exp2 (problem 3.3.7)

Percentage Accurate: 76.8% → 100.0%

Time: 5.8s

Alternatives: 10

Speedup: 2.0×

• 50.0% 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: 100.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(0.002777777777777778, {x}^{6}, \mathsf{fma}\left(x, x, \mathsf{fma}\left(0.08333333333333333, {x}^{4}, 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (- (exp x) 2.0) (exp (- x))) 5e-5)
   (fma
    0.002777777777777778
    (pow x 6.0)
    (fma
     x
     x
     (fma
      0.08333333333333333
      (pow x 4.0)
      (* 4.96031746031746e-5 (pow x 8.0)))))
   (- (* 2.0 (cosh x)) 2.0)))

C

double code(double x) {
	double tmp;
	if (((exp(x) - 2.0) + exp(-x)) <= 5e-5) {
		tmp = fma(0.002777777777777778, pow(x, 6.0), fma(x, x, fma(0.08333333333333333, pow(x, 4.0), (4.96031746031746e-5 * pow(x, 8.0)))));
	} else {
		tmp = (2.0 * cosh(x)) - 2.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) - 2.0) + exp(Float64(-x))) <= 5e-5)
		tmp = fma(0.002777777777777778, (x ^ 6.0), fma(x, x, fma(0.08333333333333333, (x ^ 4.0), Float64(4.96031746031746e-5 * (x ^ 8.0)))));
	else
		tmp = Float64(Float64(2.0 * cosh(x)) - 2.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 5e-5], N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision] + N[(x * x + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision] + N[(4.96031746031746e-5 * N[Power[x, 8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 5.00000000000000024e-5

   1. Initial program 47.2%

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

      1. associate-+l- 47.2%

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

      2. sub-neg 47.2%

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

      3. sub-neg 47.2%

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

      4. +-commutative 47.2%

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

      5. distribute-neg-in 47.2%

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

      6. remove-double-neg 47.2%

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

      7. metadata-eval 47.2%

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

   3. Simplified 47.2%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{0.002777777777777778 \cdot {x}^{6} + \left({x}^{2} + \left(0.08333333333333333 \cdot {x}^{4} + 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8}\right)\right)} \]
   5. Step-by-step derivation

      1. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, {x}^{2} + \left(0.08333333333333333 \cdot {x}^{4} + 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8}\right)\right)} \]

      2. unpow2 100.0%

         \[\leadsto \mathsf{fma}\left(0.002777777777777778, {x}^{6}, \color{blue}{x \cdot x} + \left(0.08333333333333333 \cdot {x}^{4} + 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8}\right)\right) \]

      3. fma-def 100.0%

         \[\leadsto \mathsf{fma}\left(0.002777777777777778, {x}^{6}, \color{blue}{\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4} + 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8}\right)}\right) \]

      4. fma-def 100.0%

         \[\leadsto \mathsf{fma}\left(0.002777777777777778, {x}^{6}, \mathsf{fma}\left(x, x, \color{blue}{\mathsf{fma}\left(0.08333333333333333, {x}^{4}, 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8}\right)}\right)\right) \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, \mathsf{fma}\left(x, x, \mathsf{fma}\left(0.08333333333333333, {x}^{4}, 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8}\right)\right)\right)} \]

   if 5.00000000000000024e-5 < (+.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. Step-by-step derivation

      1. associate-+r+ 100.0%

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

      2. cosh-undef 100.0%

         \[\leadsto \color{blue}{2 \cdot \cosh x} + -2 \]

      3. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, \cosh x, -2\right)} \]

      4. metadata-eval 100.0%

         \[\leadsto \mathsf{fma}\left(2, \cosh x, \color{blue}{-2}\right) \]

      5. fma-neg 100.0%

         \[\leadsto \color{blue}{2 \cdot \cosh x - 2} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{2 \cdot \cosh x - 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(0.002777777777777778, {x}^{6}, \mathsf{fma}\left(x, x, \mathsf{fma}\left(0.08333333333333333, {x}^{4}, 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \]

Alternative 2: 100.0% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (- (exp x) 2.0) (exp (- x))) 5e-5)
   (fma
    0.002777777777777778
    (pow x 6.0)
    (+ (* x x) (* 0.08333333333333333 (pow x 4.0))))
   (- (* 2.0 (cosh x)) 2.0)))

C

double code(double x) {
	double tmp;
	if (((exp(x) - 2.0) + exp(-x)) <= 5e-5) {
		tmp = fma(0.002777777777777778, pow(x, 6.0), ((x * x) + (0.08333333333333333 * pow(x, 4.0))));
	} else {
		tmp = (2.0 * cosh(x)) - 2.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) - 2.0) + exp(Float64(-x))) <= 5e-5)
		tmp = fma(0.002777777777777778, (x ^ 6.0), Float64(Float64(x * x) + Float64(0.08333333333333333 * (x ^ 4.0))));
	else
		tmp = Float64(Float64(2.0 * cosh(x)) - 2.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 5e-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], N[(N[(2.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 5.00000000000000024e-5

   1. Initial program 47.2%

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

      1. associate-+l- 47.2%

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

      2. sub-neg 47.2%

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

      3. sub-neg 47.2%

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

      4. +-commutative 47.2%

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

      5. distribute-neg-in 47.2%

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

      6. remove-double-neg 47.2%

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

      7. metadata-eval 47.2%

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

   3. Simplified 47.2%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{0.002777777777777778 \cdot {x}^{6} + \left({x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)} \]
   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 5.00000000000000024e-5 < (+.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. Step-by-step derivation

      1. associate-+r+ 100.0%

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

      2. cosh-undef 100.0%

         \[\leadsto \color{blue}{2 \cdot \cosh x} + -2 \]

      3. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, \cosh x, -2\right)} \]

      4. metadata-eval 100.0%

         \[\leadsto \mathsf{fma}\left(2, \cosh x, \color{blue}{-2}\right) \]

      5. fma-neg 100.0%

         \[\leadsto \color{blue}{2 \cdot \cosh x - 2} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{2 \cdot \cosh x - 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

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

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{x} - 2\right) + e^{-x}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x)))))
   (if (<= t_0 2e-11) (+ (* 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-11) {
		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-11) 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-11) {
		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-11:
		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-11)
		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-11)
		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-11], 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))) < 1.99999999999999988e-11

   1. Initial program 46.9%

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

      1. associate-+l- 46.9%

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

      2. sub-neg 46.9%

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

      3. sub-neg 46.9%

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

      4. +-commutative 46.9%

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

      5. distribute-neg-in 46.9%

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

      6. remove-double-neg 46.9%

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

      7. metadata-eval 46.9%

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

   3. Simplified 46.9%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   6. Simplified 100.0%

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

   if 1.99999999999999988e-11 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x} - 2\right) + e^{-x}\\ \end{array} \]

Alternative 4: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{x} - 2\right) + e^{-x}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x)))))
   (if (<= t_0 2e-11) (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 <= 2e-11) {
		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 <= 2e-11)
		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, 2e-11], 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))) < 1.99999999999999988e-11

   1. Initial program 46.9%

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

      1. associate-+l- 46.9%

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

      2. sub-neg 46.9%

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

      3. sub-neg 46.9%

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

      4. +-commutative 46.9%

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

      5. distribute-neg-in 46.9%

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

      6. remove-double-neg 46.9%

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

      7. metadata-eval 46.9%

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

   3. Simplified 46.9%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   6. Simplified 100.0%

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

      1. fma-def 100.0%

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

   8. Applied egg-rr 100.0%

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

   if 1.99999999999999988e-11 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x} - 2\right) + e^{-x}\\ \end{array} \]

Alternative 5: 99.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0056 \lor \neg \left(x \leq 0.0055\right):\\ \;\;\;\;2 \cdot \cosh x - 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot {x}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.0056) (not (<= x 0.0055)))
   (- (* 2.0 (cosh x)) 2.0)
   (+ (* x x) (* 0.08333333333333333 (pow x 4.0)))))

C

double code(double x) {
	double tmp;
	if ((x <= -0.0056) || !(x <= 0.0055)) {
		tmp = (2.0 * cosh(x)) - 2.0;
	} else {
		tmp = (x * x) + (0.08333333333333333 * pow(x, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.0056d0)) .or. (.not. (x <= 0.0055d0))) then
        tmp = (2.0d0 * cosh(x)) - 2.0d0
    else
        tmp = (x * x) + (0.08333333333333333d0 * (x ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.0056) || !(x <= 0.0055)) {
		tmp = (2.0 * Math.cosh(x)) - 2.0;
	} else {
		tmp = (x * x) + (0.08333333333333333 * Math.pow(x, 4.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.0056) or not (x <= 0.0055):
		tmp = (2.0 * math.cosh(x)) - 2.0
	else:
		tmp = (x * x) + (0.08333333333333333 * math.pow(x, 4.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.0056) || !(x <= 0.0055))
		tmp = Float64(Float64(2.0 * cosh(x)) - 2.0);
	else
		tmp = Float64(Float64(x * x) + Float64(0.08333333333333333 * (x ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.0056) || ~((x <= 0.0055)))
		tmp = (2.0 * cosh(x)) - 2.0;
	else
		tmp = (x * x) + (0.08333333333333333 * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.0056], N[Not[LessEqual[x, 0.0055]], $MachinePrecision]], N[(N[(2.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00559999999999999994 or 0.0054999999999999997 < x

   1. Initial program 99.9%

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

      1. associate-+l- 99.8%

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

      2. sub-neg 99.8%

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

      3. sub-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. remove-double-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. associate-+r+ 99.8%

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

      2. cosh-undef 99.8%

         \[\leadsto \color{blue}{2 \cdot \cosh x} + -2 \]

      3. fma-def 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, \cosh x, -2\right)} \]

      4. metadata-eval 99.8%

         \[\leadsto \mathsf{fma}\left(2, \cosh x, \color{blue}{-2}\right) \]

      5. fma-neg 99.8%

         \[\leadsto \color{blue}{2 \cdot \cosh x - 2} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{2 \cdot \cosh x - 2} \]

   if -0.00559999999999999994 < x < 0.0054999999999999997

   1. Initial program 46.9%

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

      1. associate-+l- 46.9%

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

      2. sub-neg 46.9%

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

      3. sub-neg 46.9%

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

      4. +-commutative 46.9%

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

      5. distribute-neg-in 46.9%

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

      6. remove-double-neg 46.9%

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

      7. metadata-eval 46.9%

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

   3. Simplified 46.9%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0056 \lor \neg \left(x \leq 0.0055\right):\\ \;\;\;\;2 \cdot \cosh x - 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot {x}^{4}\\ \end{array} \]

Alternative 6: 99.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0002 \lor \neg \left(x \leq 0.00018\right):\\ \;\;\;\;2 \cdot \cosh x - 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.0002) (not (<= x 0.00018)))
   (- (* 2.0 (cosh x)) 2.0)
   (* x x)))

C

double code(double x) {
	double tmp;
	if ((x <= -0.0002) || !(x <= 0.00018)) {
		tmp = (2.0 * cosh(x)) - 2.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.0002d0)) .or. (.not. (x <= 0.00018d0))) then
        tmp = (2.0d0 * cosh(x)) - 2.0d0
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.0002) || !(x <= 0.00018)) {
		tmp = (2.0 * Math.cosh(x)) - 2.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.0002) or not (x <= 0.00018):
		tmp = (2.0 * math.cosh(x)) - 2.0
	else:
		tmp = x * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.0002) || !(x <= 0.00018))
		tmp = Float64(Float64(2.0 * cosh(x)) - 2.0);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.0002) || ~((x <= 0.00018)))
		tmp = (2.0 * cosh(x)) - 2.0;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.0002], N[Not[LessEqual[x, 0.00018]], $MachinePrecision]], N[(N[(2.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0000000000000001e-4 or 1.80000000000000011e-4 < x

   1. Initial program 99.9%

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

      1. associate-+l- 99.8%

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

      2. sub-neg 99.8%

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

      3. sub-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. remove-double-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. associate-+r+ 99.8%

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

      2. cosh-undef 99.8%

         \[\leadsto \color{blue}{2 \cdot \cosh x} + -2 \]

      3. fma-def 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, \cosh x, -2\right)} \]

      4. metadata-eval 99.8%

         \[\leadsto \mathsf{fma}\left(2, \cosh x, \color{blue}{-2}\right) \]

      5. fma-neg 99.8%

         \[\leadsto \color{blue}{2 \cdot \cosh x - 2} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{2 \cdot \cosh x - 2} \]

   if -2.0000000000000001e-4 < x < 1.80000000000000011e-4

   1. Initial program 46.9%

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

      1. associate-+l- 46.9%

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

      2. sub-neg 46.9%

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

      3. sub-neg 46.9%

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

      4. +-commutative 46.9%

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

      5. distribute-neg-in 46.9%

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

      6. remove-double-neg 46.9%

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

      7. metadata-eval 46.9%

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

   3. Simplified 46.9%

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

   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%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0002 \lor \neg \left(x \leq 0.00018\right):\\ \;\;\;\;2 \cdot \cosh x - 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 7: 95.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5:\\ \;\;\;\;0.002777777777777778 \cdot {x}^{6}\\ \mathbf{elif}\;x \leq 1.66:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.5)
   (* 0.002777777777777778 (pow x 6.0))
   (if (<= x 1.66) (* x x) (expm1 x))))

C

double code(double x) {
	double tmp;
	if (x <= -4.5) {
		tmp = 0.002777777777777778 * pow(x, 6.0);
	} else if (x <= 1.66) {
		tmp = x * x;
	} else {
		tmp = expm1(x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -4.5) {
		tmp = 0.002777777777777778 * Math.pow(x, 6.0);
	} else if (x <= 1.66) {
		tmp = x * x;
	} else {
		tmp = Math.expm1(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.5:
		tmp = 0.002777777777777778 * math.pow(x, 6.0)
	elif x <= 1.66:
		tmp = x * x
	else:
		tmp = math.expm1(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.5)
		tmp = Float64(0.002777777777777778 * (x ^ 6.0));
	elseif (x <= 1.66)
		tmp = Float64(x * x);
	else
		tmp = expm1(x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -4.5], N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.66], N[(x * x), $MachinePrecision], N[(Exp[x] - 1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.5

   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 85.8%

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

      1. fma-def 85.8%

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

      2. unpow2 85.8%

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

   6. Simplified 85.8%

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

   7. Taylor expanded in x around inf 85.8%

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

   if -4.5 < x < 1.65999999999999992

   1. Initial program 47.2%

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

      1. associate-+l- 47.2%

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

      2. sub-neg 47.2%

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

      3. sub-neg 47.2%

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

      4. +-commutative 47.2%

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

      5. distribute-neg-in 47.2%

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

      6. remove-double-neg 47.2%

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

      7. metadata-eval 47.2%

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

   3. Simplified 47.2%

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

   4. Taylor expanded in x around 0 99.3%

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

      1. unpow2 99.3%

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

   6. Simplified 99.3%

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

   if 1.65999999999999992 < 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 98.9%

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

   5. Taylor expanded in x around inf 98.9%

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

      1. expm1-def 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5:\\ \;\;\;\;0.002777777777777778 \cdot {x}^{6}\\ \mathbf{elif}\;x \leq 1.66:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 8: 87.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.66:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.66) (* x x) (expm1 x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.66) {
		tmp = x * x;
	} else {
		tmp = expm1(x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.66) {
		tmp = x * x;
	} else {
		tmp = Math.expm1(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.66:
		tmp = x * x
	else:
		tmp = math.expm1(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.66)
		tmp = Float64(x * x);
	else
		tmp = expm1(x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.66], N[(x * x), $MachinePrecision], N[(Exp[x] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.65999999999999992

   1. Initial program 66.2%

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

      1. associate-+l- 66.2%

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

      2. sub-neg 66.2%

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

      3. sub-neg 66.2%

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

      4. +-commutative 66.2%

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

      5. distribute-neg-in 66.2%

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

      6. remove-double-neg 66.2%

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

      7. metadata-eval 66.2%

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

   3. Simplified 66.2%

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

   4. Taylor expanded in x around 0 82.2%

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

      1. unpow2 82.2%

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

   6. Simplified 82.2%

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

   if 1.65999999999999992 < 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 98.9%

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

   5. Taylor expanded in x around inf 98.9%

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

      1. expm1-def 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.66:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 9: 75.2% 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 75.5%

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

   1. associate-+l- 75.4%

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

   2. sub-neg 75.4%

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

   3. sub-neg 75.4%

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

   4. +-commutative 75.4%

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

   5. distribute-neg-in 75.4%

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

   6. remove-double-neg 75.4%

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

   7. metadata-eval 75.4%

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

3. Simplified 75.4%

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

4. Taylor expanded in x around 0 76.8%

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

   1. unpow2 76.8%

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

6. Simplified 76.8%

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

7. Final simplification 76.8%

   \[\leadsto x \cdot x \]

Alternative 10: 4.4% accurate, 206.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x
end function

Java

public static double code(double x) {
	return x;
}

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 75.5%

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

   1. associate-+l- 75.4%

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

   2. sub-neg 75.4%

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

   3. sub-neg 75.4%

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

   4. +-commutative 75.4%

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

   5. distribute-neg-in 75.4%

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

   6. remove-double-neg 75.4%

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

   7. metadata-eval 75.4%

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

3. Simplified 75.4%

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

4. Taylor expanded in x around 0 48.6%

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

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 2023194 
(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))))