exp2 (problem 3.3.7)

Percentage Accurate: 53.1% → 99.6%

Time: 10.5s

Alternatives: 10

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: 53.1% 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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{-x_m}\\ \mathbf{if}\;\left(e^{x_m} - 2\right) + t_0 \leq 2 \cdot 10^{-9}:\\ \;\;\;\;{x_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;e^{x_m} + \left(t_0 + -2\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (exp (- x_m))))
   (if (<= (+ (- (exp x_m) 2.0) t_0) 2e-9)
     (pow x_m 2.0)
     (+ (exp x_m) (+ t_0 -2.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = exp(-x_m);
	double tmp;
	if (((exp(x_m) - 2.0) + t_0) <= 2e-9) {
		tmp = pow(x_m, 2.0);
	} else {
		tmp = exp(x_m) + (t_0 + -2.0);
	}
	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)
    if (((exp(x_m) - 2.0d0) + t_0) <= 2d-9) then
        tmp = x_m ** 2.0d0
    else
        tmp = exp(x_m) + (t_0 + (-2.0d0))
    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);
	double tmp;
	if (((Math.exp(x_m) - 2.0) + t_0) <= 2e-9) {
		tmp = Math.pow(x_m, 2.0);
	} else {
		tmp = Math.exp(x_m) + (t_0 + -2.0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = math.exp(-x_m)
	tmp = 0
	if ((math.exp(x_m) - 2.0) + t_0) <= 2e-9:
		tmp = math.pow(x_m, 2.0)
	else:
		tmp = math.exp(x_m) + (t_0 + -2.0)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = exp(Float64(-x_m))
	tmp = 0.0
	if (Float64(Float64(exp(x_m) - 2.0) + t_0) <= 2e-9)
		tmp = x_m ^ 2.0;
	else
		tmp = Float64(exp(x_m) + Float64(t_0 + -2.0));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = exp(-x_m);
	tmp = 0.0;
	if (((exp(x_m) - 2.0) + t_0) <= 2e-9)
		tmp = x_m ^ 2.0;
	else
		tmp = exp(x_m) + (t_0 + -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Exp[(-x$95$m)], $MachinePrecision]}, If[LessEqual[N[(N[(N[Exp[x$95$m], $MachinePrecision] - 2.0), $MachinePrecision] + t$95$0), $MachinePrecision], 2e-9], N[Power[x$95$m, 2.0], $MachinePrecision], N[(N[Exp[x$95$m], $MachinePrecision] + N[(t$95$0 + -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 2.00000000000000012e-9

   1. Initial program 52.6%

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

      1. associate-+l- 52.6%

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

      2. sub-neg 52.6%

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

      3. sub-neg 52.6%

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

      4. distribute-neg-in 52.6%

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

      5. remove-double-neg 52.6%

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

      6. +-commutative 52.6%

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

      7. metadata-eval 52.6%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in x around 0 99.7%

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

   if 2.00000000000000012e-9 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 89.7%

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

      1. associate-+l- 89.4%

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

      2. sub-neg 89.4%

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

      3. sub-neg 89.4%

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

      4. distribute-neg-in 89.4%

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

      5. remove-double-neg 89.4%

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

      6. +-commutative 89.4%

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

      7. metadata-eval 89.4%

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

   3. Simplified 89.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(e^{-x} + -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% 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 53.8%

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

   1. associate-+l- 53.7%

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

   2. sub-neg 53.7%

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

   3. sub-neg 53.7%

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

   4. distribute-neg-in 53.7%

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

   5. remove-double-neg 53.7%

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

   6. +-commutative 53.7%

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

   7. metadata-eval 53.7%

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

3. Simplified 53.7%

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

5. Taylor expanded in x around 0 98.8%

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

   1. +-commutative 98.8%

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

   2. unpow2 98.8%

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

   3. fma-def 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: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {\left(x_m + \left(0.0005208333333333333 \cdot {x_m}^{5} + 0.041666666666666664 \cdot {x_m}^{3}\right)\right)}^{2} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (pow
  (+
   x_m
   (+
    (* 0.0005208333333333333 (pow x_m 5.0))
    (* 0.041666666666666664 (pow x_m 3.0))))
  2.0))

C

x_m = fabs(x);
double code(double x_m) {
	return pow((x_m + ((0.0005208333333333333 * pow(x_m, 5.0)) + (0.041666666666666664 * pow(x_m, 3.0)))), 2.0);
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (x_m + ((0.0005208333333333333d0 * (x_m ** 5.0d0)) + (0.041666666666666664d0 * (x_m ** 3.0d0)))) ** 2.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow((x_m + ((0.0005208333333333333 * Math.pow(x_m, 5.0)) + (0.041666666666666664 * Math.pow(x_m, 3.0)))), 2.0);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow((x_m + ((0.0005208333333333333 * math.pow(x_m, 5.0)) + (0.041666666666666664 * math.pow(x_m, 3.0)))), 2.0)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(x_m + Float64(Float64(0.0005208333333333333 * (x_m ^ 5.0)) + Float64(0.041666666666666664 * (x_m ^ 3.0)))) ^ 2.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (x_m + ((0.0005208333333333333 * (x_m ^ 5.0)) + (0.041666666666666664 * (x_m ^ 3.0)))) ^ 2.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Power[N[(x$95$m + N[(N[(0.0005208333333333333 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]

Derivation

1. Initial program 53.8%

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

   1. associate-+l- 53.7%

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

   2. sub-neg 53.7%

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

   3. sub-neg 53.7%

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

   4. distribute-neg-in 53.7%

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

   5. remove-double-neg 53.7%

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

   6. +-commutative 53.7%

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

   7. metadata-eval 53.7%

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

3. Simplified 53.7%

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

5. Taylor expanded in x around 0 98.8%

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

   1. +-commutative 98.8%

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

   2. unpow2 98.8%

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

   3. fma-def 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. Step-by-step derivation

   1. add-sqr-sqrt 98.8%

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

   2. pow2 98.8%

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

   3. fma-def 98.8%

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

   4. fma-def 98.8%

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

9. Applied egg-rr 98.8%

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

10. Taylor expanded in x around 0 98.7%

    \[\leadsto {\color{blue}{\left(x + \left(0.0005208333333333333 \cdot {x}^{5} + 0.041666666666666664 \cdot {x}^{3}\right)\right)}}^{2} \]

11. Final simplification 98.7%

    \[\leadsto {\left(x + \left(0.0005208333333333333 \cdot {x}^{5} + 0.041666666666666664 \cdot {x}^{3}\right)\right)}^{2} \]
12. Add Preprocessing

Alternative 4: 99.0% accurate, 0.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 53.8%

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

   1. associate-+l- 53.7%

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

   2. sub-neg 53.7%

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

   3. sub-neg 53.7%

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

   4. distribute-neg-in 53.7%

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

   5. remove-double-neg 53.7%

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

   6. +-commutative 53.7%

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

   7. metadata-eval 53.7%

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

3. Simplified 53.7%

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

5. Taylor expanded in x around 0 98.7%

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

   1. +-commutative 98.8%

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

   2. unpow2 98.8%

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

   3. fma-def 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 98.7%

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

8. Final simplification 98.7%

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

Alternative 5: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \mathsf{fma}\left(x_m, x_m, 0.08333333333333333 \cdot {x_m}^{4}\right) \end{array} \]

FPCore

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

C

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

Julia

x_m = abs(x)
function code(x_m)
	return 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[(x$95$m * x$95$m + N[(0.08333333333333333 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

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

   1. associate-+l- 53.7%

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

   2. sub-neg 53.7%

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

   3. sub-neg 53.7%

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

   4. distribute-neg-in 53.7%

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

   5. remove-double-neg 53.7%

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

   6. +-commutative 53.7%

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

   7. metadata-eval 53.7%

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

3. Simplified 53.7%

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

5. Taylor expanded in x around 0 98.3%

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

   1. +-commutative 98.8%

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

   2. unpow2 98.8%

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

   3. fma-def 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 98.3%

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

8. Final simplification 98.3%

   \[\leadsto \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right) \]
9. Add Preprocessing

Alternative 6: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.000175:\\ \;\;\;\;{x_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x_m - 2\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.000175) (pow x_m 2.0) (- (* 2.0 (cosh x_m)) 2.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.000175) {
		tmp = pow(x_m, 2.0);
	} 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 <= 0.000175d0) then
        tmp = x_m ** 2.0d0
    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 <= 0.000175) {
		tmp = Math.pow(x_m, 2.0);
	} 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 <= 0.000175:
		tmp = math.pow(x_m, 2.0)
	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 <= 0.000175)
		tmp = x_m ^ 2.0;
	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 <= 0.000175)
		tmp = x_m ^ 2.0;
	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, 0.000175], N[Power[x$95$m, 2.0], $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 < 1.74999999999999998e-4

   1. Initial program 53.3%

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

      1. associate-+l- 53.3%

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

      2. sub-neg 53.3%

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

      3. sub-neg 53.3%

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

      4. distribute-neg-in 53.3%

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

      5. remove-double-neg 53.3%

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

      6. +-commutative 53.3%

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

      7. metadata-eval 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in x around 0 98.5%

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

   if 1.74999999999999998e-4 < x

   1. Initial program 93.8%

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

      1. associate-+l- 93.7%

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

      2. sub-neg 93.7%

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

      3. sub-neg 93.7%

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

      4. distribute-neg-in 93.7%

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

      5. remove-double-neg 93.7%

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

      6. +-commutative 93.7%

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

      7. metadata-eval 93.7%

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

   3. Simplified 93.7%

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

      1. +-commutative 93.7%

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

      2. associate-+r+ 93.8%

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

      3. metadata-eval 93.8%

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

      4. sub-neg 93.8%

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

      5. +-commutative 93.8%

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

      6. associate-+r- 92.9%

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

      7. +-commutative 92.9%

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

      8. cosh-undef 92.9%

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

   6. Applied egg-rr 92.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000175:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 2.3 \cdot 10^{-103}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x_m\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 2.3e-103) 0.0 (expm1 x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.3e-103) {
		tmp = 0.0;
	} else {
		tmp = expm1(x_m);
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.3e-103) {
		tmp = 0.0;
	} else {
		tmp = Math.expm1(x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.3e-103:
		tmp = 0.0
	else:
		tmp = math.expm1(x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.3e-103)
		tmp = 0.0;
	else
		tmp = expm1(x_m);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.3e-103], 0.0, N[(Exp[x$95$m] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.3000000000000001e-103

   1. Initial program 62.0%

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

      1. associate-+l- 62.0%

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

      2. sub-neg 62.0%

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

      3. sub-neg 62.0%

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

      4. distribute-neg-in 62.0%

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

      5. remove-double-neg 62.0%

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

      6. +-commutative 62.0%

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

      7. metadata-eval 62.0%

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

   3. Simplified 62.0%

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

      1. +-commutative 62.0%

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

      2. associate-+r+ 62.0%

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

      3. metadata-eval 62.0%

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

      4. sub-neg 62.0%

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

      5. +-commutative 62.0%

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

      6. associate-+r- 62.0%

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

      7. +-commutative 62.0%

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

      8. cosh-undef 62.0%

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

   6. Applied egg-rr 62.0%

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

   7. Taylor expanded in x around 0 59.9%

      \[\leadsto 2 \cdot \color{blue}{1} - 2 \]

   if 2.3000000000000001e-103 < x

   1. Initial program 12.9%

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

      1. associate-+l- 12.8%

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

      2. sub-neg 12.8%

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

      3. sub-neg 12.8%

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

      4. distribute-neg-in 12.8%

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

      5. remove-double-neg 12.8%

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

      6. +-commutative 12.8%

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

      7. metadata-eval 12.8%

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

   3. Simplified 12.8%

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

   5. Taylor expanded in x around 0 7.7%

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

   6. Taylor expanded in x around inf 7.7%

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

      1. expm1-def 10.1%

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

   8. Simplified 10.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{-103}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.4% 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 53.8%

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

   1. associate-+l- 53.7%

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

   2. sub-neg 53.7%

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

   3. sub-neg 53.7%

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

   4. distribute-neg-in 53.7%

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

   5. remove-double-neg 53.7%

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

   6. +-commutative 53.7%

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

   7. metadata-eval 53.7%

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

3. Simplified 53.7%

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

5. Taylor expanded in x around 0 97.6%

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

6. Final simplification 97.6%

   \[\leadsto {x}^{2} \]
7. Add Preprocessing

Alternative 9: 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 53.8%

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

   1. associate-+l- 53.7%

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

   2. sub-neg 53.7%

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

   3. sub-neg 53.7%

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

   4. distribute-neg-in 53.7%

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

   5. remove-double-neg 53.7%

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

   6. +-commutative 53.7%

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

   7. metadata-eval 53.7%

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

3. Simplified 53.7%

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

5. Taylor expanded in x around 0 51.1%

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

6. Taylor expanded in x around 0 5.8%

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

7. Final simplification 5.8%

   \[\leadsto x \]
8. Add Preprocessing

Alternative 10: 50.8% 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 53.8%

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

   1. associate-+l- 53.7%

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

   2. sub-neg 53.7%

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

   3. sub-neg 53.7%

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

   4. distribute-neg-in 53.7%

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

   5. remove-double-neg 53.7%

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

   6. +-commutative 53.7%

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

   7. metadata-eval 53.7%

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

3. Simplified 53.7%

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

   1. +-commutative 53.7%

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

   2. associate-+r+ 53.8%

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

   3. metadata-eval 53.8%

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

   4. sub-neg 53.8%

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

   5. +-commutative 53.8%

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

   6. associate-+r- 53.7%

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

   7. +-commutative 53.7%

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

   8. cosh-undef 53.7%

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

6. Applied egg-rr 53.7%

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

7. Taylor expanded in x around 0 50.6%

   \[\leadsto 2 \cdot \color{blue}{1} - 2 \]

8. Final simplification 50.6%

   \[\leadsto 0 \]
9. Add Preprocessing

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 2024019 
(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))))