exp2 (problem 3.3.7)

Percentage Accurate: 53.8% → 99.9%

Time: 11.4s

Alternatives: 6

Speedup: 68.7×

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: 53.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(e^{x} - 2\right) + e^{-x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (- (exp x) 2.0) (exp (- x))))

C

double code(double x) {
	return (exp(x) - 2.0) + exp(-x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (exp(x) - 2.0d0) + exp(-x)
end function

Java

public static double code(double x) {
	return (Math.exp(x) - 2.0) + Math.exp(-x);
}

Python

def code(x):
	return (math.exp(x) - 2.0) + math.exp(-x)

Julia

function code(x)
	return Float64(Float64(exp(x) - 2.0) + exp(Float64(-x)))
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - 2.0) + exp(-x);
end

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(e^{x\_m} - 2\right) + e^{-x\_m}\\ \mathbf{if}\;t\_0 \leq 10^{-5}:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot \left(1 + \left(x\_m \cdot x\_m\right) \cdot \left(0.08333333333333333 + \left(x\_m \cdot x\_m\right) \cdot 0.002777777777777778\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ (- (exp x_m) 2.0) (exp (- x_m)))))
   (if (<= t_0 1e-5)
     (*
      (* x_m x_m)
      (+
       1.0
       (*
        (* x_m x_m)
        (+ 0.08333333333333333 (* (* x_m x_m) 0.002777777777777778)))))
     t_0)))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (exp(x_m) - 2.0) + exp(-x_m);
	double tmp;
	if (t_0 <= 1e-5) {
		tmp = (x_m * x_m) * (1.0 + ((x_m * x_m) * (0.08333333333333333 + ((x_m * x_m) * 0.002777777777777778))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (exp(x_m) - 2.0d0) + exp(-x_m)
    if (t_0 <= 1d-5) then
        tmp = (x_m * x_m) * (1.0d0 + ((x_m * x_m) * (0.08333333333333333d0 + ((x_m * x_m) * 0.002777777777777778d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = (Math.exp(x_m) - 2.0) + Math.exp(-x_m);
	double tmp;
	if (t_0 <= 1e-5) {
		tmp = (x_m * x_m) * (1.0 + ((x_m * x_m) * (0.08333333333333333 + ((x_m * x_m) * 0.002777777777777778))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = (math.exp(x_m) - 2.0) + math.exp(-x_m)
	tmp = 0
	if t_0 <= 1e-5:
		tmp = (x_m * x_m) * (1.0 + ((x_m * x_m) * (0.08333333333333333 + ((x_m * x_m) * 0.002777777777777778))))
	else:
		tmp = t_0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(exp(x_m) - 2.0) + exp(Float64(-x_m)))
	tmp = 0.0
	if (t_0 <= 1e-5)
		tmp = Float64(Float64(x_m * x_m) * Float64(1.0 + Float64(Float64(x_m * x_m) * Float64(0.08333333333333333 + Float64(Float64(x_m * x_m) * 0.002777777777777778)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = (exp(x_m) - 2.0) + exp(-x_m);
	tmp = 0.0;
	if (t_0 <= 1e-5)
		tmp = (x_m * x_m) * (1.0 + ((x_m * x_m) * (0.08333333333333333 + ((x_m * x_m) * 0.002777777777777778))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[x$95$m], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x$95$m)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-5], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.08333333333333333 + N[(N[(x$95$m * x$95$m), $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))) < 1.00000000000000008e-5

   1. Initial program 54.1%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

      1. unpow2 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. unpow2 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. unpow2 100.0%

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

   11. Applied egg-rr 100.0%

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

   if 1.00000000000000008e-5 < (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x)))

   1. Initial program 98.3%

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

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left(e^{x\_m} - 2\right) + e^{-x\_m} \leq 10^{-5}:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot \left(1 + \left(x\_m \cdot x\_m\right) \cdot \left(0.08333333333333333 + \left(x\_m \cdot x\_m\right) \cdot 0.002777777777777778\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{2 \cdot \cosh x\_m + -2}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (+ (- (exp x_m) 2.0) (exp (- x_m))) 1e-5)
   (*
    (* x_m x_m)
    (+
     1.0
     (*
      (* x_m x_m)
      (+ 0.08333333333333333 (* (* x_m x_m) 0.002777777777777778)))))
   (pow (cbrt (+ (* 2.0 (cosh x_m)) -2.0)) 3.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (((exp(x_m) - 2.0) + exp(-x_m)) <= 1e-5) {
		tmp = (x_m * x_m) * (1.0 + ((x_m * x_m) * (0.08333333333333333 + ((x_m * x_m) * 0.002777777777777778))));
	} else {
		tmp = pow(cbrt(((2.0 * cosh(x_m)) + -2.0)), 3.0);
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (((Math.exp(x_m) - 2.0) + Math.exp(-x_m)) <= 1e-5) {
		tmp = (x_m * x_m) * (1.0 + ((x_m * x_m) * (0.08333333333333333 + ((x_m * x_m) * 0.002777777777777778))));
	} else {
		tmp = Math.pow(Math.cbrt(((2.0 * Math.cosh(x_m)) + -2.0)), 3.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (Float64(Float64(exp(x_m) - 2.0) + exp(Float64(-x_m))) <= 1e-5)
		tmp = Float64(Float64(x_m * x_m) * Float64(1.0 + Float64(Float64(x_m * x_m) * Float64(0.08333333333333333 + Float64(Float64(x_m * x_m) * 0.002777777777777778)))));
	else
		tmp = cbrt(Float64(Float64(2.0 * cosh(x_m)) + -2.0)) ^ 3.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(N[(N[Exp[x$95$m], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x$95$m)], $MachinePrecision]), $MachinePrecision], 1e-5], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.08333333333333333 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(N[(2.0 * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x))) < 1.00000000000000008e-5

   1. Initial program 54.1%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

      1. unpow2 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. unpow2 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. unpow2 100.0%

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

   11. Applied egg-rr 100.0%

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

   if 1.00000000000000008e-5 < (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x)))

   1. Initial program 98.3%

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

      1. add-cube-cbrt 97.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(e^{x} - 2\right) + e^{-x}} \cdot \sqrt[3]{\left(e^{x} - 2\right) + e^{-x}}\right) \cdot \sqrt[3]{\left(e^{x} - 2\right) + e^{-x}}} \]

      2. pow3 97.9%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(e^{x} - 2\right) + e^{-x}}\right)}^{3}} \]

   4. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{2 \cdot \cosh x + -2}\right)}^{3}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.032:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot \left(1 + \left(x\_m \cdot x\_m\right) \cdot \left(0.08333333333333333 + \left(x\_m \cdot x\_m\right) \cdot 0.002777777777777778\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x\_m - 2\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.032)
   (*
    (* x_m x_m)
    (+
     1.0
     (*
      (* x_m x_m)
      (+ 0.08333333333333333 (* (* x_m x_m) 0.002777777777777778)))))
   (- (* 2.0 (cosh x_m)) 2.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.032) {
		tmp = (x_m * x_m) * (1.0 + ((x_m * x_m) * (0.08333333333333333 + ((x_m * x_m) * 0.002777777777777778))));
	} else {
		tmp = (2.0 * cosh(x_m)) - 2.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.032d0) then
        tmp = (x_m * x_m) * (1.0d0 + ((x_m * x_m) * (0.08333333333333333d0 + ((x_m * x_m) * 0.002777777777777778d0))))
    else
        tmp = (2.0d0 * cosh(x_m)) - 2.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.032) {
		tmp = (x_m * x_m) * (1.0 + ((x_m * x_m) * (0.08333333333333333 + ((x_m * x_m) * 0.002777777777777778))));
	} else {
		tmp = (2.0 * Math.cosh(x_m)) - 2.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.032:
		tmp = (x_m * x_m) * (1.0 + ((x_m * x_m) * (0.08333333333333333 + ((x_m * x_m) * 0.002777777777777778))))
	else:
		tmp = (2.0 * math.cosh(x_m)) - 2.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.032)
		tmp = Float64(Float64(x_m * x_m) * Float64(1.0 + Float64(Float64(x_m * x_m) * Float64(0.08333333333333333 + Float64(Float64(x_m * x_m) * 0.002777777777777778)))));
	else
		tmp = Float64(Float64(2.0 * cosh(x_m)) - 2.0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.032)
		tmp = (x_m * x_m) * (1.0 + ((x_m * x_m) * (0.08333333333333333 + ((x_m * x_m) * 0.002777777777777778))));
	else
		tmp = (2.0 * cosh(x_m)) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.032], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.08333333333333333 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.032000000000000001

   1. Initial program 54.9%

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

   3. Taylor expanded in x around 0 98.7%

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

      1. *-commutative 98.7%

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

   5. Simplified 98.7%

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

      1. unpow2 98.7%

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

   7. Applied egg-rr 98.7%

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

      1. unpow2 98.7%

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

   9. Applied egg-rr 98.7%

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

       1. unpow2 98.7%

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

   11. Applied egg-rr 98.7%

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

   if 0.032000000000000001 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r- 100.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 18.7%

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

      5. sqr-neg 18.7%

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

      6. sqrt-unprod 18.7%

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

      7. add-sqr-sqrt 18.7%

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

      8. add-sqr-sqrt 18.7%

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

      9. sqrt-unprod 18.7%

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

      10. sqr-neg 18.7%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 100.0%

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

      13. cosh-undef 100.0%

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

   4. Applied egg-rr 100.0%

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

Alternative 4: 99.0% accurate, 12.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \left(x\_m \cdot x\_m\right) \cdot \left(1 + \left(x\_m \cdot x\_m\right) \cdot \left(0.08333333333333333 + \left(x\_m \cdot x\_m\right) \cdot 0.002777777777777778\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (* x_m x_m)
  (+
   1.0
   (*
    (* x_m x_m)
    (+ 0.08333333333333333 (* (* x_m x_m) 0.002777777777777778))))))

C

x_m = fabs(x);
double code(double x_m) {
	return (x_m * x_m) * (1.0 + ((x_m * x_m) * (0.08333333333333333 + ((x_m * x_m) * 0.002777777777777778))));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (x_m * x_m) * (1.0d0 + ((x_m * x_m) * (0.08333333333333333d0 + ((x_m * x_m) * 0.002777777777777778d0))))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (x_m * x_m) * (1.0 + ((x_m * x_m) * (0.08333333333333333 + ((x_m * x_m) * 0.002777777777777778))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (x_m * x_m) * (1.0 + ((x_m * x_m) * (0.08333333333333333 + ((x_m * x_m) * 0.002777777777777778))))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(x_m * x_m) * Float64(1.0 + Float64(Float64(x_m * x_m) * Float64(0.08333333333333333 + Float64(Float64(x_m * x_m) * 0.002777777777777778)))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (x_m * x_m) * (1.0 + ((x_m * x_m) * (0.08333333333333333 + ((x_m * x_m) * 0.002777777777777778))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.08333333333333333 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.1%

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

3. Taylor expanded in x around 0 98.3%

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

   1. *-commutative 98.3%

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

5. Simplified 98.3%

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

   1. unpow2 98.3%

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

7. Applied egg-rr 98.3%

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

   1. unpow2 98.3%

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

9. Applied egg-rr 98.3%

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

    1. unpow2 98.3%

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

11. Applied egg-rr 98.3%

    \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot 0.002777777777777778\right)\right) \]
12. Add Preprocessing

Alternative 5: 98.8% accurate, 18.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \left(x\_m \cdot x\_m\right) \cdot \left(1 + \left(x\_m \cdot x\_m\right) \cdot 0.08333333333333333\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (* (* x_m x_m) (+ 1.0 (* (* x_m x_m) 0.08333333333333333))))

C

x_m = fabs(x);
double code(double x_m) {
	return (x_m * x_m) * (1.0 + ((x_m * x_m) * 0.08333333333333333));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (x_m * x_m) * (1.0d0 + ((x_m * x_m) * 0.08333333333333333d0))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (x_m * x_m) * (1.0 + ((x_m * x_m) * 0.08333333333333333));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (x_m * x_m) * (1.0 + ((x_m * x_m) * 0.08333333333333333))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(x_m * x_m) * Float64(1.0 + Float64(Float64(x_m * x_m) * 0.08333333333333333)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (x_m * x_m) * (1.0 + ((x_m * x_m) * 0.08333333333333333));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.1%

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

3. Taylor expanded in x around 0 98.3%

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

   1. *-commutative 98.3%

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

5. Simplified 98.3%

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

6. Taylor expanded in x around 0 98.1%

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

   1. +-commutative 98.1%

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

8. Simplified 98.1%

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

   1. unpow2 98.3%

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

10. Applied egg-rr 98.1%

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

    1. unpow2 98.3%

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

12. Applied egg-rr 98.1%

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

13. Final simplification 98.1%

    \[\leadsto \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot 0.08333333333333333\right) \]
14. Add Preprocessing

Alternative 6: 98.3% accurate, 68.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x\_m \cdot x\_m \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* x_m x_m))

C

x_m = fabs(x);
double code(double x_m) {
	return x_m * x_m;
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return x_m * x_m;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return x_m * x_m

Julia

x_m = abs(x)
function code(x_m)
	return Float64(x_m * x_m)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = x_m * x_m;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * x$95$m), $MachinePrecision]

Derivation

1. Initial program 55.1%

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

3. Taylor expanded in x around 0 97.7%

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

   1. unpow2 98.3%

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

5. Applied egg-rr 97.7%

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

Developer Target 1: 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 2024118 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64
  :pre (<= (fabs x) 710.0)

  :alt
  (! :herbie-platform default (* 4 (* (sinh (/ x 2)) (sinh (/ x 2)))))

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