exp2 (problem 3.3.7)

Percentage Accurate: 76.4% → 100.0%

Time: 6.4s

Alternatives: 10

Speedup: 2.0×

• 49.5% 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: 76.4% 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

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (cosh x))))
   (if (<= (+ (- (exp x) 2.0) (exp (- x))) 4e-5)
     (+
      (* 0.002777777777777778 (pow x 6.0))
      (fma x x (* 0.08333333333333333 (pow x 4.0))))
     (fma (* 2.0 t_0) t_0 -2.0))))

C

double code(double x) {
	double t_0 = sqrt(cosh(x));
	double tmp;
	if (((exp(x) - 2.0) + exp(-x)) <= 4e-5) {
		tmp = (0.002777777777777778 * pow(x, 6.0)) + fma(x, x, (0.08333333333333333 * pow(x, 4.0)));
	} else {
		tmp = fma((2.0 * t_0), t_0, -2.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(cosh(x))
	tmp = 0.0
	if (Float64(Float64(exp(x) - 2.0) + exp(Float64(-x))) <= 4e-5)
		tmp = Float64(Float64(0.002777777777777778 * (x ^ 6.0)) + fma(x, x, Float64(0.08333333333333333 * (x ^ 4.0))));
	else
		tmp = fma(Float64(2.0 * t_0), t_0, -2.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[Cosh[x], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 4e-5], N[(N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(x * x + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * t$95$0), $MachinePrecision] * t$95$0 + -2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 4.00000000000000033e-5

   1. Initial program 58.0%

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

      1. associate-+l- 58.0%

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

      2. sub-neg 58.0%

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

      3. sub-neg 58.0%

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

      4. distribute-neg-in 58.0%

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

      5. remove-double-neg 58.0%

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

      6. +-commutative 58.0%

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

      7. metadata-eval 58.0%

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

   3. Simplified 58.0%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. fma-def 100.0%

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

   6. Applied egg-rr 100.0%

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

   if 4.00000000000000033e-5 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. sub-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-neg-in 99.9%

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

      5. remove-double-neg 99.9%

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

      6. +-commutative 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+r+ 99.9%

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

      3. metadata-eval 99.9%

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

      4. sub-neg 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+r- 99.9%

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

      7. +-commutative 99.9%

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

      8. cosh-undef 99.9%

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

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{2 \cdot \cosh x - 2} \]
   6. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. add-sqr-sqrt 99.9%

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

      3. associate-*r* 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   7. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \sqrt{\cosh x}, \sqrt{\cosh x}, -2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x)))))
   (if (<= t_0 4e-5)
     (+
      (* 0.002777777777777778 (pow x 6.0))
      (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 <= 4e-5) {
		tmp = (0.002777777777777778 * pow(x, 6.0)) + 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 <= 4e-5)
		tmp = Float64(Float64(0.002777777777777778 * (x ^ 6.0)) + 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, 4e-5], N[(N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(x * x + 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))) < 4.00000000000000033e-5

   1. Initial program 58.0%

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

      1. associate-+l- 58.0%

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

      2. sub-neg 58.0%

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

      3. sub-neg 58.0%

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

      4. distribute-neg-in 58.0%

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

      5. remove-double-neg 58.0%

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

      6. +-commutative 58.0%

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

      7. metadata-eval 58.0%

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

   3. Simplified 58.0%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. fma-def 100.0%

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

   6. Applied egg-rr 100.0%

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

   if 4.00000000000000033e-5 < (+.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 100.0%

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

Alternative 3: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{x} - 2\right) + e^{-x}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x)))))
   (if (<= t_0 0.0) (pow x 2.0) t_0)))

C

double code(double x) {
	double t_0 = (exp(x) - 2.0) + exp(-x);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = pow(x, 2.0);
	} 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 <= 0.0d0) then
        tmp = x ** 2.0d0
    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 <= 0.0) {
		tmp = Math.pow(x, 2.0);
	} 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 <= 0.0:
		tmp = math.pow(x, 2.0)
	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 <= 0.0)
		tmp = x ^ 2.0;
	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 <= 0.0)
		tmp = x ^ 2.0;
	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, 0.0], N[Power[x, 2.0], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 0.0

   1. Initial program 57.7%

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

      1. associate-+l- 57.7%

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

      2. sub-neg 57.7%

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

      3. sub-neg 57.7%

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

      4. distribute-neg-in 57.7%

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

      5. remove-double-neg 57.7%

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

      6. +-commutative 57.7%

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

      7. metadata-eval 57.7%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 0.0 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 0:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x} - 2\right) + e^{-x}\\ \end{array} \]

Alternative 4: 100.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x)))))
   (if (<= t_0 5e-6) (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-6) {
		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-6)
		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-6], 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))) < 5.00000000000000041e-6

   1. Initial program 57.8%

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

      1. associate-+l- 57.8%

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

      2. sub-neg 57.8%

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

      3. sub-neg 57.8%

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

      4. distribute-neg-in 57.8%

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

      5. remove-double-neg 57.8%

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

      6. +-commutative 57.8%

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

      7. metadata-eval 57.8%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. +-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. fma-def 100.0%

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

   6. Applied egg-rr 99.9%

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

   if 5.00000000000000041e-6 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 99.8%

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

4. Final simplification 99.9%

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

Alternative 5: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00015:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(e^{-x} + -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.00015) (pow x 2.0) (+ (exp x) (+ (exp (- x)) -2.0))))

C

double code(double x) {
	double tmp;
	if (x <= 0.00015) {
		tmp = pow(x, 2.0);
	} else {
		tmp = exp(x) + (exp(-x) + -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.00015d0) then
        tmp = x ** 2.0d0
    else
        tmp = exp(x) + (exp(-x) + (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.00015) {
		tmp = Math.pow(x, 2.0);
	} else {
		tmp = Math.exp(x) + (Math.exp(-x) + -2.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.00015:
		tmp = math.pow(x, 2.0)
	else:
		tmp = math.exp(x) + (math.exp(-x) + -2.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.00015)
		tmp = x ^ 2.0;
	else
		tmp = Float64(exp(x) + Float64(exp(Float64(-x)) + -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.00015)
		tmp = x ^ 2.0;
	else
		tmp = exp(x) + (exp(-x) + -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.00015], N[Power[x, 2.0], $MachinePrecision], N[(N[Exp[x], $MachinePrecision] + N[(N[Exp[(-x)], $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.49999999999999987e-4

   1. Initial program 72.1%

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

      1. associate-+l- 72.1%

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

      2. sub-neg 72.1%

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

      3. sub-neg 72.1%

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

      4. distribute-neg-in 72.1%

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

      5. remove-double-neg 72.1%

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

      6. +-commutative 72.1%

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

      7. metadata-eval 72.1%

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

   3. Simplified 72.1%

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

   4. Taylor expanded in x around 0 86.1%

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

   if 1.49999999999999987e-4 < x

   1. Initial program 99.5%

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

      1. associate-+l- 99.4%

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

      2. sub-neg 99.4%

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

      3. sub-neg 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. remove-double-neg 99.4%

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

      6. +-commutative 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00015:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(e^{-x} + -2\right)\\ \end{array} \]

Alternative 6: 87.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.000175:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.000175) (pow x 2.0) (- (* 2.0 (cosh x)) 2.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.000175) {
		tmp = pow(x, 2.0);
	} else {
		tmp = (2.0 * cosh(x)) - 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.000175d0) then
        tmp = x ** 2.0d0
    else
        tmp = (2.0d0 * cosh(x)) - 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.000175) {
		tmp = Math.pow(x, 2.0);
	} else {
		tmp = (2.0 * Math.cosh(x)) - 2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.000175:
		tmp = math.pow(x, 2.0)
	else:
		tmp = (2.0 * math.cosh(x)) - 2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.000175)
		tmp = x ^ 2.0;
	else
		tmp = Float64(Float64(2.0 * cosh(x)) - 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.000175)
		tmp = x ^ 2.0;
	else
		tmp = (2.0 * cosh(x)) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.000175], N[Power[x, 2.0], $MachinePrecision], N[(N[(2.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.74999999999999998e-4

   1. Initial program 72.1%

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

      1. associate-+l- 72.1%

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

      2. sub-neg 72.1%

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

      3. sub-neg 72.1%

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

      4. distribute-neg-in 72.1%

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

      5. remove-double-neg 72.1%

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

      6. +-commutative 72.1%

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

      7. metadata-eval 72.1%

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

   3. Simplified 72.1%

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

   4. Taylor expanded in x around 0 86.1%

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

   if 1.74999999999999998e-4 < x

   1. Initial program 99.5%

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

      1. associate-+l- 99.4%

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

      2. sub-neg 99.4%

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

      3. sub-neg 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. remove-double-neg 99.4%

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

      6. +-commutative 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. +-commutative 99.4%

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

      2. associate-+r+ 99.5%

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

      3. metadata-eval 99.5%

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

      4. sub-neg 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r- 99.3%

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

      7. +-commutative 99.3%

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

      8. cosh-undef 99.3%

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

   5. Applied egg-rr 99.3%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000175:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \]

Alternative 7: 87.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.65:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.65) (pow x 2.0) (expm1 x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.65) {
		tmp = pow(x, 2.0);
	} else {
		tmp = expm1(x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.65) {
		tmp = Math.pow(x, 2.0);
	} else {
		tmp = Math.expm1(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.65:
		tmp = math.pow(x, 2.0)
	else:
		tmp = math.expm1(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.65)
		tmp = x ^ 2.0;
	else
		tmp = expm1(x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.65], N[Power[x, 2.0], $MachinePrecision], N[(Exp[x] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6499999999999999

   1. Initial program 72.3%

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

      1. associate-+l- 72.2%

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

      2. sub-neg 72.2%

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

      3. sub-neg 72.2%

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

      4. distribute-neg-in 72.2%

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

      5. remove-double-neg 72.2%

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

      6. +-commutative 72.2%

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

      7. metadata-eval 72.2%

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

   3. Simplified 72.2%

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

   4. Taylor expanded in x around 0 85.4%

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

   if 1.6499999999999999 < 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. distribute-neg-in 100.0%

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

      5. remove-double-neg 100.0%

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

      6. +-commutative 100.0%

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

      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 0 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. expm1-def 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 8: 50.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.25 \cdot 10^{-103}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 2.25e-103) 0.0 (expm1 x)))

C

double code(double x) {
	double tmp;
	if (x <= 2.25e-103) {
		tmp = 0.0;
	} else {
		tmp = expm1(x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.25e-103) {
		tmp = 0.0;
	} else {
		tmp = Math.expm1(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.25e-103:
		tmp = 0.0
	else:
		tmp = math.expm1(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.25e-103)
		tmp = 0.0;
	else
		tmp = expm1(x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.25e-103], 0.0, N[(Exp[x] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.25e-103

   1. Initial program 77.0%

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

      1. associate-+l- 77.0%

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

      2. sub-neg 77.0%

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

      3. sub-neg 77.0%

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

      4. distribute-neg-in 77.0%

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

      5. remove-double-neg 77.0%

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

      6. +-commutative 77.0%

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

      7. metadata-eval 77.0%

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

   3. Simplified 77.0%

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

      1. +-commutative 77.0%

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

      2. associate-+r+ 77.0%

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

      3. metadata-eval 77.0%

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

      4. sub-neg 77.0%

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

      5. add-log-exp 77.0%

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

      6. sub-neg 77.0%

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

      7. metadata-eval 77.0%

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

      8. associate-+r+ 77.0%

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

      9. +-commutative 77.0%

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

      10. associate-+r+ 77.0%

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

      11. +-commutative 77.0%

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

      12. cosh-undef 77.0%

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

   5. Applied egg-rr 77.0%

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

      1. +-commutative 77.0%

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

      2. add-log-exp 77.0%

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

      3. add-sqr-sqrt 77.0%

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

      4. associate-*r* 77.0%

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

      5. add-sqr-sqrt 38.9%

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

      6. associate-*l* 38.9%

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

      7. fma-def 38.9%

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

      8. pow1/2 38.9%

         \[\leadsto \mathsf{fma}\left(\sqrt{2 \cdot \sqrt{\cosh x}}, \color{blue}{{\left(2 \cdot \sqrt{\cosh x}\right)}^{0.5}} \cdot \sqrt{\cosh x}, -2\right) \]

      9. pow1/2 38.9%

         \[\leadsto \mathsf{fma}\left(\sqrt{2 \cdot \sqrt{\cosh x}}, {\left(2 \cdot \sqrt{\cosh x}\right)}^{0.5} \cdot \color{blue}{{\cosh x}^{0.5}}, -2\right) \]

      10. pow-prod-down 38.9%

          \[\leadsto \mathsf{fma}\left(\sqrt{2 \cdot \sqrt{\cosh x}}, \color{blue}{{\left(\left(2 \cdot \sqrt{\cosh x}\right) \cdot \cosh x\right)}^{0.5}}, -2\right) \]

   7. Applied egg-rr 38.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{2 \cdot \sqrt{\cosh x}}, {\left(\left(2 \cdot \sqrt{\cosh x}\right) \cdot \cosh x\right)}^{0.5}, -2\right)} \]
   8. Step-by-step derivation

      1. unpow1/2 38.9%

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

      2. associate-*l* 38.9%

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

   9. Simplified 38.9%

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

   10. Taylor expanded in x around 0 3.5%

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

       1. unpow2 3.5%

          \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{2}} - 2 \]

       2. rem-square-sqrt 41.1%

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

       3. metadata-eval 41.1%

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

   12. Simplified 41.1%

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

   if 2.25e-103 < x

   1. Initial program 87.1%

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

      1. associate-+l- 87.0%

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

      2. sub-neg 87.0%

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

      3. sub-neg 87.0%

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

      4. distribute-neg-in 87.0%

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

      5. remove-double-neg 87.0%

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

      6. +-commutative 87.0%

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

      7. metadata-eval 87.0%

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

   3. Simplified 87.0%

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

   4. Taylor expanded in x around 0 84.8%

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

   5. Taylor expanded in x around inf 84.8%

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

      1. expm1-def 85.1%

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

   7. Simplified 85.1%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.25 \cdot 10^{-103}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 9: 27.9% accurate, 67.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.25 \cdot 10^{-103}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 2.25e-103) 0.0 x))

C

double code(double x) {
	double tmp;
	if (x <= 2.25e-103) {
		tmp = 0.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.25d-103) then
        tmp = 0.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.25e-103) {
		tmp = 0.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.25e-103:
		tmp = 0.0
	else:
		tmp = x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.25e-103)
		tmp = 0.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.25e-103)
		tmp = 0.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.25e-103], 0.0, x]

Derivation

1. Split input into 2 regimes

2. if x < 2.25e-103

   1. Initial program 77.0%

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

      1. associate-+l- 77.0%

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

      2. sub-neg 77.0%

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

      3. sub-neg 77.0%

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

      4. distribute-neg-in 77.0%

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

      5. remove-double-neg 77.0%

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

      6. +-commutative 77.0%

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

      7. metadata-eval 77.0%

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

   3. Simplified 77.0%

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

      1. +-commutative 77.0%

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

      2. associate-+r+ 77.0%

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

      3. metadata-eval 77.0%

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

      4. sub-neg 77.0%

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

      5. add-log-exp 77.0%

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

      6. sub-neg 77.0%

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

      7. metadata-eval 77.0%

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

      8. associate-+r+ 77.0%

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

      9. +-commutative 77.0%

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

      10. associate-+r+ 77.0%

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

      11. +-commutative 77.0%

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

      12. cosh-undef 77.0%

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

   5. Applied egg-rr 77.0%

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

      1. +-commutative 77.0%

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

      2. add-log-exp 77.0%

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

      3. add-sqr-sqrt 77.0%

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

      4. associate-*r* 77.0%

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

      5. add-sqr-sqrt 38.9%

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

      6. associate-*l* 38.9%

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

      7. fma-def 38.9%

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

      8. pow1/2 38.9%

         \[\leadsto \mathsf{fma}\left(\sqrt{2 \cdot \sqrt{\cosh x}}, \color{blue}{{\left(2 \cdot \sqrt{\cosh x}\right)}^{0.5}} \cdot \sqrt{\cosh x}, -2\right) \]

      9. pow1/2 38.9%

         \[\leadsto \mathsf{fma}\left(\sqrt{2 \cdot \sqrt{\cosh x}}, {\left(2 \cdot \sqrt{\cosh x}\right)}^{0.5} \cdot \color{blue}{{\cosh x}^{0.5}}, -2\right) \]

      10. pow-prod-down 38.9%

          \[\leadsto \mathsf{fma}\left(\sqrt{2 \cdot \sqrt{\cosh x}}, \color{blue}{{\left(\left(2 \cdot \sqrt{\cosh x}\right) \cdot \cosh x\right)}^{0.5}}, -2\right) \]

   7. Applied egg-rr 38.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{2 \cdot \sqrt{\cosh x}}, {\left(\left(2 \cdot \sqrt{\cosh x}\right) \cdot \cosh x\right)}^{0.5}, -2\right)} \]
   8. Step-by-step derivation

      1. unpow1/2 38.9%

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

      2. associate-*l* 38.9%

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

   9. Simplified 38.9%

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

   10. Taylor expanded in x around 0 3.5%

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

       1. unpow2 3.5%

          \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{2}} - 2 \]

       2. rem-square-sqrt 41.1%

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

       3. metadata-eval 41.1%

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

   12. Simplified 41.1%

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

   if 2.25e-103 < x

   1. Initial program 87.1%

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

      1. associate-+l- 87.0%

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

      2. sub-neg 87.0%

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

      3. sub-neg 87.0%

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

      4. distribute-neg-in 87.0%

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

      5. remove-double-neg 87.0%

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

      6. +-commutative 87.0%

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

      7. metadata-eval 87.0%

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

   3. Simplified 87.0%

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

   4. Taylor expanded in x around 0 84.8%

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

   5. Taylor expanded in x around 0 5.8%

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

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.25 \cdot 10^{-103}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10: 26.6% 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 80.6%

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

   1. associate-+l- 80.6%

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

   2. sub-neg 80.6%

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

   3. sub-neg 80.6%

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

   4. distribute-neg-in 80.6%

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

   5. remove-double-neg 80.6%

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

   6. +-commutative 80.6%

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

   7. metadata-eval 80.6%

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

3. Simplified 80.6%

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

   1. +-commutative 80.6%

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

   2. associate-+r+ 80.6%

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

   3. metadata-eval 80.6%

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

   4. sub-neg 80.6%

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

   5. add-log-exp 79.9%

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

   6. sub-neg 79.9%

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

   7. metadata-eval 79.9%

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

   8. associate-+r+ 79.8%

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

   9. +-commutative 79.8%

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

   10. associate-+r+ 79.8%

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

   11. +-commutative 79.8%

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

   12. cosh-undef 79.8%

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

5. Applied egg-rr 79.8%

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

   1. +-commutative 79.8%

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

   2. add-log-exp 80.6%

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

   3. add-sqr-sqrt 80.5%

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

   4. associate-*r* 80.5%

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

   5. add-sqr-sqrt 56.2%

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

   6. associate-*l* 56.2%

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

   7. fma-def 56.2%

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

   8. pow1/2 56.2%

      \[\leadsto \mathsf{fma}\left(\sqrt{2 \cdot \sqrt{\cosh x}}, \color{blue}{{\left(2 \cdot \sqrt{\cosh x}\right)}^{0.5}} \cdot \sqrt{\cosh x}, -2\right) \]

   9. pow1/2 56.2%

      \[\leadsto \mathsf{fma}\left(\sqrt{2 \cdot \sqrt{\cosh x}}, {\left(2 \cdot \sqrt{\cosh x}\right)}^{0.5} \cdot \color{blue}{{\cosh x}^{0.5}}, -2\right) \]

   10. pow-prod-down 55.9%

       \[\leadsto \mathsf{fma}\left(\sqrt{2 \cdot \sqrt{\cosh x}}, \color{blue}{{\left(\left(2 \cdot \sqrt{\cosh x}\right) \cdot \cosh x\right)}^{0.5}}, -2\right) \]

7. Applied egg-rr 55.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{2 \cdot \sqrt{\cosh x}}, {\left(\left(2 \cdot \sqrt{\cosh x}\right) \cdot \cosh x\right)}^{0.5}, -2\right)} \]
8. Step-by-step derivation

   1. unpow1/2 55.9%

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

   2. associate-*l* 55.9%

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

9. Simplified 55.9%

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

10. Taylor expanded in x around 0 3.6%

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

    1. unpow2 3.6%

       \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{2}} - 2 \]

    2. rem-square-sqrt 27.0%

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

    3. metadata-eval 27.0%

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

12. Simplified 27.0%

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

13. Final simplification 27.0%

    \[\leadsto 0 \]

Developer target: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 4 \cdot {\sinh \left(\frac{x}{2}\right)}^{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 4.0 (pow (sinh (/ x 2.0)) 2.0)))

C

double code(double x) {
	return 4.0 * pow(sinh((x / 2.0)), 2.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 4.0d0 * (sinh((x / 2.0d0)) ** 2.0d0)
end function

Java

public static double code(double x) {
	return 4.0 * Math.pow(Math.sinh((x / 2.0)), 2.0);
}

Python

def code(x):
	return 4.0 * math.pow(math.sinh((x / 2.0)), 2.0)

Julia

function code(x)
	return Float64(4.0 * (sinh(Float64(x / 2.0)) ^ 2.0))
end

MATLAB

function tmp = code(x)
	tmp = 4.0 * (sinh((x / 2.0)) ^ 2.0);
end

Wolfram

code[x_] := N[(4.0 * N[Power[N[Sinh[N[(x / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023319 
(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))))