exp2 (problem 3.3.7)

Percentage Accurate: 53.6% → 99.9%

Time: 5.9s

Alternatives: 6

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: 53.6% 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

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

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 2e-5)
     (+ (* 0.08333333333333333 (pow x_m 4.0)) (pow x_m 2.0))
     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 <= 2e-5) {
		tmp = (0.08333333333333333 * pow(x_m, 4.0)) + pow(x_m, 2.0);
	} 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 <= 2d-5) then
        tmp = (0.08333333333333333d0 * (x_m ** 4.0d0)) + (x_m ** 2.0d0)
    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 <= 2e-5) {
		tmp = (0.08333333333333333 * Math.pow(x_m, 4.0)) + Math.pow(x_m, 2.0);
	} 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 <= 2e-5:
		tmp = (0.08333333333333333 * math.pow(x_m, 4.0)) + math.pow(x_m, 2.0)
	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 <= 2e-5)
		tmp = Float64(Float64(0.08333333333333333 * (x_m ^ 4.0)) + (x_m ^ 2.0));
	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 <= 2e-5)
		tmp = (0.08333333333333333 * (x_m ^ 4.0)) + (x_m ^ 2.0);
	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, 2e-5], N[(N[(0.08333333333333333 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + N[Power[x$95$m, 2.0], $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))) < 2.00000000000000016e-5

   1. Initial program 52.6%

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

   2. Taylor expanded in x around 0 99.9%

      \[\leadsto \color{blue}{0.08333333333333333 \cdot {x}^{4} + {x}^{2}} \]

   if 2.00000000000000016e-5 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 94.6%

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

4. Final simplification 99.8%

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

Alternative 2: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 4.96031746031746 \cdot 10^{-5} \cdot {x_m}^{8} + \left(0.002777777777777778 \cdot {x_m}^{6} + \left(0.08333333333333333 \cdot {x_m}^{4} + {x_m}^{2}\right)\right) \end{array} \]

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)) (pow x_m 2.0)))))

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)) + pow(x_m, 2.0)));
}

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 ** 2.0d0)))
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)) + Math.pow(x_m, 2.0)));
}

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)) + math.pow(x_m, 2.0)))

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)) + (x_m ^ 2.0))))
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 ^ 2.0)));
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[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.9%

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

2. Taylor expanded in x around 0 98.7%

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

3. Final simplification 98.7%

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

Alternative 3: 99.6% accurate, 0.5× speedup

Math

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

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 5e-10) (pow x_m 2.0) 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 <= 5e-10) {
		tmp = pow(x_m, 2.0);
	} 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 <= 5d-10) then
        tmp = x_m ** 2.0d0
    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 <= 5e-10) {
		tmp = Math.pow(x_m, 2.0);
	} 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 <= 5e-10:
		tmp = math.pow(x_m, 2.0)
	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 <= 5e-10)
		tmp = x_m ^ 2.0;
	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 <= 5e-10)
		tmp = x_m ^ 2.0;
	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, 5e-10], N[Power[x$95$m, 2.0], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 5.00000000000000031e-10

   1. Initial program 52.5%

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

   2. Taylor expanded in x around 0 99.9%

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

   if 5.00000000000000031e-10 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 92.5%

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

4. Final simplification 99.6%

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

Alternative 4: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0.002777777777777778 \cdot {x_m}^{6} + \left(0.08333333333333333 \cdot {x_m}^{4} + {x_m}^{2}\right) \end{array} \]

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)) (pow x_m 2.0))))

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)) + pow(x_m, 2.0));
}

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 ** 2.0d0))
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)) + Math.pow(x_m, 2.0));
}

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)) + math.pow(x_m, 2.0))

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)) + (x_m ^ 2.0)))
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 ^ 2.0));
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[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.9%

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

2. Taylor expanded in x around 0 98.5%

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

3. Final simplification 98.5%

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

Alternative 5: 98.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {x_m}^{2} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (pow x_m 2.0))

C

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

Fortran

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

Java

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

Python

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

Julia

x_m = abs(x)
function code(x_m)
	return x_m ^ 2.0
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Power[x$95$m, 2.0], $MachinePrecision]

Derivation

1. Initial program 53.9%

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

2. Taylor expanded in x around 0 97.4%

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

3. Final simplification 97.4%

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

Alternative 6: 51.4% accurate, 206.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.0;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 0.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.0

Julia

x_m = abs(x)
function code(x_m)
	return 0.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 0.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.0

Derivation

1. Initial program 53.9%

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

3. Applied egg-rr 50.4%

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

4. Final simplification 50.4%

   \[\leadsto 0 \]

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 2023334 
(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))))