exp2 (problem 3.3.7)

Percentage Accurate: 76.8% → 99.9%

Time: 4.3s

Alternatives: 8

Speedup: 1.9×

• 50.1% 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: 76.8% 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: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{x} - 2\right) + e^{-x}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x)))))
   (if (<= t_0 5e-9) (fma 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 <= 5e-9) {
		tmp = fma(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 <= 5e-9)
		tmp = fma(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, 5e-9], N[(x * x + 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))) < 5.0000000000000001e-9

   1. Initial program 47.8%

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

   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%

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

      2. fma-def 100.0%

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

   4. Simplified 100.0%

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

   if 5.0000000000000001e-9 < (+.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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x} - 2\right) + e^{-x}\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{x} - 2\right) + e^{-x}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;0.08333333333333333 \cdot {x}^{4} + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x)))))
   (if (<= t_0 5e-9) (+ (* 0.08333333333333333 (pow x 4.0)) (* x x)) t_0)))

C

double code(double x) {
	double t_0 = (exp(x) - 2.0) + exp(-x);
	double tmp;
	if (t_0 <= 5e-9) {
		tmp = (0.08333333333333333 * pow(x, 4.0)) + (x * x);
	} 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 <= 5d-9) then
        tmp = (0.08333333333333333d0 * (x ** 4.0d0)) + (x * x)
    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 <= 5e-9) {
		tmp = (0.08333333333333333 * Math.pow(x, 4.0)) + (x * x);
	} 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 <= 5e-9:
		tmp = (0.08333333333333333 * math.pow(x, 4.0)) + (x * x)
	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 <= 5e-9)
		tmp = Float64(Float64(0.08333333333333333 * (x ^ 4.0)) + Float64(x * x));
	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 <= 5e-9)
		tmp = (0.08333333333333333 * (x ^ 4.0)) + (x * x);
	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, 5e-9], N[(N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x * x), $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))) < 5.0000000000000001e-9

   1. Initial program 47.8%

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

   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 5.0000000000000001e-9 < (+.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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;0.08333333333333333 \cdot {x}^{4} + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x} - 2\right) + e^{-x}\\ \end{array} \]

Alternative 3: 93.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5000000000000:\\ \;\;\;\;0.08333333333333333 \cdot {x}^{4} + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;{\left(3.3489797668038406 \cdot 10^{-7} \cdot {x}^{24}\right)}^{0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5000000000000.0)
   (+ (* 0.08333333333333333 (pow x 4.0)) (* x x))
   (pow (* 3.3489797668038406e-7 (pow x 24.0)) 0.16666666666666666)))

C

double code(double x) {
	double tmp;
	if (x <= 5000000000000.0) {
		tmp = (0.08333333333333333 * pow(x, 4.0)) + (x * x);
	} else {
		tmp = pow((3.3489797668038406e-7 * pow(x, 24.0)), 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5000000000000.0d0) then
        tmp = (0.08333333333333333d0 * (x ** 4.0d0)) + (x * x)
    else
        tmp = (3.3489797668038406d-7 * (x ** 24.0d0)) ** 0.16666666666666666d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5000000000000.0) {
		tmp = (0.08333333333333333 * Math.pow(x, 4.0)) + (x * x);
	} else {
		tmp = Math.pow((3.3489797668038406e-7 * Math.pow(x, 24.0)), 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5000000000000.0:
		tmp = (0.08333333333333333 * math.pow(x, 4.0)) + (x * x)
	else:
		tmp = math.pow((3.3489797668038406e-7 * math.pow(x, 24.0)), 0.16666666666666666)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5000000000000.0)
		tmp = Float64(Float64(0.08333333333333333 * (x ^ 4.0)) + Float64(x * x));
	else
		tmp = Float64(3.3489797668038406e-7 * (x ^ 24.0)) ^ 0.16666666666666666;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5000000000000.0)
		tmp = (0.08333333333333333 * (x ^ 4.0)) + (x * x);
	else
		tmp = (3.3489797668038406e-7 * (x ^ 24.0)) ^ 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5000000000000.0], N[(N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[Power[N[(3.3489797668038406e-7 * N[Power[x, 24.0], $MachinePrecision]), $MachinePrecision], 0.16666666666666666], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e12

   1. Initial program 66.0%

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

   2. Taylor expanded in x around 0 91.7%

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

      1. unpow2 91.7%

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

   4. Simplified 91.7%

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

   if 5e12 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 79.9%

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

      1. unpow2 79.9%

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

   4. Simplified 79.9%

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

   5. Taylor expanded in x around inf 79.9%

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

      1. add-cbrt-cube 95.3%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(0.08333333333333333 \cdot {x}^{4}\right) \cdot \left(0.08333333333333333 \cdot {x}^{4}\right)\right) \cdot \left(0.08333333333333333 \cdot {x}^{4}\right)}} \]

      2. pow3 95.3%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(0.08333333333333333 \cdot {x}^{4}\right)}^{3}}} \]

      3. *-commutative 95.3%

         \[\leadsto \sqrt[3]{{\color{blue}{\left({x}^{4} \cdot 0.08333333333333333\right)}}^{3}} \]

      4. unpow-prod-down 95.3%

         \[\leadsto \sqrt[3]{\color{blue}{{\left({x}^{4}\right)}^{3} \cdot {0.08333333333333333}^{3}}} \]

      5. pow-pow 95.3%

         \[\leadsto \sqrt[3]{\color{blue}{{x}^{\left(4 \cdot 3\right)}} \cdot {0.08333333333333333}^{3}} \]

      6. metadata-eval 95.3%

         \[\leadsto \sqrt[3]{{x}^{\color{blue}{12}} \cdot {0.08333333333333333}^{3}} \]

      7. metadata-eval 95.3%

         \[\leadsto \sqrt[3]{{x}^{12} \cdot \color{blue}{0.0005787037037037037}} \]

   7. Applied egg-rr 95.3%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{12} \cdot 0.0005787037037037037}} \]
   8. Step-by-step derivation

      1. pow1/3 95.3%

         \[\leadsto \color{blue}{{\left({x}^{12} \cdot 0.0005787037037037037\right)}^{0.3333333333333333}} \]

      2. sqr-pow 95.3%

         \[\leadsto \color{blue}{{\left({x}^{12} \cdot 0.0005787037037037037\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \cdot {\left({x}^{12} \cdot 0.0005787037037037037\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \]

      3. pow-prod-down 100.0%

         \[\leadsto \color{blue}{{\left(\left({x}^{12} \cdot 0.0005787037037037037\right) \cdot \left({x}^{12} \cdot 0.0005787037037037037\right)\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \]

      4. *-commutative 100.0%

         \[\leadsto {\left(\color{blue}{\left(0.0005787037037037037 \cdot {x}^{12}\right)} \cdot \left({x}^{12} \cdot 0.0005787037037037037\right)\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto {\left(\left(0.0005787037037037037 \cdot {x}^{12}\right) \cdot \color{blue}{\left(0.0005787037037037037 \cdot {x}^{12}\right)}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

      6. swap-sqr 100.0%

         \[\leadsto {\color{blue}{\left(\left(0.0005787037037037037 \cdot 0.0005787037037037037\right) \cdot \left({x}^{12} \cdot {x}^{12}\right)\right)}}^{\left(\frac{0.3333333333333333}{2}\right)} \]

      7. metadata-eval 100.0%

         \[\leadsto {\left(\color{blue}{3.3489797668038406 \cdot 10^{-7}} \cdot \left({x}^{12} \cdot {x}^{12}\right)\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

      8. pow-prod-up 100.0%

         \[\leadsto {\left(3.3489797668038406 \cdot 10^{-7} \cdot \color{blue}{{x}^{\left(12 + 12\right)}}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

      9. metadata-eval 100.0%

         \[\leadsto {\left(3.3489797668038406 \cdot 10^{-7} \cdot {x}^{\color{blue}{24}}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

      10. metadata-eval 100.0%

          \[\leadsto {\left(3.3489797668038406 \cdot 10^{-7} \cdot {x}^{24}\right)}^{\color{blue}{0.16666666666666666}} \]

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(3.3489797668038406 \cdot 10^{-7} \cdot {x}^{24}\right)}^{0.16666666666666666}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5000000000000:\\ \;\;\;\;0.08333333333333333 \cdot {x}^{4} + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;{\left(3.3489797668038406 \cdot 10^{-7} \cdot {x}^{24}\right)}^{0.16666666666666666}\\ \end{array} \]

Alternative 4: 92.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 20000000000000:\\ \;\;\;\;0.08333333333333333 \cdot {x}^{4} + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{x}^{12} \cdot 0.0005787037037037037}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 20000000000000.0)
   (+ (* 0.08333333333333333 (pow x 4.0)) (* x x))
   (cbrt (* (pow x 12.0) 0.0005787037037037037))))

C

double code(double x) {
	double tmp;
	if (x <= 20000000000000.0) {
		tmp = (0.08333333333333333 * pow(x, 4.0)) + (x * x);
	} else {
		tmp = cbrt((pow(x, 12.0) * 0.0005787037037037037));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 20000000000000.0) {
		tmp = (0.08333333333333333 * Math.pow(x, 4.0)) + (x * x);
	} else {
		tmp = Math.cbrt((Math.pow(x, 12.0) * 0.0005787037037037037));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 20000000000000.0)
		tmp = Float64(Float64(0.08333333333333333 * (x ^ 4.0)) + Float64(x * x));
	else
		tmp = cbrt(Float64((x ^ 12.0) * 0.0005787037037037037));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 20000000000000.0], N[(N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[x, 12.0], $MachinePrecision] * 0.0005787037037037037), $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e13

   1. Initial program 66.0%

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

   2. Taylor expanded in x around 0 91.7%

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

      1. unpow2 91.7%

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

   4. Simplified 91.7%

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

   if 2e13 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 79.9%

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

      1. unpow2 79.9%

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

   4. Simplified 79.9%

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

   5. Taylor expanded in x around inf 79.9%

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

      1. add-cbrt-cube 95.3%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(0.08333333333333333 \cdot {x}^{4}\right) \cdot \left(0.08333333333333333 \cdot {x}^{4}\right)\right) \cdot \left(0.08333333333333333 \cdot {x}^{4}\right)}} \]

      2. pow3 95.3%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(0.08333333333333333 \cdot {x}^{4}\right)}^{3}}} \]

      3. *-commutative 95.3%

         \[\leadsto \sqrt[3]{{\color{blue}{\left({x}^{4} \cdot 0.08333333333333333\right)}}^{3}} \]

      4. unpow-prod-down 95.3%

         \[\leadsto \sqrt[3]{\color{blue}{{\left({x}^{4}\right)}^{3} \cdot {0.08333333333333333}^{3}}} \]

      5. pow-pow 95.3%

         \[\leadsto \sqrt[3]{\color{blue}{{x}^{\left(4 \cdot 3\right)}} \cdot {0.08333333333333333}^{3}} \]

      6. metadata-eval 95.3%

         \[\leadsto \sqrt[3]{{x}^{\color{blue}{12}} \cdot {0.08333333333333333}^{3}} \]

      7. metadata-eval 95.3%

         \[\leadsto \sqrt[3]{{x}^{12} \cdot \color{blue}{0.0005787037037037037}} \]

   7. Applied egg-rr 95.3%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{12} \cdot 0.0005787037037037037}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 20000000000000:\\ \;\;\;\;0.08333333333333333 \cdot {x}^{4} + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{x}^{12} \cdot 0.0005787037037037037}\\ \end{array} \]

Alternative 5: 88.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

2. Taylor expanded in x around 0 88.9%

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

   1. unpow2 88.9%

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

4. Simplified 88.9%

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

5. Final simplification 88.9%

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

Alternative 6: 82.1% 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 65.6%

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

   2. Taylor expanded in x around 0 82.6%

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

      1. unpow2 82.6%

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

   4. Simplified 82.6%

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

   if 3.5 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 77.5%

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

      1. unpow2 77.5%

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

   4. Simplified 77.5%

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

   5. Taylor expanded in x around inf 77.5%

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

4. Final simplification 81.3%

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

Alternative 7: 76.2% 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 74.1%

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

2. Taylor expanded in x around 0 74.7%

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

   1. unpow2 74.7%

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

4. Simplified 74.7%

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

5. Final simplification 74.7%

   \[\leadsto x \cdot x \]

Alternative 8: 26.5% 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 74.1%

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

3. Applied egg-rr 24.1%

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

4. Final simplification 24.1%

   \[\leadsto 0 \]

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 2023201 
(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))))