exp2 (problem 3.3.7)

Percentage Accurate: 77.2% → 99.3%

Time: 4.7s

Alternatives: 5

Speedup: 0.7×

• 49.9% 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: 77.2% 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, 0.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= (+ (- (exp x) 2.0) (exp (- x))) 2e-8)
   (fma x x (* 0.08333333333333333 (pow x 4.0)))
   (expm1 x)))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (((exp(x) - 2.0) + exp(-x)) <= 2e-8) {
		tmp = fma(x, x, (0.08333333333333333 * pow(x, 4.0)));
	} else {
		tmp = expm1(x);
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) - 2.0) + exp(Float64(-x))) <= 2e-8)
		tmp = fma(x, x, Float64(0.08333333333333333 * (x ^ 4.0)));
	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], 2e-8], N[(x * x + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $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))) < 2e-8

   1. Initial program 51.5%

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

      1. associate-+l- 51.5%

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

      2. sub-neg 51.5%

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

      3. sub-neg 51.5%

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

      4. +-commutative 51.5%

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

      5. distribute-neg-in 51.5%

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

      6. remove-double-neg 51.5%

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

      7. metadata-eval 51.5%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. fma-def 100.0%

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

   6. Simplified 100.0%

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

   if 2e-8 < (+.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. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. 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 45.8%

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

   5. Taylor expanded in x around inf 45.8%

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

      1. expm1-def 45.8%

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

   7. Simplified 45.8%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 2: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= (+ (- (exp x) 2.0) (exp (- x))) 2e-8) (* x x) (expm1 x)))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (((exp(x) - 2.0) + exp(-x)) <= 2e-8) {
		tmp = x * x;
	} 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)) <= 2e-8) {
		tmp = x * x;
	} 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)) <= 2e-8:
		tmp = x * x
	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))) <= 2e-8)
		tmp = Float64(x * x);
	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], 2e-8], N[(x * x), $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))) < 2e-8

   1. Initial program 51.5%

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

      1. associate-+l- 51.5%

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

      2. sub-neg 51.5%

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

      3. sub-neg 51.5%

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

      4. +-commutative 51.5%

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

      5. distribute-neg-in 51.5%

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

      6. remove-double-neg 51.5%

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

      7. metadata-eval 51.5%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. unpow2 99.7%

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

   6. Simplified 99.7%

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

   if 2e-8 < (+.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. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. 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 45.8%

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

   5. Taylor expanded in x around inf 45.8%

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

      1. expm1-def 45.8%

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

   7. Simplified 45.8%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 3: 76.5% accurate, 68.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

x = abs(x)
function code(x)
	return Float64(x * x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.4%

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

   1. associate-+l- 74.4%

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

   2. sub-neg 74.4%

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

   3. sub-neg 74.4%

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

   4. +-commutative 74.4%

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

   5. distribute-neg-in 74.4%

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

   6. remove-double-neg 74.4%

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

   7. 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 78.7%

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

   1. unpow2 78.7%

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

6. Simplified 78.7%

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

7. Final simplification 78.7%

   \[\leadsto x \cdot x \]

Alternative 4: 3.7% accurate, 206.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ 2 \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 2.0)

C

x = abs(x);
double code(double x) {
	return 2.0;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0
end function

Java

x = Math.abs(x);
public static double code(double x) {
	return 2.0;
}

Python

x = abs(x)
def code(x):
	return 2.0

Julia

x = abs(x)
function code(x)
	return 2.0
end

MATLAB

x = abs(x)
function tmp = code(x)
	tmp = 2.0;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := 2.0

Derivation

1. Initial program 74.4%

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

   1. associate-+l- 74.4%

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

   2. sub-neg 74.4%

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

   3. sub-neg 74.4%

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

   4. +-commutative 74.4%

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

   5. distribute-neg-in 74.4%

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

   6. remove-double-neg 74.4%

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

   7. 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. Step-by-step derivation

   1. +-commutative 74.4%

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

   2. metadata-eval 74.4%

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

   3. sub-neg 74.4%

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

   4. associate--r- 74.4%

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

   5. add-sqr-sqrt 39.8%

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

   6. sqrt-unprod 74.3%

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

   7. sqr-neg 74.3%

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

   8. sqrt-unprod 34.4%

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

   9. add-sqr-sqrt 48.7%

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

5. Applied egg-rr 48.7%

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

   1. associate--r- 48.7%

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

   2. sub-neg 48.7%

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

   3. metadata-eval 48.7%

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

   4. +-commutative 48.7%

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

   5. rem-square-sqrt 21.5%

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

   6. fabs-sqr 21.5%

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

   7. rem-square-sqrt 24.5%

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

   8. metadata-eval 24.5%

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

   9. sub-neg 24.5%

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

   10. fabs-sub 24.5%

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

   11. rem-square-sqrt 3.0%

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

   12. fabs-sqr 3.0%

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

   13. rem-square-sqrt 3.0%

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

   14. remove-double-neg 3.0%

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

   15. remove-double-neg 3.0%

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

   16. distribute-neg-out 3.0%

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

   17. +-commutative 3.0%

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

   18. neg-sub0 3.0%

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

   19. associate--r- 3.0%

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

   20. metadata-eval 3.0%

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

7. Simplified 3.7%

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

8. Final simplification 3.7%

   \[\leadsto 2 \]

Alternative 5: 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. associate-+l- 74.4%

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

   2. sub-neg 74.4%

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

   3. sub-neg 74.4%

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

   4. +-commutative 74.4%

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

   5. distribute-neg-in 74.4%

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

   6. remove-double-neg 74.4%

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

   7. 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 48.7%

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

5. Taylor expanded in x around 0 4.4%

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

6. Final simplification 4.4%

   \[\leadsto x \]

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 2023271 
(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))))