exp2 (problem 3.3.7)

Percentage Accurate: 76.7% → 99.7%

Time: 3.4s

Alternatives: 6

Speedup: 1.9×

• 49.6% 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: 76.7% 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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{x} - 2\right) + e^{-x}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x)))))
   (if (<= t_0 2e-13) (+ (* x x) (* 0.08333333333333333 (pow x 4.0))) t_0)))

C

double code(double x) {
	double t_0 = (exp(x) - 2.0) + exp(-x);
	double tmp;
	if (t_0 <= 2e-13) {
		tmp = (x * x) + (0.08333333333333333 * pow(x, 4.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (exp(x) - 2.0d0) + exp(-x)
    if (t_0 <= 2d-13) then
        tmp = (x * x) + (0.08333333333333333d0 * (x ** 4.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (Math.exp(x) - 2.0) + Math.exp(-x);
	double tmp;
	if (t_0 <= 2e-13) {
		tmp = (x * x) + (0.08333333333333333 * Math.pow(x, 4.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (math.exp(x) - 2.0) + math.exp(-x)
	tmp = 0
	if t_0 <= 2e-13:
		tmp = (x * x) + (0.08333333333333333 * math.pow(x, 4.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(exp(x) - 2.0) + exp(Float64(-x)))
	tmp = 0.0
	if (t_0 <= 2e-13)
		tmp = Float64(Float64(x * x) + Float64(0.08333333333333333 * (x ^ 4.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (exp(x) - 2.0) + exp(-x);
	tmp = 0.0;
	if (t_0 <= 2e-13)
		tmp = (x * x) + (0.08333333333333333 * (x ^ 4.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-13], N[(N[(x * x), $MachinePrecision] + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 2.0000000000000001e-13

   1. Initial program 54.5%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   if 2.0000000000000001e-13 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

      \[\left(e^{x} - 2\right) + e^{-x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x} - 2\right) + e^{-x}\\ \end{array} \]

Alternative 2: 91.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 500000:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{8} \cdot 0.006944444444444444}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 500000.0)
   (+ (* x x) (* 0.08333333333333333 (pow x 4.0)))
   (sqrt (* (pow x 8.0) 0.006944444444444444))))

C

double code(double x) {
	double tmp;
	if (x <= 500000.0) {
		tmp = (x * x) + (0.08333333333333333 * pow(x, 4.0));
	} else {
		tmp = sqrt((pow(x, 8.0) * 0.006944444444444444));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 500000.0d0) then
        tmp = (x * x) + (0.08333333333333333d0 * (x ** 4.0d0))
    else
        tmp = sqrt(((x ** 8.0d0) * 0.006944444444444444d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 500000.0) {
		tmp = (x * x) + (0.08333333333333333 * Math.pow(x, 4.0));
	} else {
		tmp = Math.sqrt((Math.pow(x, 8.0) * 0.006944444444444444));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 500000.0:
		tmp = (x * x) + (0.08333333333333333 * math.pow(x, 4.0))
	else:
		tmp = math.sqrt((math.pow(x, 8.0) * 0.006944444444444444))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 500000.0)
		tmp = Float64(Float64(x * x) + Float64(0.08333333333333333 * (x ^ 4.0)));
	else
		tmp = sqrt(Float64((x ^ 8.0) * 0.006944444444444444));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 500000.0)
		tmp = (x * x) + (0.08333333333333333 * (x ^ 4.0));
	else
		tmp = sqrt(((x ^ 8.0) * 0.006944444444444444));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 500000.0], N[(N[(x * x), $MachinePrecision] + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[x, 8.0], $MachinePrecision] * 0.006944444444444444), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e5

   1. Initial program 72.5%

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

   2. Taylor expanded in x around 0 88.6%

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

      1. unpow2 88.6%

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

   4. Simplified 88.6%

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

   if 5e5 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 83.8%

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

      1. unpow2 83.8%

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

   4. Simplified 83.8%

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

   5. Taylor expanded in x around inf 83.8%

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

      1. add-sqr-sqrt 83.8%

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

      2. sqrt-unprod 87.0%

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

      3. *-commutative 87.0%

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

      4. *-commutative 87.0%

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

      5. swap-sqr 87.0%

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

      6. pow-prod-up 87.0%

         \[\leadsto \sqrt{\color{blue}{{x}^{\left(4 + 4\right)}} \cdot \left(0.08333333333333333 \cdot 0.08333333333333333\right)} \]

      7. metadata-eval 87.0%

         \[\leadsto \sqrt{{x}^{\color{blue}{8}} \cdot \left(0.08333333333333333 \cdot 0.08333333333333333\right)} \]

      8. metadata-eval 87.0%

         \[\leadsto \sqrt{{x}^{8} \cdot \color{blue}{0.006944444444444444}} \]

   7. Applied egg-rr 87.0%

      \[\leadsto \color{blue}{\sqrt{{x}^{8} \cdot 0.006944444444444444}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 500000:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{8} \cdot 0.006944444444444444}\\ \end{array} \]

Alternative 3: 88.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

2. Taylor expanded in x around 0 87.5%

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

   1. unpow2 87.5%

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

4. Simplified 87.5%

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

5. Final simplification 87.5%

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

Alternative 4: 81.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 \cdot {x}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.5) (* x x) (* 0.08333333333333333 (pow x 4.0))))

C

double code(double x) {
	double tmp;
	if (x <= 3.5) {
		tmp = x * x;
	} else {
		tmp = 0.08333333333333333 * pow(x, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.5d0) then
        tmp = x * x
    else
        tmp = 0.08333333333333333d0 * (x ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.5) {
		tmp = x * x;
	} else {
		tmp = 0.08333333333333333 * Math.pow(x, 4.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.5:
		tmp = x * x
	else:
		tmp = 0.08333333333333333 * math.pow(x, 4.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.5)
		tmp = Float64(x * x);
	else
		tmp = Float64(0.08333333333333333 * (x ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.5)
		tmp = x * x;
	else
		tmp = 0.08333333333333333 * (x ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.5], N[(x * x), $MachinePrecision], N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5

   1. Initial program 72.1%

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

   2. Taylor expanded in x around 0 82.4%

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

      1. unpow2 82.4%

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

   4. Simplified 82.4%

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

   if 3.5 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 80.1%

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

      1. unpow2 80.1%

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

   4. Simplified 80.1%

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

   5. Taylor expanded in x around inf 80.1%

      \[\leadsto \color{blue}{0.08333333333333333 \cdot {x}^{4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 \cdot {x}^{4}\\ \end{array} \]

Alternative 5: 75.9% 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 78.8%

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

2. Taylor expanded in x around 0 76.0%

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

   1. unpow2 76.0%

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

4. Simplified 76.0%

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

5. Final simplification 76.0%

   \[\leadsto x \cdot x \]

Alternative 6: 26.7% 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 78.8%

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

3. Applied egg-rr 26.0%

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

4. Final simplification 26.0%

   \[\leadsto 0 \]

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 2023257 
(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))))