exp2 (problem 3.3.7)

Percentage Accurate: 76.1% → 98.6%

Time: 3.9s

Alternatives: 3

Speedup: 0.7×

• 49.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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: 76.1% 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: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 10^{-15}:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= (+ (- (exp x) 2.0) (exp (- x))) 1e-15) (pow x 2.0) (expm1 x)))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (((exp(x) - 2.0) + exp(-x)) <= 1e-15) {
		tmp = pow(x, 2.0);
	} else {
		tmp = expm1(x);
	}
	return tmp;
}

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (((Math.exp(x) - 2.0) + Math.exp(-x)) <= 1e-15) {
		tmp = Math.pow(x, 2.0);
	} else {
		tmp = Math.expm1(x);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if ((math.exp(x) - 2.0) + math.exp(-x)) <= 1e-15:
		tmp = math.pow(x, 2.0)
	else:
		tmp = math.expm1(x)
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) - 2.0) + exp(Float64(-x))) <= 1e-15)
		tmp = x ^ 2.0;
	else
		tmp = expm1(x);
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 1e-15], N[Power[x, 2.0], $MachinePrecision], N[(Exp[x] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 1.0000000000000001e-15

   1. Initial program 47.2%

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

      1. +-commutative 47.2%

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

      2. associate-+r- 47.1%

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

      3. metadata-eval 47.1%

         \[\leadsto \left(e^{-x} + e^{x}\right) - \color{blue}{\left(2 + 0\right)} \]

      4. associate--l- 47.1%

         \[\leadsto \color{blue}{\left(\left(e^{-x} + e^{x}\right) - 2\right) - 0} \]

      5. associate-+r- 47.2%

         \[\leadsto \color{blue}{\left(e^{-x} + \left(e^{x} - 2\right)\right)} - 0 \]

      6. +-commutative 47.2%

         \[\leadsto \color{blue}{\left(\left(e^{x} - 2\right) + e^{-x}\right)} - 0 \]

      7. associate-+r- 47.2%

         \[\leadsto \color{blue}{\left(e^{x} - 2\right) + \left(e^{-x} - 0\right)} \]

      8. sub-neg 47.2%

         \[\leadsto \color{blue}{\left(e^{x} + \left(-2\right)\right)} + \left(e^{-x} - 0\right) \]

      9. associate-+r+ 47.2%

         \[\leadsto \color{blue}{e^{x} + \left(\left(-2\right) + \left(e^{-x} - 0\right)\right)} \]

      10. associate--l+ 47.2%

          \[\leadsto e^{x} + \color{blue}{\left(\left(\left(-2\right) + e^{-x}\right) - 0\right)} \]

      11. +-commutative 47.2%

          \[\leadsto e^{x} + \left(\color{blue}{\left(e^{-x} + \left(-2\right)\right)} - 0\right) \]

      12. associate--l+ 47.2%

          \[\leadsto e^{x} + \color{blue}{\left(e^{-x} + \left(\left(-2\right) - 0\right)\right)} \]

      13. metadata-eval 47.2%

          \[\leadsto e^{x} + \left(e^{-x} + \left(\color{blue}{-2} - 0\right)\right) \]

      14. metadata-eval 47.2%

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

   3. Simplified 47.2%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 1.0000000000000001e-15 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r- 100.0%

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

      3. metadata-eval 100.0%

         \[\leadsto \left(e^{-x} + e^{x}\right) - \color{blue}{\left(2 + 0\right)} \]

      4. associate--l- 100.0%

         \[\leadsto \color{blue}{\left(\left(e^{-x} + e^{x}\right) - 2\right) - 0} \]

      5. associate-+r- 100.0%

         \[\leadsto \color{blue}{\left(e^{-x} + \left(e^{x} - 2\right)\right)} - 0 \]

      6. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(\left(e^{x} - 2\right) + e^{-x}\right)} - 0 \]

      7. associate-+r- 100.0%

         \[\leadsto \color{blue}{\left(e^{x} - 2\right) + \left(e^{-x} - 0\right)} \]

      8. sub-neg 100.0%

         \[\leadsto \color{blue}{\left(e^{x} + \left(-2\right)\right)} + \left(e^{-x} - 0\right) \]

      9. associate-+r+ 100.0%

         \[\leadsto \color{blue}{e^{x} + \left(\left(-2\right) + \left(e^{-x} - 0\right)\right)} \]

      10. associate--l+ 100.0%

          \[\leadsto e^{x} + \color{blue}{\left(\left(\left(-2\right) + e^{-x}\right) - 0\right)} \]

      11. +-commutative 100.0%

          \[\leadsto e^{x} + \left(\color{blue}{\left(e^{-x} + \left(-2\right)\right)} - 0\right) \]

      12. associate--l+ 100.0%

          \[\leadsto e^{x} + \color{blue}{\left(e^{-x} + \left(\left(-2\right) - 0\right)\right)} \]

      13. metadata-eval 100.0%

          \[\leadsto e^{x} + \left(e^{-x} + \left(\color{blue}{-2} - 0\right)\right) \]

      14. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 45.1%

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

   5. Taylor expanded in x around inf 45.1%

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

      1. expm1-def 45.1%

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

   7. Simplified 45.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 10^{-15}:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 2: 53.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \mathsf{expm1}\left(x\right) \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (expm1 x))

C

x = abs(x);
double code(double x) {
	return expm1(x);
}

Java

x = Math.abs(x);
public static double code(double x) {
	return Math.expm1(x);
}

Python

x = abs(x)
def code(x):
	return math.expm1(x)

Julia

x = abs(x)
function code(x)
	return expm1(x)
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[(Exp[x] - 1), $MachinePrecision]

Derivation

1. Initial program 74.4%

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

   1. +-commutative 74.4%

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

   2. associate-+r- 74.4%

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

   3. metadata-eval 74.4%

      \[\leadsto \left(e^{-x} + e^{x}\right) - \color{blue}{\left(2 + 0\right)} \]

   4. associate--l- 74.4%

      \[\leadsto \color{blue}{\left(\left(e^{-x} + e^{x}\right) - 2\right) - 0} \]

   5. associate-+r- 74.4%

      \[\leadsto \color{blue}{\left(e^{-x} + \left(e^{x} - 2\right)\right)} - 0 \]

   6. +-commutative 74.4%

      \[\leadsto \color{blue}{\left(\left(e^{x} - 2\right) + e^{-x}\right)} - 0 \]

   7. associate-+r- 74.4%

      \[\leadsto \color{blue}{\left(e^{x} - 2\right) + \left(e^{-x} - 0\right)} \]

   8. sub-neg 74.4%

      \[\leadsto \color{blue}{\left(e^{x} + \left(-2\right)\right)} + \left(e^{-x} - 0\right) \]

   9. associate-+r+ 74.4%

      \[\leadsto \color{blue}{e^{x} + \left(\left(-2\right) + \left(e^{-x} - 0\right)\right)} \]

   10. associate--l+ 74.4%

       \[\leadsto e^{x} + \color{blue}{\left(\left(\left(-2\right) + e^{-x}\right) - 0\right)} \]

   11. +-commutative 74.4%

       \[\leadsto e^{x} + \left(\color{blue}{\left(e^{-x} + \left(-2\right)\right)} - 0\right) \]

   12. associate--l+ 74.4%

       \[\leadsto e^{x} + \color{blue}{\left(e^{-x} + \left(\left(-2\right) - 0\right)\right)} \]

   13. metadata-eval 74.4%

       \[\leadsto e^{x} + \left(e^{-x} + \left(\color{blue}{-2} - 0\right)\right) \]

   14. metadata-eval 74.4%

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

3. Simplified 74.4%

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

4. Taylor expanded in x around 0 46.1%

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

5. Taylor expanded in x around inf 46.1%

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

   1. expm1-def 26.1%

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

7. Simplified 26.1%

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

8. Final simplification 26.1%

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

Alternative 3: 6.2% accurate, 206.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ x \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 x)

C

x = abs(x);
double code(double x) {
	return x;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = x
end function

Java

x = Math.abs(x);
public static double code(double x) {
	return x;
}

Python

x = abs(x)
def code(x):
	return x

Julia

x = abs(x)
function code(x)
	return x
end

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := x

Derivation

1. Initial program 74.4%

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

   1. +-commutative 74.4%

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

   2. associate-+r- 74.4%

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

   3. metadata-eval 74.4%

      \[\leadsto \left(e^{-x} + e^{x}\right) - \color{blue}{\left(2 + 0\right)} \]

   4. associate--l- 74.4%

      \[\leadsto \color{blue}{\left(\left(e^{-x} + e^{x}\right) - 2\right) - 0} \]

   5. associate-+r- 74.4%

      \[\leadsto \color{blue}{\left(e^{-x} + \left(e^{x} - 2\right)\right)} - 0 \]

   6. +-commutative 74.4%

      \[\leadsto \color{blue}{\left(\left(e^{x} - 2\right) + e^{-x}\right)} - 0 \]

   7. associate-+r- 74.4%

      \[\leadsto \color{blue}{\left(e^{x} - 2\right) + \left(e^{-x} - 0\right)} \]

   8. sub-neg 74.4%

      \[\leadsto \color{blue}{\left(e^{x} + \left(-2\right)\right)} + \left(e^{-x} - 0\right) \]

   9. associate-+r+ 74.4%

      \[\leadsto \color{blue}{e^{x} + \left(\left(-2\right) + \left(e^{-x} - 0\right)\right)} \]

   10. associate--l+ 74.4%

       \[\leadsto e^{x} + \color{blue}{\left(\left(\left(-2\right) + e^{-x}\right) - 0\right)} \]

   11. +-commutative 74.4%

       \[\leadsto e^{x} + \left(\color{blue}{\left(e^{-x} + \left(-2\right)\right)} - 0\right) \]

   12. associate--l+ 74.4%

       \[\leadsto e^{x} + \color{blue}{\left(e^{-x} + \left(\left(-2\right) - 0\right)\right)} \]

   13. metadata-eval 74.4%

       \[\leadsto e^{x} + \left(e^{-x} + \left(\color{blue}{-2} - 0\right)\right) \]

   14. metadata-eval 74.4%

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

3. Simplified 74.4%

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

4. Taylor expanded in x around 0 46.1%

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

5. Taylor expanded in x around 0 4.2%

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

6. Final simplification 4.2%

   \[\leadsto x \]

Developer target: 100.0% 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 2023305 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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