exp2 (problem 3.3.7)

Percentage Accurate: 53.2% → 100.0%

Time: 9.9s

Alternatives: 12

Speedup: 2.0×

Specification

\[\left|x\right| \leq 710\]

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: 53.2% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 0.002:\\ \;\;\;\;{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + {x}^{2} \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (- (exp x) 2.0) (exp (- x))) 0.002)
   (*
    (pow x 2.0)
    (+
     1.0
     (*
      (pow x 2.0)
      (+
       0.08333333333333333
       (*
        (pow x 2.0)
        (+ 0.002777777777777778 (* (pow x 2.0) 4.96031746031746e-5)))))))
   (- (* 2.0 (cosh x)) 2.0)))

C

double code(double x) {
	double tmp;
	if (((exp(x) - 2.0) + exp(-x)) <= 0.002) {
		tmp = pow(x, 2.0) * (1.0 + (pow(x, 2.0) * (0.08333333333333333 + (pow(x, 2.0) * (0.002777777777777778 + (pow(x, 2.0) * 4.96031746031746e-5))))));
	} 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 (((exp(x) - 2.0d0) + exp(-x)) <= 0.002d0) then
        tmp = (x ** 2.0d0) * (1.0d0 + ((x ** 2.0d0) * (0.08333333333333333d0 + ((x ** 2.0d0) * (0.002777777777777778d0 + ((x ** 2.0d0) * 4.96031746031746d-5))))))
    else
        tmp = (2.0d0 * cosh(x)) - 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((Math.exp(x) - 2.0) + Math.exp(-x)) <= 0.002) {
		tmp = Math.pow(x, 2.0) * (1.0 + (Math.pow(x, 2.0) * (0.08333333333333333 + (Math.pow(x, 2.0) * (0.002777777777777778 + (Math.pow(x, 2.0) * 4.96031746031746e-5))))));
	} else {
		tmp = (2.0 * Math.cosh(x)) - 2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((math.exp(x) - 2.0) + math.exp(-x)) <= 0.002:
		tmp = math.pow(x, 2.0) * (1.0 + (math.pow(x, 2.0) * (0.08333333333333333 + (math.pow(x, 2.0) * (0.002777777777777778 + (math.pow(x, 2.0) * 4.96031746031746e-5))))))
	else:
		tmp = (2.0 * math.cosh(x)) - 2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) - 2.0) + exp(Float64(-x))) <= 0.002)
		tmp = Float64((x ^ 2.0) * Float64(1.0 + Float64((x ^ 2.0) * Float64(0.08333333333333333 + Float64((x ^ 2.0) * Float64(0.002777777777777778 + Float64((x ^ 2.0) * 4.96031746031746e-5)))))));
	else
		tmp = Float64(Float64(2.0 * cosh(x)) - 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((exp(x) - 2.0) + exp(-x)) <= 0.002)
		tmp = (x ^ 2.0) * (1.0 + ((x ^ 2.0) * (0.08333333333333333 + ((x ^ 2.0) * (0.002777777777777778 + ((x ^ 2.0) * 4.96031746031746e-5))))));
	else
		tmp = (2.0 * cosh(x)) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 0.002], N[(N[Power[x, 2.0], $MachinePrecision] * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.08333333333333333 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.002777777777777778 + N[(N[Power[x, 2.0], $MachinePrecision] * 4.96031746031746e-5), $MachinePrecision]), $MachinePrecision]), $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) #s(literal 2 binary64)) (exp.f64 (neg.f64 x))) < 2e-3

   1. Initial program 51.8%

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

      1. associate-+l- 51.8%

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

      2. sub-neg 51.8%

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

      3. sub-neg 51.8%

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

      4. distribute-neg-in 51.8%

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

      5. remove-double-neg 51.8%

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

      6. +-commutative 51.8%

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

      7. metadata-eval 51.8%

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

   3. Simplified 51.8%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + 4.96031746031746 \cdot 10^{-5} \cdot {x}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + \color{blue}{{x}^{2} \cdot 4.96031746031746 \cdot 10^{-5}}\right)\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + {x}^{2} \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right)} \]

   if 2e-3 < (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x)))

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. sub-neg 99.7%

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

      3. sub-neg 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. remove-double-neg 99.7%

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

      6. +-commutative 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. +-commutative 99.7%

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

      2. associate-+r+ 99.7%

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

      3. metadata-eval 99.7%

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

      4. sub-neg 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r- 99.7%

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

      7. +-commutative 99.7%

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

      8. cosh-undef 99.7%

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

   6. Applied egg-rr 99.7%

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

Alternative 2: 100.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{x} - 2\right) + e^{-x}\\ \mathbf{if}\;t\_0 \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(x, x, {x}^{4} \cdot \mathsf{fma}\left({x}^{2}, 0.002777777777777778, 0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x)))))
   (if (<= t_0 0.0002)
     (fma
      x
      x
      (*
       (pow x 4.0)
       (fma (pow x 2.0) 0.002777777777777778 0.08333333333333333)))
     t_0)))

C

double code(double x) {
	double t_0 = (exp(x) - 2.0) + exp(-x);
	double tmp;
	if (t_0 <= 0.0002) {
		tmp = fma(x, x, (pow(x, 4.0) * fma(pow(x, 2.0), 0.002777777777777778, 0.08333333333333333)));
	} 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 <= 0.0002)
		tmp = fma(x, x, Float64((x ^ 4.0) * fma((x ^ 2.0), 0.002777777777777778, 0.08333333333333333)));
	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, 0.0002], N[(x * x + N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[Power[x, 2.0], $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x))) < 2.0000000000000001e-4

   1. Initial program 51.6%

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

      1. associate-+l- 51.7%

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

      2. sub-neg 51.7%

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

      3. sub-neg 51.7%

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

      4. distribute-neg-in 51.7%

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

      5. remove-double-neg 51.7%

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

      6. +-commutative 51.7%

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

      7. metadata-eval 51.7%

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

   3. Simplified 51.7%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {x}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + \color{blue}{{x}^{2} \cdot 0.002777777777777778}\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot 0.002777777777777778\right)\right)} \]
   8. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{1 \cdot {x}^{2} + \left({x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot 0.002777777777777778\right)\right) \cdot {x}^{2}} \]

      2. *-un-lft-identity 100.0%

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

      3. unpow2 100.0%

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

      4. fma-define 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*r* 100.0%

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

      7. pow-sqr 100.0%

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

      8. metadata-eval 100.0%

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

      9. +-commutative 100.0%

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

      10. fma-define 100.0%

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

   9. Applied egg-rr 100.0%

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

   if 2.0000000000000001e-4 < (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x)))

   1. Initial program 97.9%

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

Alternative 3: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{x} - 2\right) + e^{-x}\\ \mathbf{if}\;t\_0 \leq 0.0002:\\ \;\;\;\;{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot 0.002777777777777778\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x)))))
   (if (<= t_0 0.0002)
     (*
      (pow x 2.0)
      (+
       1.0
       (*
        (pow x 2.0)
        (+ 0.08333333333333333 (* (pow x 2.0) 0.002777777777777778)))))
     t_0)))

C

double code(double x) {
	double t_0 = (exp(x) - 2.0) + exp(-x);
	double tmp;
	if (t_0 <= 0.0002) {
		tmp = pow(x, 2.0) * (1.0 + (pow(x, 2.0) * (0.08333333333333333 + (pow(x, 2.0) * 0.002777777777777778))));
	} 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 <= 0.0002d0) then
        tmp = (x ** 2.0d0) * (1.0d0 + ((x ** 2.0d0) * (0.08333333333333333d0 + ((x ** 2.0d0) * 0.002777777777777778d0))))
    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 <= 0.0002) {
		tmp = Math.pow(x, 2.0) * (1.0 + (Math.pow(x, 2.0) * (0.08333333333333333 + (Math.pow(x, 2.0) * 0.002777777777777778))));
	} 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 <= 0.0002:
		tmp = math.pow(x, 2.0) * (1.0 + (math.pow(x, 2.0) * (0.08333333333333333 + (math.pow(x, 2.0) * 0.002777777777777778))))
	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 <= 0.0002)
		tmp = Float64((x ^ 2.0) * Float64(1.0 + Float64((x ^ 2.0) * Float64(0.08333333333333333 + Float64((x ^ 2.0) * 0.002777777777777778)))));
	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 <= 0.0002)
		tmp = (x ^ 2.0) * (1.0 + ((x ^ 2.0) * (0.08333333333333333 + ((x ^ 2.0) * 0.002777777777777778))));
	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, 0.0002], N[(N[Power[x, 2.0], $MachinePrecision] * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.08333333333333333 + N[(N[Power[x, 2.0], $MachinePrecision] * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x))) < 2.0000000000000001e-4

   1. Initial program 51.6%

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

      1. associate-+l- 51.7%

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

      2. sub-neg 51.7%

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

      3. sub-neg 51.7%

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

      4. distribute-neg-in 51.7%

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

      5. remove-double-neg 51.7%

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

      6. +-commutative 51.7%

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

      7. metadata-eval 51.7%

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

   3. Simplified 51.7%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {x}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + \color{blue}{{x}^{2} \cdot 0.002777777777777778}\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot 0.002777777777777778\right)\right)} \]

   if 2.0000000000000001e-4 < (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x)))

   1. Initial program 97.9%

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

Alternative 4: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{x} - 2\right) + e^{-x}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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) #s(literal 2 binary64)) (exp.f64 (neg.f64 x))) < 5.00000000000000041e-6

   1. Initial program 51.5%

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

      1. associate-+l- 51.5%

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

      2. sub-neg 51.5%

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

      3. sub-neg 51.5%

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

      4. distribute-neg-in 51.5%

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

      5. remove-double-neg 51.5%

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

      6. +-commutative 51.5%

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

      7. metadata-eval 51.5%

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

   3. Simplified 51.5%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {x}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + \color{blue}{{x}^{2} \cdot 0.002777777777777778}\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot 0.002777777777777778\right)\right)} \]
   8. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{1 \cdot {x}^{2} + \left({x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot 0.002777777777777778\right)\right) \cdot {x}^{2}} \]

      2. *-un-lft-identity 100.0%

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

      3. unpow2 100.0%

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

      4. fma-define 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*r* 100.0%

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

      7. pow-sqr 100.0%

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

      8. metadata-eval 100.0%

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

      9. +-commutative 100.0%

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

      10. fma-define 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around 0 100.0%

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

   if 5.00000000000000041e-6 < (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x)))

   1. Initial program 96.2%

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

4. Final simplification 99.8%

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

Alternative 5: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{x} - 2\right) + e^{-x}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-10}:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x)))))
   (if (<= t_0 5e-10) (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 <= 5e-10) {
		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 <= 5d-10) 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 <= 5e-10) {
		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 <= 5e-10:
		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 <= 5e-10)
		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 <= 5e-10)
		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, 5e-10], N[Power[x, 2.0], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x))) < 5.00000000000000031e-10

   1. Initial program 51.4%

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

      1. associate-+l- 51.5%

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

      2. sub-neg 51.5%

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

      3. sub-neg 51.5%

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

      4. distribute-neg-in 51.5%

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

      5. remove-double-neg 51.5%

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

      6. +-commutative 51.5%

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

      7. metadata-eval 51.5%

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

   3. Simplified 51.5%

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

   5. Taylor expanded in x around 0 99.7%

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

   if 5.00000000000000031e-10 < (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x)))

   1. Initial program 93.3%

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

Alternative 6: 99.0% 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 52.0%

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

      1. associate-+l- 52.1%

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

      2. sub-neg 52.1%

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

      3. sub-neg 52.1%

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

      4. distribute-neg-in 52.1%

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

      5. remove-double-neg 52.1%

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

      6. +-commutative 52.1%

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

      7. metadata-eval 52.1%

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

   3. Simplified 52.1%

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

   5. Taylor expanded in x around 0 98.6%

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

   if 1.74999999999999998e-4 < x

   1. Initial program 96.9%

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

      1. associate-+l- 96.4%

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

      2. sub-neg 96.4%

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

      3. sub-neg 96.4%

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

      4. distribute-neg-in 96.4%

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

      5. remove-double-neg 96.4%

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

      6. +-commutative 96.4%

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

      7. metadata-eval 96.4%

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

   3. Simplified 96.4%

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

      1. +-commutative 96.4%

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

      2. associate-+r+ 96.9%

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

      3. metadata-eval 96.9%

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

      4. sub-neg 96.9%

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

      5. +-commutative 96.9%

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

      6. associate-+r- 96.4%

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

      7. +-commutative 96.4%

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

      8. cosh-undef 96.4%

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

   6. Applied egg-rr 96.4%

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

Alternative 7: 98.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {x}^{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow x 2.0))

C

double code(double x) {
	return pow(x, 2.0);
}

Fortran

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

Java

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

Python

def code(x):
	return math.pow(x, 2.0)

Julia

function code(x)
	return x ^ 2.0
end

MATLAB

function tmp = code(x)
	tmp = x ^ 2.0;
end

Wolfram

code[x_] := N[Power[x, 2.0], $MachinePrecision]

Derivation

1. Initial program 52.9%

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

   1. associate-+l- 52.9%

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

   2. sub-neg 52.9%

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

   3. sub-neg 52.9%

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

   4. distribute-neg-in 52.9%

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

   5. remove-double-neg 52.9%

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

   6. +-commutative 52.9%

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

   7. metadata-eval 52.9%

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

3. Simplified 52.9%

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

5. Taylor expanded in x around 0 97.0%

   \[\leadsto \color{blue}{{x}^{2}} \]
6. Add Preprocessing

Alternative 8: 6.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (expm1 x))

C

double code(double x) {
	return expm1(x);
}

Java

public static double code(double x) {
	return Math.expm1(x);
}

Python

def code(x):
	return math.expm1(x)

Julia

function code(x)
	return expm1(x)
end

Wolfram

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

Derivation

1. Initial program 52.9%

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

   1. associate-+l- 52.9%

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

   2. sub-neg 52.9%

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

   3. sub-neg 52.9%

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

   4. distribute-neg-in 52.9%

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

   5. remove-double-neg 52.9%

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

   6. +-commutative 52.9%

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

   7. metadata-eval 52.9%

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

3. Simplified 52.9%

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

5. Taylor expanded in x around 0 49.9%

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

6. Taylor expanded in x around inf 49.9%

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

   1. expm1-define 6.4%

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

8. Simplified 6.4%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right)} \]
9. Add Preprocessing

Alternative 9: 6.0% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (+
   1.0
   (* x (+ 0.5 (* x (+ 0.16666666666666666 (* x 0.041666666666666664))))))))

C

double code(double x) {
	return x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + (x * (0.5d0 + (x * (0.16666666666666666d0 + (x * 0.041666666666666664d0))))))
end function

Java

public static double code(double x) {
	return x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))));
}

Python

def code(x):
	return x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(0.16666666666666666 + Float64(x * 0.041666666666666664)))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * N[(0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.9%

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

   1. associate-+l- 52.9%

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

   2. sub-neg 52.9%

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

   3. sub-neg 52.9%

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

   4. distribute-neg-in 52.9%

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

   5. remove-double-neg 52.9%

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

   6. +-commutative 52.9%

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

   7. metadata-eval 52.9%

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

3. Simplified 52.9%

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

5. Taylor expanded in x around 0 49.9%

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

6. Taylor expanded in x around 0 5.7%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + 0.041666666666666664 \cdot x\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative 5.7%

      \[\leadsto x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + \color{blue}{x \cdot 0.041666666666666664}\right)\right)\right) \]

8. Simplified 5.7%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)} \]
9. Add Preprocessing

Alternative 10: 6.0% accurate, 18.7× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666))))))

C

double code(double x) {
	return x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0))))
end function

Java

public static double code(double x) {
	return x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))));
}

Python

def code(x):
	return x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666)))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.9%

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

   1. associate-+l- 52.9%

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

   2. sub-neg 52.9%

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

   3. sub-neg 52.9%

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

   4. distribute-neg-in 52.9%

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

   5. remove-double-neg 52.9%

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

   6. +-commutative 52.9%

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

   7. metadata-eval 52.9%

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

3. Simplified 52.9%

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

5. Taylor expanded in x around 0 49.9%

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

6. Taylor expanded in x around 0 5.6%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(0.5 + 0.16666666666666666 \cdot x\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative 5.6%

      \[\leadsto x \cdot \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.16666666666666666}\right)\right) \]

8. Simplified 5.6%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)} \]
9. Add Preprocessing

Alternative 11: 6.0% accurate, 29.4× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + x \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (+ 1.0 (* x 0.5))))

C

double code(double x) {
	return x * (1.0 + (x * 0.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + (x * 0.5d0))
end function

Java

public static double code(double x) {
	return x * (1.0 + (x * 0.5));
}

Python

def code(x):
	return x * (1.0 + (x * 0.5))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(x * 0.5)))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + (x * 0.5));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.9%

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

   1. associate-+l- 52.9%

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

   2. sub-neg 52.9%

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

   3. sub-neg 52.9%

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

   4. distribute-neg-in 52.9%

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

   5. remove-double-neg 52.9%

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

   6. +-commutative 52.9%

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

   7. metadata-eval 52.9%

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

3. Simplified 52.9%

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

5. Taylor expanded in x around 0 49.9%

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

6. Taylor expanded in x around 0 5.7%

   \[\leadsto \color{blue}{x \cdot \left(1 + 0.5 \cdot x\right)} \]
7. Step-by-step derivation

   1. *-commutative 5.7%

      \[\leadsto x \cdot \left(1 + \color{blue}{x \cdot 0.5}\right) \]

8. Simplified 5.7%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot 0.5\right)} \]
9. Add Preprocessing

Alternative 12: 6.0% 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 52.9%

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

   1. associate-+l- 52.9%

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

   2. sub-neg 52.9%

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

   3. sub-neg 52.9%

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

   4. distribute-neg-in 52.9%

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

   5. remove-double-neg 52.9%

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

   6. +-commutative 52.9%

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

   7. metadata-eval 52.9%

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

3. Simplified 52.9%

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

5. Taylor expanded in x around 0 49.9%

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

6. Taylor expanded in x around 0 5.6%

   \[\leadsto \color{blue}{x} \]
7. Add Preprocessing

Developer target: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sinh \left(\frac{x}{2}\right)\\ 4 \cdot \left(t\_0 \cdot t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sinh (/ x 2.0)))) (* 4.0 (* t_0 t_0))))

C

double code(double x) {
	double t_0 = sinh((x / 2.0));
	return 4.0 * (t_0 * t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sinh((x / 2.0d0))
    code = 4.0d0 * (t_0 * t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.sinh((x / 2.0));
	return 4.0 * (t_0 * t_0);
}

Python

def code(x):
	t_0 = math.sinh((x / 2.0))
	return 4.0 * (t_0 * t_0)

Julia

function code(x)
	t_0 = sinh(Float64(x / 2.0))
	return Float64(4.0 * Float64(t_0 * t_0))
end

MATLAB

function tmp = code(x)
	t_0 = sinh((x / 2.0));
	tmp = 4.0 * (t_0 * t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sinh[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024108 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64
  :pre (<= (fabs x) 710.0)

  :alt
  (* 4.0 (* (sinh (/ x 2.0)) (sinh (/ x 2.0))))

  (+ (- (exp x) 2.0) (exp (- x))))