exp2 (problem 3.3.7)

Percentage Accurate: 76.9% → 100.0%

Time: 6.1s

Alternatives: 7

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: 76.9% 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: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{x} - 2\right) + e^{-x}\\ \mathbf{if}\;t_0 \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(0.002777777777777778, {x}^{6}, x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x)))))
   (if (<= t_0 1e-5)
     (fma
      0.002777777777777778
      (pow x 6.0)
      (+ (* 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 <= 1e-5) {
		tmp = fma(0.002777777777777778, pow(x, 6.0), ((x * x) + (0.08333333333333333 * 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 <= 1e-5)
		tmp = fma(0.002777777777777778, (x ^ 6.0), Float64(Float64(x * x) + Float64(0.08333333333333333 * (x ^ 4.0))));
	else
		tmp = t_0;
	end
	return 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, 1e-5], N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision] + N[(N[(x * x), $MachinePrecision] + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $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))) < 1.00000000000000008e-5

   1. Initial program 51.9%

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

      1. +-commutative 51.9%

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

      2. associate-+r- 51.7%

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

      3. +-commutative 51.7%

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

      4. associate-+r- 52.1%

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

      5. +-commutative 52.1%

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

      6. associate-+l- 51.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{0.002777777777777778 \cdot {x}^{6} + \left({x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)} \]
   5. Step-by-step derivation

      1. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, {x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)} \]

      2. unpow2 100.0%

         \[\leadsto \mathsf{fma}\left(0.002777777777777778, {x}^{6}, \color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4}\right) \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)} \]

   if 1.00000000000000008e-5 < (+.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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(0.002777777777777778, {x}^{6}, x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x} - 2\right) + e^{-x}\\ \end{array} \]

Alternative 2: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{x} - 2\right) + e^{-x}\\ \mathbf{if}\;t_0 \leq 10^{-5}:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x)))))
   (if (<= t_0 1e-5) (+ (* 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 <= 1e-5) {
		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 <= 1d-5) 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 <= 1e-5) {
		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 <= 1e-5:
		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 <= 1e-5)
		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 <= 1e-5)
		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, 1e-5], 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))) < 1.00000000000000008e-5

   1. Initial program 51.9%

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

      1. +-commutative 51.9%

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

      2. associate-+r- 51.7%

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

      3. +-commutative 51.7%

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

      4. associate-+r- 52.1%

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

      5. +-commutative 52.1%

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

      6. associate-+l- 51.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. unpow2 99.9%

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

   6. Simplified 99.9%

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

   if 1.00000000000000008e-5 < (+.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 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 10^{-5}:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x} - 2\right) + e^{-x}\\ \end{array} \]

Alternative 3: 92.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.4:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{12} \cdot 7.71604938271605 \cdot 10^{-6}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 6.4)
   (+ (* x x) (* 0.08333333333333333 (pow x 4.0)))
   (sqrt (* (pow x 12.0) 7.71604938271605e-6))))

C

double code(double x) {
	double tmp;
	if (x <= 6.4) {
		tmp = (x * x) + (0.08333333333333333 * pow(x, 4.0));
	} else {
		tmp = sqrt((pow(x, 12.0) * 7.71604938271605e-6));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 6.4d0) then
        tmp = (x * x) + (0.08333333333333333d0 * (x ** 4.0d0))
    else
        tmp = sqrt(((x ** 12.0d0) * 7.71604938271605d-6))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 6.4) {
		tmp = (x * x) + (0.08333333333333333 * Math.pow(x, 4.0));
	} else {
		tmp = Math.sqrt((Math.pow(x, 12.0) * 7.71604938271605e-6));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 6.4:
		tmp = (x * x) + (0.08333333333333333 * math.pow(x, 4.0))
	else:
		tmp = math.sqrt((math.pow(x, 12.0) * 7.71604938271605e-6))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 6.4)
		tmp = Float64(Float64(x * x) + Float64(0.08333333333333333 * (x ^ 4.0)));
	else
		tmp = sqrt(Float64((x ^ 12.0) * 7.71604938271605e-6));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 6.4)
		tmp = (x * x) + (0.08333333333333333 * (x ^ 4.0));
	else
		tmp = sqrt(((x ^ 12.0) * 7.71604938271605e-6));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 6.4], N[(N[(x * x), $MachinePrecision] + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[x, 12.0], $MachinePrecision] * 7.71604938271605e-6), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.4000000000000004

   1. Initial program 68.0%

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

      1. +-commutative 68.0%

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

      2. associate-+r- 67.9%

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

      3. +-commutative 67.9%

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

      4. associate-+r- 68.2%

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

      5. +-commutative 68.2%

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

      6. associate-+l- 68.0%

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

   3. Simplified 68.0%

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

   4. Taylor expanded in x around 0 90.8%

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

      1. unpow2 90.8%

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

   6. Simplified 90.8%

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

   if 6.4000000000000004 < 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. +-commutative 100.0%

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

      4. associate-+r- 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 84.7%

      \[\leadsto \color{blue}{0.002777777777777778 \cdot {x}^{6} + \left({x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)} \]
   5. Step-by-step derivation

      1. fma-def 84.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, {x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)} \]

      2. unpow2 84.7%

         \[\leadsto \mathsf{fma}\left(0.002777777777777778, {x}^{6}, \color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4}\right) \]

   6. Simplified 84.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)} \]

   7. Taylor expanded in x around inf 84.7%

      \[\leadsto \color{blue}{0.002777777777777778 \cdot {x}^{6}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 84.7%

         \[\leadsto \color{blue}{\sqrt{0.002777777777777778 \cdot {x}^{6}} \cdot \sqrt{0.002777777777777778 \cdot {x}^{6}}} \]

      2. sqrt-unprod 95.4%

         \[\leadsto \color{blue}{\sqrt{\left(0.002777777777777778 \cdot {x}^{6}\right) \cdot \left(0.002777777777777778 \cdot {x}^{6}\right)}} \]

      3. *-commutative 95.4%

         \[\leadsto \sqrt{\color{blue}{\left({x}^{6} \cdot 0.002777777777777778\right)} \cdot \left(0.002777777777777778 \cdot {x}^{6}\right)} \]

      4. *-commutative 95.4%

         \[\leadsto \sqrt{\left({x}^{6} \cdot 0.002777777777777778\right) \cdot \color{blue}{\left({x}^{6} \cdot 0.002777777777777778\right)}} \]

      5. swap-sqr 95.4%

         \[\leadsto \sqrt{\color{blue}{\left({x}^{6} \cdot {x}^{6}\right) \cdot \left(0.002777777777777778 \cdot 0.002777777777777778\right)}} \]

      6. pow-prod-up 95.4%

         \[\leadsto \sqrt{\color{blue}{{x}^{\left(6 + 6\right)}} \cdot \left(0.002777777777777778 \cdot 0.002777777777777778\right)} \]

      7. metadata-eval 95.4%

         \[\leadsto \sqrt{{x}^{\color{blue}{12}} \cdot \left(0.002777777777777778 \cdot 0.002777777777777778\right)} \]

      8. metadata-eval 95.4%

         \[\leadsto \sqrt{{x}^{12} \cdot \color{blue}{7.71604938271605 \cdot 10^{-6}}} \]

   9. Applied egg-rr 95.4%

      \[\leadsto \color{blue}{\sqrt{{x}^{12} \cdot 7.71604938271605 \cdot 10^{-6}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.4:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{12} \cdot 7.71604938271605 \cdot 10^{-6}}\\ \end{array} \]

Alternative 4: 90.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.4:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;0.002777777777777778 \cdot {x}^{6}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 6.4)
   (+ (* x x) (* 0.08333333333333333 (pow x 4.0)))
   (* 0.002777777777777778 (pow x 6.0))))

C

double code(double x) {
	double tmp;
	if (x <= 6.4) {
		tmp = (x * x) + (0.08333333333333333 * pow(x, 4.0));
	} else {
		tmp = 0.002777777777777778 * pow(x, 6.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 6.4d0) then
        tmp = (x * x) + (0.08333333333333333d0 * (x ** 4.0d0))
    else
        tmp = 0.002777777777777778d0 * (x ** 6.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 6.4) {
		tmp = (x * x) + (0.08333333333333333 * Math.pow(x, 4.0));
	} else {
		tmp = 0.002777777777777778 * Math.pow(x, 6.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 6.4:
		tmp = (x * x) + (0.08333333333333333 * math.pow(x, 4.0))
	else:
		tmp = 0.002777777777777778 * math.pow(x, 6.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 6.4)
		tmp = Float64(Float64(x * x) + Float64(0.08333333333333333 * (x ^ 4.0)));
	else
		tmp = Float64(0.002777777777777778 * (x ^ 6.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 6.4)
		tmp = (x * x) + (0.08333333333333333 * (x ^ 4.0));
	else
		tmp = 0.002777777777777778 * (x ^ 6.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 6.4], N[(N[(x * x), $MachinePrecision] + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.4000000000000004

   1. Initial program 68.0%

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

      1. +-commutative 68.0%

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

      2. associate-+r- 67.9%

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

      3. +-commutative 67.9%

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

      4. associate-+r- 68.2%

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

      5. +-commutative 68.2%

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

      6. associate-+l- 68.0%

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

   3. Simplified 68.0%

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

   4. Taylor expanded in x around 0 90.8%

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

      1. unpow2 90.8%

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

   6. Simplified 90.8%

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

   if 6.4000000000000004 < 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. +-commutative 100.0%

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

      4. associate-+r- 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 84.7%

      \[\leadsto \color{blue}{0.002777777777777778 \cdot {x}^{6} + \left({x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)} \]
   5. Step-by-step derivation

      1. fma-def 84.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, {x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)} \]

      2. unpow2 84.7%

         \[\leadsto \mathsf{fma}\left(0.002777777777777778, {x}^{6}, \color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4}\right) \]

   6. Simplified 84.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)} \]

   7. Taylor expanded in x around inf 84.7%

      \[\leadsto \color{blue}{0.002777777777777778 \cdot {x}^{6}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.4:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;0.002777777777777778 \cdot {x}^{6}\\ \end{array} \]

Alternative 5: 84.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.2:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.002777777777777778 \cdot {x}^{6}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.2) (* x x) (* 0.002777777777777778 (pow x 6.0))))

C

double code(double x) {
	double tmp;
	if (x <= 4.2) {
		tmp = x * x;
	} else {
		tmp = 0.002777777777777778 * pow(x, 6.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4.2d0) then
        tmp = x * x
    else
        tmp = 0.002777777777777778d0 * (x ** 6.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.2) {
		tmp = x * x;
	} else {
		tmp = 0.002777777777777778 * Math.pow(x, 6.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.2:
		tmp = x * x
	else:
		tmp = 0.002777777777777778 * math.pow(x, 6.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.2)
		tmp = Float64(x * x);
	else
		tmp = Float64(0.002777777777777778 * (x ^ 6.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.2)
		tmp = x * x;
	else
		tmp = 0.002777777777777778 * (x ^ 6.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.2], N[(x * x), $MachinePrecision], N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.20000000000000018

   1. Initial program 68.0%

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

      1. +-commutative 68.0%

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

      2. associate-+r- 67.9%

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

      3. +-commutative 67.9%

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

      4. associate-+r- 68.2%

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

      5. +-commutative 68.2%

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

      6. associate-+l- 68.0%

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

   3. Simplified 68.0%

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

   4. Taylor expanded in x around 0 84.5%

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

      1. unpow2 84.5%

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

   6. Simplified 84.5%

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

   if 4.20000000000000018 < 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. +-commutative 100.0%

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

      4. associate-+r- 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 84.7%

      \[\leadsto \color{blue}{0.002777777777777778 \cdot {x}^{6} + \left({x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)} \]
   5. Step-by-step derivation

      1. fma-def 84.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, {x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)} \]

      2. unpow2 84.7%

         \[\leadsto \mathsf{fma}\left(0.002777777777777778, {x}^{6}, \color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4}\right) \]

   6. Simplified 84.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)} \]

   7. Taylor expanded in x around inf 84.7%

      \[\leadsto \color{blue}{0.002777777777777778 \cdot {x}^{6}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.2:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.002777777777777778 \cdot {x}^{6}\\ \end{array} \]

Alternative 6: 76.6% 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 75.8%

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

   1. +-commutative 75.8%

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

   2. associate-+r- 75.7%

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

   3. +-commutative 75.7%

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

   4. associate-+r- 75.9%

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

   5. +-commutative 75.9%

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

   6. associate-+l- 75.8%

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

3. Simplified 75.8%

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

4. Taylor expanded in x around 0 76.9%

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

   1. unpow2 76.9%

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

6. Simplified 76.9%

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

7. Final simplification 76.9%

   \[\leadsto x \cdot x \]

Alternative 7: 27.0% 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 75.8%

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

   1. +-commutative 75.8%

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

   2. associate-+r- 75.7%

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

   3. +-commutative 75.7%

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

   4. associate-+r- 75.9%

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

   5. +-commutative 75.9%

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

   6. associate-+l- 75.8%

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

3. Simplified 75.8%

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

5. Applied egg-rr 26.4%

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

6. Final simplification 26.4%

   \[\leadsto 0 \]

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 2023279 
(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))))