exp2 (problem 3.3.7)

Percentage Accurate: 53.5% → 99.6%

Time: 12.9s

Alternatives: 6

Speedup: 68.7×

Specification

\[\left|x\right| \leq 710\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.5% 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, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.000195:\\ \;\;\;\;x\_m \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x\_m - 2\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.000195) (* 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 <= 0.000195) {
		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 <= 0.000195d0) 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 <= 0.000195) {
		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 <= 0.000195:
		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 <= 0.000195)
		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 <= 0.000195)
		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, 0.000195], 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 < 1.94999999999999996e-4

   1. Initial program 51.8%

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

   3. Taylor expanded in x around 0 98.9%

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

      1. unpow2 99.3%

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

   5. Applied egg-rr 98.9%

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

   if 1.94999999999999996e-4 < x

   1. Initial program 92.0%

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

      1. +-commutative 92.0%

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

      2. associate-+r- 90.4%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 16.1%

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

      5. sqr-neg 16.1%

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

      6. sqrt-unprod 16.1%

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

      7. add-sqr-sqrt 16.1%

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

      8. add-sqr-sqrt 16.1%

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

      9. sqrt-unprod 16.1%

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

      10. sqr-neg 16.1%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 90.4%

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

      13. cosh-undef 90.4%

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

   4. Applied egg-rr 90.4%

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

Alternative 2: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {\left(x\_m \cdot \left(1 + {x\_m}^{2} \cdot \left(0.041666666666666664 + {x\_m}^{2} \cdot \left(0.0005208333333333333 + {x\_m}^{2} \cdot 3.1001984126984127 \cdot 10^{-6}\right)\right)\right)\right)}^{2} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (pow
  (*
   x_m
   (+
    1.0
    (*
     (pow x_m 2.0)
     (+
      0.041666666666666664
      (*
       (pow x_m 2.0)
       (+ 0.0005208333333333333 (* (pow x_m 2.0) 3.1001984126984127e-6)))))))
  2.0))

C

x_m = fabs(x);
double code(double x_m) {
	return pow((x_m * (1.0 + (pow(x_m, 2.0) * (0.041666666666666664 + (pow(x_m, 2.0) * (0.0005208333333333333 + (pow(x_m, 2.0) * 3.1001984126984127e-6))))))), 2.0);
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (x_m * (1.0d0 + ((x_m ** 2.0d0) * (0.041666666666666664d0 + ((x_m ** 2.0d0) * (0.0005208333333333333d0 + ((x_m ** 2.0d0) * 3.1001984126984127d-6))))))) ** 2.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow((x_m * (1.0 + (Math.pow(x_m, 2.0) * (0.041666666666666664 + (Math.pow(x_m, 2.0) * (0.0005208333333333333 + (Math.pow(x_m, 2.0) * 3.1001984126984127e-6))))))), 2.0);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow((x_m * (1.0 + (math.pow(x_m, 2.0) * (0.041666666666666664 + (math.pow(x_m, 2.0) * (0.0005208333333333333 + (math.pow(x_m, 2.0) * 3.1001984126984127e-6))))))), 2.0)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(x_m * Float64(1.0 + Float64((x_m ^ 2.0) * Float64(0.041666666666666664 + Float64((x_m ^ 2.0) * Float64(0.0005208333333333333 + Float64((x_m ^ 2.0) * 3.1001984126984127e-6))))))) ^ 2.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (x_m * (1.0 + ((x_m ^ 2.0) * (0.041666666666666664 + ((x_m ^ 2.0) * (0.0005208333333333333 + ((x_m ^ 2.0) * 3.1001984126984127e-6))))))) ^ 2.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Power[N[(x$95$m * N[(1.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.041666666666666664 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.0005208333333333333 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 3.1001984126984127e-6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]

Derivation

1. Initial program 52.2%

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

   1. add-sqr-sqrt 52.2%

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

   2. pow2 52.2%

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

4. Applied egg-rr 52.2%

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

5. Taylor expanded in x around 0 99.2%

   \[\leadsto {\color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(0.041666666666666664 + {x}^{2} \cdot \left(0.0005208333333333333 + 3.1001984126984127 \cdot 10^{-6} \cdot {x}^{2}\right)\right)\right)\right)}}^{2} \]
6. Step-by-step derivation

   1. *-commutative 99.2%

      \[\leadsto {\left(x \cdot \left(1 + {x}^{2} \cdot \left(0.041666666666666664 + {x}^{2} \cdot \left(0.0005208333333333333 + \color{blue}{{x}^{2} \cdot 3.1001984126984127 \cdot 10^{-6}}\right)\right)\right)\right)}^{2} \]

7. Simplified 99.2%

   \[\leadsto {\color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(0.041666666666666664 + {x}^{2} \cdot \left(0.0005208333333333333 + {x}^{2} \cdot 3.1001984126984127 \cdot 10^{-6}\right)\right)\right)\right)}}^{2} \]
8. Add Preprocessing

Alternative 3: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

3. Taylor expanded in x around 0 98.9%

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

   1. distribute-rgt-in 98.9%

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

   2. *-lft-identity 98.9%

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

   3. *-commutative 98.9%

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

   4. associate-*l* 98.9%

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

   5. +-commutative 98.9%

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

   6. *-commutative 98.9%

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

   7. fma-define 98.9%

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

   8. pow-sqr 98.9%

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

   9. metadata-eval 98.9%

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

5. Simplified 98.9%

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

   1. unpow2 98.9%

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

7. Applied egg-rr 98.9%

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

   1. unpow2 98.9%

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

9. Applied egg-rr 98.9%

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

Alternative 4: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	return pow((x_m + (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 + (0.041666666666666664d0 * (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 + (0.041666666666666664 * Math.pow(x_m, 3.0))), 2.0);
}

Python

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

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (x_m + (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[(0.041666666666666664 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]

Derivation

1. Initial program 52.2%

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

   1. add-sqr-sqrt 52.2%

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

   2. pow2 52.2%

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

4. Applied egg-rr 52.2%

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

5. Taylor expanded in x around 0 98.6%

   \[\leadsto {\color{blue}{\left(x \cdot \left(1 + 0.041666666666666664 \cdot {x}^{2}\right)\right)}}^{2} \]
6. Step-by-step derivation

   1. distribute-rgt-in 98.6%

      \[\leadsto {\color{blue}{\left(1 \cdot x + \left(0.041666666666666664 \cdot {x}^{2}\right) \cdot x\right)}}^{2} \]

   2. *-lft-identity 98.6%

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

   3. associate-*l* 98.6%

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

   4. pow-plus 98.6%

      \[\leadsto {\left(x + 0.041666666666666664 \cdot \color{blue}{{x}^{\left(2 + 1\right)}}\right)}^{2} \]

   5. metadata-eval 98.6%

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

7. Simplified 98.6%

   \[\leadsto {\color{blue}{\left(x + 0.041666666666666664 \cdot {x}^{3}\right)}}^{2} \]
8. Add Preprocessing

Alternative 5: 98.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

3. Taylor expanded in x around 0 98.6%

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

   1. distribute-rgt-in 98.6%

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

   2. *-lft-identity 98.6%

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

   3. associate-*l* 98.6%

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

   4. pow-sqr 98.6%

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

   5. metadata-eval 98.6%

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

5. Simplified 98.6%

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

   1. unpow2 98.9%

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

7. Applied egg-rr 98.6%

   \[\leadsto \color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4} \]
8. Add Preprocessing

Alternative 6: 98.4% 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 52.2%

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

3. Taylor expanded in x around 0 98.1%

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

   1. unpow2 98.9%

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

5. Applied egg-rr 98.1%

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

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sinh \left(\frac{x}{2}\right)\\ 4 \cdot \left(t\_0 \cdot t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sinh (/ x 2.0)))) (* 4.0 (* t_0 t_0))))

C

double code(double x) {
	double t_0 = sinh((x / 2.0));
	return 4.0 * (t_0 * t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sinh((x / 2.0d0))
    code = 4.0d0 * (t_0 * t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.sinh((x / 2.0));
	return 4.0 * (t_0 * t_0);
}

Python

def code(x):
	t_0 = math.sinh((x / 2.0))
	return 4.0 * (t_0 * t_0)

Julia

function code(x)
	t_0 = sinh(Float64(x / 2.0))
	return Float64(4.0 * Float64(t_0 * t_0))
end

MATLAB

function tmp = code(x)
	t_0 = sinh((x / 2.0));
	tmp = 4.0 * (t_0 * t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sinh[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024117 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64
  :pre (<= (fabs x) 710.0)

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

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