exp2 (problem 3.3.7)

Percentage Accurate: 54.0% → 99.5%

Time: 12.6s

Alternatives: 8

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: 54.0% 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: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(e^{x\_m} - 2\right) + e^{-x\_m}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;x\_m \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 5e-11) (* 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 <= 5e-11) {
		tmp = 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 <= 5d-11) then
        tmp = 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 <= 5e-11) {
		tmp = 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 <= 5e-11:
		tmp = 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 <= 5e-11)
		tmp = 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 <= 5e-11)
		tmp = 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, 5e-11], N[(x$95$m * x$95$m), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 5.00000000000000018e-11

   1. Initial program 52.3%

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

      1. sub-neg 52.3%

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

      2. associate-+l+ 52.3%

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

      3. +-commutative 52.3%

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

      4. metadata-eval 52.3%

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

   3. Simplified 52.3%

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

      1. +-commutative 52.3%

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

      2. associate-+r+ 52.3%

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

      3. metadata-eval 52.3%

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

      4. sub-neg 52.3%

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

      5. add-exp-log 52.3%

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

      6. sub-neg 52.3%

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

      7. metadata-eval 52.3%

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

      8. associate-+r+ 52.3%

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

      9. +-commutative 52.3%

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

      10. associate-+r+ 52.2%

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

      11. +-commutative 52.2%

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

      12. cosh-undef 52.2%

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

   6. Applied egg-rr 52.2%

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

   7. Taylor expanded in x around 0 52.5%

      \[\leadsto e^{\color{blue}{2 \cdot \log x}} \]
   8. Step-by-step derivation

      1. *-commutative 52.5%

         \[\leadsto e^{\color{blue}{\log x \cdot 2}} \]

      2. pow-to-exp 99.8%

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

      3. unpow2 99.8%

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

   9. Applied egg-rr 99.8%

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

   if 5.00000000000000018e-11 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 92.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x} - 2\right) + e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.5× speedup

Math

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

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))
   (fma x_m x_m (* 0.08333333333333333 (pow x_m 4.0))))))

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)) + fma(x_m, x_m, (0.08333333333333333 * pow(x_m, 4.0))));
}

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)) + fma(x_m, x_m, Float64(0.08333333333333333 * (x_m ^ 4.0)))))
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[(x$95$m * x$95$m + N[(0.08333333333333333 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.4%

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

   1. sub-neg 53.4%

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

   2. associate-+l+ 53.4%

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

   3. +-commutative 53.4%

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

   4. metadata-eval 53.4%

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

3. Simplified 53.4%

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

5. Taylor expanded in x around 0 98.7%

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

   1. +-commutative 98.7%

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

   2. unpow2 98.7%

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

   3. fma-define 98.8%

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

7. Applied egg-rr 98.8%

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

8. Final simplification 98.8%

   \[\leadsto 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\right) \]
9. Add Preprocessing

Alternative 3: 98.9% 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 53.4%

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

   1. sub-neg 53.4%

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

   2. associate-+l+ 53.4%

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

   3. +-commutative 53.4%

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

   4. metadata-eval 53.4%

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

3. Simplified 53.4%

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

5. Taylor expanded in x around 0 98.6%

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

6. Final simplification 98.6%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.000185:\\ \;\;\;\;x\_m \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;e^{x\_m} + \left(e^{-x\_m} + -2\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.000185) (* x_m x_m) (+ (exp x_m) (+ (exp (- x_m)) -2.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.000185) {
		tmp = x_m * x_m;
	} else {
		tmp = exp(x_m) + (exp(-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 <= 0.000185d0) then
        tmp = x_m * x_m
    else
        tmp = exp(x_m) + (exp(-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 <= 0.000185) {
		tmp = x_m * x_m;
	} else {
		tmp = Math.exp(x_m) + (Math.exp(-x_m) + -2.0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.000185:
		tmp = x_m * x_m
	else:
		tmp = math.exp(x_m) + (math.exp(-x_m) + -2.0)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.000185)
		tmp = Float64(x_m * x_m);
	else
		tmp = Float64(exp(x_m) + Float64(exp(Float64(-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 <= 0.000185)
		tmp = x_m * x_m;
	else
		tmp = exp(x_m) + (exp(-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, 0.000185], N[(x$95$m * x$95$m), $MachinePrecision], N[(N[Exp[x$95$m], $MachinePrecision] + N[(N[Exp[(-x$95$m)], $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.85e-4

   1. Initial program 52.8%

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

      1. sub-neg 52.8%

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

      2. associate-+l+ 52.8%

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

      3. +-commutative 52.8%

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

      4. metadata-eval 52.8%

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

   3. Simplified 52.8%

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

      1. +-commutative 52.8%

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

      2. associate-+r+ 52.8%

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

      3. metadata-eval 52.8%

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

      4. sub-neg 52.8%

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

      5. add-exp-log 52.8%

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

      6. sub-neg 52.8%

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

      7. metadata-eval 52.8%

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

      8. associate-+r+ 52.8%

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

      9. +-commutative 52.8%

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

      10. associate-+r+ 52.7%

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

      11. +-commutative 52.7%

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

      12. cosh-undef 52.7%

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

   6. Applied egg-rr 52.7%

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

   7. Taylor expanded in x around 0 51.9%

      \[\leadsto e^{\color{blue}{2 \cdot \log x}} \]
   8. Step-by-step derivation

      1. *-commutative 51.9%

         \[\leadsto e^{\color{blue}{\log x \cdot 2}} \]

      2. pow-to-exp 98.9%

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

      3. unpow2 98.9%

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

   9. Applied egg-rr 98.9%

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

   if 1.85e-4 < x

   1. Initial program 92.3%

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

      1. sub-neg 92.3%

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

      2. associate-+l+ 92.3%

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

      3. +-commutative 92.3%

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

      4. metadata-eval 92.3%

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

   3. Simplified 92.3%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000185:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(e^{-x} + -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (+ (* 0.08333333333333333 (pow x_m 4.0)) (pow x_m 2.0)))

C

x_m = fabs(x);
double code(double x_m) {
	return (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 = (0.08333333333333333d0 * (x_m ** 4.0d0)) + (x_m ** 2.0d0)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (0.08333333333333333 * Math.pow(x_m, 4.0)) + Math.pow(x_m, 2.0);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (0.08333333333333333 * math.pow(x_m, 4.0)) + math.pow(x_m, 2.0)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(0.08333333333333333 * (x_m ^ 4.0)) + (x_m ^ 2.0))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (0.08333333333333333 * (x_m ^ 4.0)) + (x_m ^ 2.0);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(0.08333333333333333 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.4%

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

   1. sub-neg 53.4%

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

   2. associate-+l+ 53.4%

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

   3. +-commutative 53.4%

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

   4. metadata-eval 53.4%

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

3. Simplified 53.4%

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

5. Taylor expanded in x around 0 98.3%

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

6. Final simplification 98.3%

   \[\leadsto 0.08333333333333333 \cdot {x}^{4} + {x}^{2} \]
7. Add Preprocessing

Alternative 6: 99.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.000205:\\ \;\;\;\;x\_m \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x\_m - 2\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.000205) (* 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 <= 0.000205) {
		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 <= 0.000205d0) 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 <= 0.000205) {
		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 <= 0.000205:
		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 <= 0.000205)
		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 <= 0.000205)
		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, 0.000205], 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 < 2.05e-4

   1. Initial program 52.8%

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

      1. sub-neg 52.8%

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

      2. associate-+l+ 52.8%

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

      3. +-commutative 52.8%

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

      4. metadata-eval 52.8%

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

   3. Simplified 52.8%

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

      1. +-commutative 52.8%

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

      2. associate-+r+ 52.8%

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

      3. metadata-eval 52.8%

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

      4. sub-neg 52.8%

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

      5. add-exp-log 52.8%

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

      6. sub-neg 52.8%

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

      7. metadata-eval 52.8%

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

      8. associate-+r+ 52.8%

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

      9. +-commutative 52.8%

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

      10. associate-+r+ 52.7%

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

      11. +-commutative 52.7%

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

      12. cosh-undef 52.7%

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

   6. Applied egg-rr 52.7%

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

   7. Taylor expanded in x around 0 51.9%

      \[\leadsto e^{\color{blue}{2 \cdot \log x}} \]
   8. Step-by-step derivation

      1. *-commutative 51.9%

         \[\leadsto e^{\color{blue}{\log x \cdot 2}} \]

      2. pow-to-exp 98.9%

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

      3. unpow2 98.9%

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

   9. Applied egg-rr 98.9%

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

   if 2.05e-4 < x

   1. Initial program 92.3%

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

      1. sub-neg 92.3%

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

      2. associate-+l+ 92.3%

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

      3. +-commutative 92.3%

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

      4. metadata-eval 92.3%

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

   3. Simplified 92.3%

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

      1. +-commutative 92.3%

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

      2. associate-+r+ 92.3%

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

      3. metadata-eval 92.3%

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

      4. sub-neg 92.3%

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

      5. +-commutative 92.3%

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

      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%

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

   6. Applied egg-rr 92.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000205:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 53.4%

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

   1. sub-neg 53.4%

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

   2. associate-+l+ 53.4%

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

   3. +-commutative 53.4%

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

   4. metadata-eval 53.4%

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

3. Simplified 53.4%

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

   1. +-commutative 53.4%

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

   2. associate-+r+ 53.4%

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

   3. metadata-eval 53.4%

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

   4. sub-neg 53.4%

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

   5. add-exp-log 53.4%

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

   6. sub-neg 53.4%

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

   7. metadata-eval 53.4%

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

   8. associate-+r+ 53.4%

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

   9. +-commutative 53.4%

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

   10. associate-+r+ 53.3%

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

   11. +-commutative 53.3%

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

   12. cosh-undef 53.3%

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

6. Applied egg-rr 53.3%

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

7. Taylor expanded in x around 0 51.5%

   \[\leadsto e^{\color{blue}{2 \cdot \log x}} \]
8. Step-by-step derivation

   1. *-commutative 51.5%

      \[\leadsto e^{\color{blue}{\log x \cdot 2}} \]

   2. pow-to-exp 97.8%

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

   3. unpow2 97.8%

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

9. Applied egg-rr 97.8%

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

10. Final simplification 97.8%

    \[\leadsto x \cdot x \]
11. Add Preprocessing

Alternative 8: 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 53.4%

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

   1. sub-neg 53.4%

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

   2. associate-+l+ 53.4%

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

   3. +-commutative 53.4%

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

   4. metadata-eval 53.4%

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

3. Simplified 53.4%

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

5. Taylor expanded in x around 0 51.2%

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

6. Taylor expanded in x around 0 5.8%

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

7. Final simplification 5.8%

   \[\leadsto x \]
8. 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 2024034 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64
  :pre (<= (fabs x) 710.0)

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

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