0} \]

9. Simplified 53.9%

   \[\leadsto \color{blue}{0} \]

10. Final simplification 53.9%

    \[\leadsto 0 \]
11. 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 2024067 
(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))))