exp2 (problem 3.3.7)

Percentage Accurate: 53.8% → 98.9%

Time: 11.3s

Alternatives: 3

Speedup: 34.8×

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: 53.8% 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: 98.9% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)\right)} \]

   4. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\frac{1}{12} \cdot {x}^{2} + 1\right)}\right) \]

   5. distribute-lft-in N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{12} \cdot {x}^{2}\right) + x \cdot 1\right)} \]

   6. *-rgt-identity N/A

      \[\leadsto x \cdot \left(x \cdot \left(\frac{1}{12} \cdot {x}^{2}\right) + \color{blue}{x}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{12} \cdot {x}^{2}, x\right)} \]

   8. *-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{1}{12}}, x\right) \]

   9. lower-*.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{1}{12}}, x\right) \]

   10. unpow2 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{12}, x\right) \]

   11. lower-*.f64 99.6

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(x, \color{blue}{x}, x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot 0.08333333333333333\right)\right)\right) \]
8. Add Preprocessing

Alternative 2: 98.9% accurate, 9.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)\right)} \]

   4. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\frac{1}{12} \cdot {x}^{2} + 1\right)}\right) \]

   5. distribute-lft-in N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{12} \cdot {x}^{2}\right) + x \cdot 1\right)} \]

   6. *-rgt-identity N/A

      \[\leadsto x \cdot \left(x \cdot \left(\frac{1}{12} \cdot {x}^{2}\right) + \color{blue}{x}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{12} \cdot {x}^{2}, x\right)} \]

   8. *-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{1}{12}}, x\right) \]

   9. lower-*.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{1}{12}}, x\right) \]

   10. unpow2 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{12}, x\right) \]

   11. lower-*.f64 99.6

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

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.08333333333333333, x\right)} \]
6. Add Preprocessing

Alternative 3: 98.4% accurate, 34.8× 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 55.4%

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

3. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 99.2

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

5. Applied rewrites 99.2%

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

Developer Target 1: 99.9% accurate, 0.9× 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 2024233 
(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))))