exp2 (problem 3.3.7)

Percentage Accurate: 53.7% → 99.3%

Time: 9.8s

Alternatives: 3

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: 53.7% 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: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ {x}^{2} + {x}^{4} \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot \left(0.002777777777777778 + \left(x \cdot x\right) \cdot 4.96031746031746 \cdot 10^{-5}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  (pow x 2.0)
  (*
   (pow x 4.0)
   (+
    0.08333333333333333
    (* (* x x) (+ 0.002777777777777778 (* (* x x) 4.96031746031746e-5)))))))

C

double code(double x) {
	return pow(x, 2.0) + (pow(x, 4.0) * (0.08333333333333333 + ((x * x) * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** 2.0d0) + ((x ** 4.0d0) * (0.08333333333333333d0 + ((x * x) * (0.002777777777777778d0 + ((x * x) * 4.96031746031746d-5)))))
end function

Java

public static double code(double x) {
	return Math.pow(x, 2.0) + (Math.pow(x, 4.0) * (0.08333333333333333 + ((x * x) * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5)))));
}

Python

def code(x):
	return math.pow(x, 2.0) + (math.pow(x, 4.0) * (0.08333333333333333 + ((x * x) * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5)))))

Julia

function code(x)
	return Float64((x ^ 2.0) + Float64((x ^ 4.0) * Float64(0.08333333333333333 + Float64(Float64(x * x) * Float64(0.002777777777777778 + Float64(Float64(x * x) * 4.96031746031746e-5))))))
end

MATLAB

function tmp = code(x)
	tmp = (x ^ 2.0) + ((x ^ 4.0) * (0.08333333333333333 + ((x * x) * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5)))));
end

Wolfram

code[x_] := N[(N[Power[x, 2.0], $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * N[(0.08333333333333333 + N[(N[(x * x), $MachinePrecision] * N[(0.002777777777777778 + N[(N[(x * x), $MachinePrecision] * 4.96031746031746e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.0%

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

3. Taylor expanded in x around 0 99.4%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + 4.96031746031746 \cdot 10^{-5} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. distribute-rgt-in 99.4%

      \[\leadsto \color{blue}{1 \cdot {x}^{2} + \left({x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + 4.96031746031746 \cdot 10^{-5} \cdot {x}^{2}\right)\right)\right) \cdot {x}^{2}} \]

   2. *-lft-identity 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-*l* 99.4%

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

   5. +-commutative 99.4%

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

   6. fma-define 99.4%

      \[\leadsto {x}^{2} + \color{blue}{\mathsf{fma}\left({x}^{2}, 0.002777777777777778 + 4.96031746031746 \cdot 10^{-5} \cdot {x}^{2}, 0.08333333333333333\right)} \cdot \left({x}^{2} \cdot {x}^{2}\right) \]

   7. +-commutative 99.4%

      \[\leadsto {x}^{2} + \mathsf{fma}\left({x}^{2}, \color{blue}{4.96031746031746 \cdot 10^{-5} \cdot {x}^{2} + 0.002777777777777778}, 0.08333333333333333\right) \cdot \left({x}^{2} \cdot {x}^{2}\right) \]

   8. *-commutative 99.4%

      \[\leadsto {x}^{2} + \mathsf{fma}\left({x}^{2}, \color{blue}{{x}^{2} \cdot 4.96031746031746 \cdot 10^{-5}} + 0.002777777777777778, 0.08333333333333333\right) \cdot \left({x}^{2} \cdot {x}^{2}\right) \]

   9. fma-define 99.4%

      \[\leadsto {x}^{2} + \mathsf{fma}\left({x}^{2}, \color{blue}{\mathsf{fma}\left({x}^{2}, 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right)}, 0.08333333333333333\right) \cdot \left({x}^{2} \cdot {x}^{2}\right) \]

   10. pow-sqr 99.4%

       \[\leadsto {x}^{2} + \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left({x}^{2}, 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right) \cdot \color{blue}{{x}^{\left(2 \cdot 2\right)}} \]

   11. metadata-eval 99.4%

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

5. Simplified 99.4%

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

6. Taylor expanded in x around 0 99.4%

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

   1. unpow2 99.4%

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

8. Applied egg-rr 99.4%

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

   1. unpow2 99.4%

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

10. Applied egg-rr 99.4%

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

11. Final simplification 99.4%

    \[\leadsto {x}^{2} + {x}^{4} \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot \left(0.002777777777777778 + \left(x \cdot x\right) \cdot 4.96031746031746 \cdot 10^{-5}\right)\right) \]
12. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.0%

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

3. Taylor expanded in x around 0 99.4%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + 4.96031746031746 \cdot 10^{-5} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. distribute-rgt-in 99.4%

      \[\leadsto \color{blue}{1 \cdot {x}^{2} + \left({x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + 4.96031746031746 \cdot 10^{-5} \cdot {x}^{2}\right)\right)\right) \cdot {x}^{2}} \]

   2. *-lft-identity 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-*l* 99.4%

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

   5. +-commutative 99.4%

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

   6. fma-define 99.4%

      \[\leadsto {x}^{2} + \color{blue}{\mathsf{fma}\left({x}^{2}, 0.002777777777777778 + 4.96031746031746 \cdot 10^{-5} \cdot {x}^{2}, 0.08333333333333333\right)} \cdot \left({x}^{2} \cdot {x}^{2}\right) \]

   7. +-commutative 99.4%

      \[\leadsto {x}^{2} + \mathsf{fma}\left({x}^{2}, \color{blue}{4.96031746031746 \cdot 10^{-5} \cdot {x}^{2} + 0.002777777777777778}, 0.08333333333333333\right) \cdot \left({x}^{2} \cdot {x}^{2}\right) \]

   8. *-commutative 99.4%

      \[\leadsto {x}^{2} + \mathsf{fma}\left({x}^{2}, \color{blue}{{x}^{2} \cdot 4.96031746031746 \cdot 10^{-5}} + 0.002777777777777778, 0.08333333333333333\right) \cdot \left({x}^{2} \cdot {x}^{2}\right) \]

   9. fma-define 99.4%

      \[\leadsto {x}^{2} + \mathsf{fma}\left({x}^{2}, \color{blue}{\mathsf{fma}\left({x}^{2}, 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right)}, 0.08333333333333333\right) \cdot \left({x}^{2} \cdot {x}^{2}\right) \]

   10. pow-sqr 99.4%

       \[\leadsto {x}^{2} + \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left({x}^{2}, 4.96031746031746 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right) \cdot \color{blue}{{x}^{\left(2 \cdot 2\right)}} \]

   11. metadata-eval 99.4%

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

5. Simplified 99.4%

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

6. Taylor expanded in x around inf 12.2%

   \[\leadsto \color{blue}{{x}^{8} \cdot \left(4.96031746031746 \cdot 10^{-5} + \left(\frac{0.08333333333333333}{{x}^{4}} + \left(0.002777777777777778 \cdot \frac{1}{{x}^{2}} + \frac{1}{{x}^{6}}\right)\right)\right)} \]

7. Taylor expanded in x around 0 99.2%

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

   1. distribute-lft-in 99.2%

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

   2. *-rgt-identity 99.2%

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

   3. unpow2 99.2%

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

   4. fma-define 99.2%

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

   5. *-commutative 99.2%

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

   6. associate-*l* 99.2%

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

   7. pow-sqr 99.2%

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

   8. metadata-eval 99.2%

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

9. Simplified 99.2%

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

10. Final simplification 99.2%

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

Alternative 3: 98.5% 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 54.0%

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

3. Taylor expanded in x around 0 98.4%

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

   1. unpow2 99.4%

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

5. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{x \cdot x} \]
6. Add Preprocessing

Developer Target 1: 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 2024172 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64
  :pre (<= (fabs x) 710.0)

  :alt
  (! :herbie-platform default (* 4 (* (sinh (/ x 2)) (sinh (/ x 2)))))

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