exp2 (problem 3.3.7)

Percentage Accurate: 53.9% → 99.9%

Time: 9.6s

Alternatives: 7

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: 53.9% 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

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

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)
     (+ (* 0.08333333333333333 (pow x_m 4.0)) (* x_m x_m))
     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 = (0.08333333333333333 * pow(x_m, 4.0)) + (x_m * x_m);
	} 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 = (0.08333333333333333d0 * (x_m ** 4.0d0)) + (x_m * x_m)
    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 = (0.08333333333333333 * Math.pow(x_m, 4.0)) + (x_m * x_m);
	} 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 = (0.08333333333333333 * math.pow(x_m, 4.0)) + (x_m * x_m)
	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(0.08333333333333333 * (x_m ^ 4.0)) + Float64(x_m * x_m));
	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 = (0.08333333333333333 * (x_m ^ 4.0)) + (x_m * x_m);
	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[(0.08333333333333333 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 1.00000000000000008e-5

   1. Initial program 48.4%

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

      1. associate-+l- 48.4%

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

      2. sub-neg 48.4%

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

      3. sub-neg 48.4%

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

      4. distribute-neg-in 48.4%

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

      5. remove-double-neg 48.4%

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

      6. +-commutative 48.4%

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

      7. metadata-eval 48.4%

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

   3. Simplified 48.4%

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

   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. unpow2 99.9%

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

   6. Applied egg-rr 99.9%

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

   if 1.00000000000000008e-5 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 96.3%

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

4. Final simplification 99.8%

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

Alternative 2: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 4.96031746031746 \cdot 10^{-5} \cdot {x_m}^{8} + \left(0.002777777777777778 \cdot {x_m}^{6} + \left(0.08333333333333333 \cdot {x_m}^{4} + x_m \cdot x_m\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (+
  (* 4.96031746031746e-5 (pow x_m 8.0))
  (+
   (* 0.002777777777777778 (pow x_m 6.0))
   (+ (* 0.08333333333333333 (pow x_m 4.0)) (* x_m x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	return (4.96031746031746e-5 * pow(x_m, 8.0)) + ((0.002777777777777778 * pow(x_m, 6.0)) + ((0.08333333333333333 * pow(x_m, 4.0)) + (x_m * x_m)));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (4.96031746031746d-5 * (x_m ** 8.0d0)) + ((0.002777777777777778d0 * (x_m ** 6.0d0)) + ((0.08333333333333333d0 * (x_m ** 4.0d0)) + (x_m * x_m)))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (4.96031746031746e-5 * Math.pow(x_m, 8.0)) + ((0.002777777777777778 * Math.pow(x_m, 6.0)) + ((0.08333333333333333 * Math.pow(x_m, 4.0)) + (x_m * x_m)));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (4.96031746031746e-5 * math.pow(x_m, 8.0)) + ((0.002777777777777778 * math.pow(x_m, 6.0)) + ((0.08333333333333333 * math.pow(x_m, 4.0)) + (x_m * x_m)))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(4.96031746031746e-5 * (x_m ^ 8.0)) + Float64(Float64(0.002777777777777778 * (x_m ^ 6.0)) + Float64(Float64(0.08333333333333333 * (x_m ^ 4.0)) + Float64(x_m * x_m))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (4.96031746031746e-5 * (x_m ^ 8.0)) + ((0.002777777777777778 * (x_m ^ 6.0)) + ((0.08333333333333333 * (x_m ^ 4.0)) + (x_m * x_m)));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(4.96031746031746e-5 * N[Power[x$95$m, 8.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.002777777777777778 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.08333333333333333 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.5%

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

   1. associate-+l- 49.5%

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

   2. sub-neg 49.5%

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

   3. sub-neg 49.5%

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

   4. distribute-neg-in 49.5%

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

   5. remove-double-neg 49.5%

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

   6. +-commutative 49.5%

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

   7. metadata-eval 49.5%

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

3. Simplified 49.5%

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

4. Taylor expanded in x around 0 98.6%

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

   1. unpow2 98.2%

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

6. Applied egg-rr 98.6%

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

7. Final simplification 98.6%

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

Alternative 3: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0.002777777777777778 \cdot {x_m}^{6} + \left(0.08333333333333333 \cdot {x_m}^{4} + x_m \cdot x_m\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (+
  (* 0.002777777777777778 (pow x_m 6.0))
  (+ (* 0.08333333333333333 (pow x_m 4.0)) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	return (0.002777777777777778 * pow(x_m, 6.0)) + ((0.08333333333333333 * pow(x_m, 4.0)) + (x_m * x_m));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (0.002777777777777778d0 * (x_m ** 6.0d0)) + ((0.08333333333333333d0 * (x_m ** 4.0d0)) + (x_m * x_m))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (0.002777777777777778 * Math.pow(x_m, 6.0)) + ((0.08333333333333333 * Math.pow(x_m, 4.0)) + (x_m * x_m));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (0.002777777777777778 * math.pow(x_m, 6.0)) + ((0.08333333333333333 * math.pow(x_m, 4.0)) + (x_m * x_m))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(0.002777777777777778 * (x_m ^ 6.0)) + Float64(Float64(0.08333333333333333 * (x_m ^ 4.0)) + Float64(x_m * x_m)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (0.002777777777777778 * (x_m ^ 6.0)) + ((0.08333333333333333 * (x_m ^ 4.0)) + (x_m * x_m));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(0.002777777777777778 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.08333333333333333 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.5%

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

   1. associate-+l- 49.5%

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

   2. sub-neg 49.5%

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

   3. sub-neg 49.5%

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

   4. distribute-neg-in 49.5%

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

   5. remove-double-neg 49.5%

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

   6. +-commutative 49.5%

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

   7. metadata-eval 49.5%

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

3. Simplified 49.5%

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

4. Taylor expanded in x around 0 98.4%

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

   1. unpow2 98.2%

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

6. Applied egg-rr 98.4%

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

7. Final simplification 98.4%

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

Alternative 4: 99.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.006:\\ \;\;\;\;0.08333333333333333 \cdot {x_m}^{4} + x_m \cdot x_m\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x_m - 2\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.006)
   (+ (* 0.08333333333333333 (pow x_m 4.0)) (* x_m x_m))
   (- (* 2.0 (cosh x_m)) 2.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.006) {
		tmp = (0.08333333333333333 * pow(x_m, 4.0)) + (x_m * x_m);
	} 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 <= 0.006d0) then
        tmp = (0.08333333333333333d0 * (x_m ** 4.0d0)) + (x_m * x_m)
    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 <= 0.006) {
		tmp = (0.08333333333333333 * Math.pow(x_m, 4.0)) + (x_m * x_m);
	} 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 <= 0.006:
		tmp = (0.08333333333333333 * math.pow(x_m, 4.0)) + (x_m * x_m)
	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 <= 0.006)
		tmp = Float64(Float64(0.08333333333333333 * (x_m ^ 4.0)) + Float64(x_m * x_m));
	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 <= 0.006)
		tmp = (0.08333333333333333 * (x_m ^ 4.0)) + (x_m * x_m);
	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, 0.006], N[(N[(0.08333333333333333 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * x$95$m), $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 < 0.0060000000000000001

   1. Initial program 48.8%

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

      1. associate-+l- 48.8%

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

      2. sub-neg 48.8%

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

      3. sub-neg 48.8%

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

      4. distribute-neg-in 48.8%

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

      5. remove-double-neg 48.8%

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

      6. +-commutative 48.8%

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

      7. metadata-eval 48.8%

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

   3. Simplified 48.8%

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

   4. Taylor expanded in x around 0 99.3%

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

      1. unpow2 99.3%

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

   6. Applied egg-rr 99.3%

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

   if 0.0060000000000000001 < x

   1. Initial program 96.6%

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

      1. associate-+l- 96.6%

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

      2. sub-neg 96.6%

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

      3. sub-neg 96.6%

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

      4. distribute-neg-in 96.6%

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

      5. remove-double-neg 96.6%

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

      6. +-commutative 96.6%

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

      7. metadata-eval 96.6%

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

   3. Simplified 96.6%

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

      1. +-commutative 96.6%

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

      2. associate-+r+ 96.6%

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

      3. metadata-eval 96.6%

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

      4. sub-neg 96.6%

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

      5. +-commutative 96.6%

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

      6. associate-+r- 96.6%

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

      7. +-commutative 96.6%

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

      8. cosh-undef 96.6%

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

   5. Applied egg-rr 96.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.006:\\ \;\;\;\;0.08333333333333333 \cdot {x}^{4} + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \]

Alternative 5: 99.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.00022:\\ \;\;\;\;x_m \cdot x_m\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x_m - 2\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.00022) (* x_m x_m) (- (* 2.0 (cosh x_m)) 2.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.00022) {
		tmp = x_m * x_m;
	} 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 <= 0.00022d0) then
        tmp = x_m * x_m
    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 <= 0.00022) {
		tmp = x_m * x_m;
	} 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 <= 0.00022:
		tmp = x_m * x_m
	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 <= 0.00022)
		tmp = Float64(x_m * x_m);
	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 <= 0.00022)
		tmp = x_m * x_m;
	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, 0.00022], N[(x$95$m * x$95$m), $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 < 2.20000000000000008e-4

   1. Initial program 48.7%

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

      1. associate-+l- 48.7%

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

      2. sub-neg 48.7%

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

      3. sub-neg 48.7%

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

      4. distribute-neg-in 48.7%

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

      5. remove-double-neg 48.7%

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

      6. +-commutative 48.7%

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

      7. metadata-eval 48.7%

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

   3. Simplified 48.7%

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

   4. Taylor expanded in x around 0 99.0%

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

      1. unpow2 99.4%

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

   6. Applied egg-rr 99.0%

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

   if 2.20000000000000008e-4 < x

   1. Initial program 93.1%

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

      1. associate-+l- 92.0%

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

      2. sub-neg 92.0%

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

      3. sub-neg 92.0%

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

      4. distribute-neg-in 92.0%

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

      5. remove-double-neg 92.0%

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

      6. +-commutative 92.0%

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

      7. metadata-eval 92.0%

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

   3. Simplified 92.0%

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

      1. +-commutative 92.0%

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

      2. associate-+r+ 93.1%

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

      3. metadata-eval 93.1%

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

      4. sub-neg 93.1%

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

      5. +-commutative 93.1%

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

      6. associate-+r- 92.0%

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

      7. +-commutative 92.0%

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

      8. cosh-undef 92.0%

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

   5. Applied egg-rr 92.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00022:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \]

Alternative 6: 98.1% 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 49.5%

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

   1. associate-+l- 49.5%

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

   2. sub-neg 49.5%

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

   3. sub-neg 49.5%

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

   4. distribute-neg-in 49.5%

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

   5. remove-double-neg 49.5%

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

   6. +-commutative 49.5%

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

   7. metadata-eval 49.5%

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

3. Simplified 49.5%

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

4. Taylor expanded in x around 0 97.6%

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

   1. unpow2 98.2%

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

6. Applied egg-rr 97.6%

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

7. Final simplification 97.6%

   \[\leadsto x \cdot x \]

Alternative 7: 7.0% accurate, 206.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.5%

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

   1. associate-+l- 49.5%

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

   2. sub-neg 49.5%

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

   3. sub-neg 49.5%

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

   4. distribute-neg-in 49.5%

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

   5. remove-double-neg 49.5%

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

   6. +-commutative 49.5%

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

   7. metadata-eval 49.5%

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

3. Simplified 49.5%

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

4. Taylor expanded in x around 0 47.2%

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

5. Taylor expanded in x around 0 5.6%

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

6. Final simplification 5.6%

   \[\leadsto x \]

Developer target: 99.9% 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 2023335 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64
  :pre (<= (fabs x) 710.0)

  :herbie-target
  (* 4.0 (pow (sinh (/ x 2.0)) 2.0))

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