exp2 (problem 3.3.7)

Percentage Accurate: 51.9% → 99.6%

Time: 10.1s

Alternatives: 5

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: 51.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.00019:\\ \;\;\;\;x\_m \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x\_m - 2\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.00019) (* 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 <= 0.00019) {
		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 <= 0.00019d0) 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 <= 0.00019) {
		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 <= 0.00019:
		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 <= 0.00019)
		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 <= 0.00019)
		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, 0.00019], 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 < 1.9000000000000001e-4

   1. Initial program 57.7%

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

      1. associate-+l- 57.7%

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

      2. sub-neg 57.7%

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

      3. sub-neg 57.7%

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

      4. distribute-neg-in 57.7%

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

      5. remove-double-neg 57.7%

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

      6. +-commutative 57.7%

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

      7. metadata-eval 57.7%

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

   3. Simplified 57.7%

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

   5. Taylor expanded in x around 0 98.8%

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

      1. unpow2 99.4%

         \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + \color{blue}{\left(x \cdot x\right)} \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right) \]

   7. Applied egg-rr 98.8%

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

   if 1.9000000000000001e-4 < x

   1. Initial program 98.2%

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

      1. associate-+l- 98.2%

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

      2. sub-neg 98.2%

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

      3. sub-neg 98.2%

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

      4. distribute-neg-in 98.2%

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

      5. remove-double-neg 98.2%

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

      6. +-commutative 98.2%

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

      7. metadata-eval 98.2%

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

   3. Simplified 98.2%

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

      1. +-commutative 98.2%

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

      2. associate-+r+ 98.2%

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

      3. metadata-eval 98.2%

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

      4. sub-neg 98.2%

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

      5. +-commutative 98.2%

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

      6. associate-+r- 98.2%

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

      7. +-commutative 98.2%

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

      8. cosh-undef 98.2%

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

   6. Applied egg-rr 98.2%

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

Alternative 2: 99.1% accurate, 0.3× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (+
  (pow x_m 2.0)
  (*
   (fma
    (pow x_m 2.0)
    (fma (pow x_m 2.0) 4.96031746031746e-5 0.002777777777777778)
    0.08333333333333333)
   (pow x_m 4.0))))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(x_m, 2.0) + (fma(pow(x_m, 2.0), fma(pow(x_m, 2.0), 4.96031746031746e-5, 0.002777777777777778), 0.08333333333333333) * pow(x_m, 4.0));
}

Julia

x_m = abs(x)
function code(x_m)
	return Float64((x_m ^ 2.0) + Float64(fma((x_m ^ 2.0), fma((x_m ^ 2.0), 4.96031746031746e-5, 0.002777777777777778), 0.08333333333333333) * (x_m ^ 4.0)))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[x$95$m, 2.0], $MachinePrecision] + N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 4.96031746031746e-5 + 0.002777777777777778), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.0%

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

   1. associate-+l- 58.0%

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

   2. sub-neg 58.0%

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

   3. sub-neg 58.0%

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

   4. distribute-neg-in 58.0%

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

   5. remove-double-neg 58.0%

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

   6. +-commutative 58.0%

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

   7. metadata-eval 58.0%

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

3. Simplified 58.0%

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

5. Taylor expanded in x around 0 98.9%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + 4.96031746031746 \cdot 10^{-5} \cdot {x}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-in 98.9%

      \[\leadsto \color{blue}{1 \cdot {x}^{2} + \left({x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + 4.96031746031746 \cdot 10^{-5} \cdot {x}^{2}\right)\right)\right) \cdot {x}^{2}} \]

   2. *-lft-identity 98.9%

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

   3. *-commutative 98.9%

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

   4. associate-*l* 98.9%

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

   5. +-commutative 98.9%

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

   6. fma-define 98.9%

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

   7. +-commutative 98.9%

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

   8. *-commutative 98.9%

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

   9. fma-define 98.9%

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

   10. pow-sqr 98.9%

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

   11. metadata-eval 98.9%

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

7. Simplified 98.9%

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

Alternative 3: 99.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {x\_m}^{2} \cdot \left(1 + \left(x\_m \cdot x\_m\right) \cdot \left(0.08333333333333333 + \left(x\_m \cdot x\_m\right) \cdot \left(0.002777777777777778 + 4.96031746031746 \cdot 10^{-5} \cdot \left(x\_m \cdot x\_m\right)\right)\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (pow x_m 2.0)
  (+
   1.0
   (*
    (* x_m x_m)
    (+
     0.08333333333333333
     (*
      (* x_m x_m)
      (+ 0.002777777777777778 (* 4.96031746031746e-5 (* x_m x_m)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(x_m, 2.0) * (1.0 + ((x_m * x_m) * (0.08333333333333333 + ((x_m * x_m) * (0.002777777777777778 + (4.96031746031746e-5 * (x_m * x_m)))))));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (x_m ** 2.0d0) * (1.0d0 + ((x_m * x_m) * (0.08333333333333333d0 + ((x_m * x_m) * (0.002777777777777778d0 + (4.96031746031746d-5 * (x_m * x_m)))))))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(x_m, 2.0) * (1.0 + ((x_m * x_m) * (0.08333333333333333 + ((x_m * x_m) * (0.002777777777777778 + (4.96031746031746e-5 * (x_m * x_m)))))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(x_m, 2.0) * (1.0 + ((x_m * x_m) * (0.08333333333333333 + ((x_m * x_m) * (0.002777777777777778 + (4.96031746031746e-5 * (x_m * x_m)))))))

Julia

x_m = abs(x)
function code(x_m)
	return Float64((x_m ^ 2.0) * Float64(1.0 + Float64(Float64(x_m * x_m) * Float64(0.08333333333333333 + Float64(Float64(x_m * x_m) * Float64(0.002777777777777778 + Float64(4.96031746031746e-5 * Float64(x_m * x_m))))))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (x_m ^ 2.0) * (1.0 + ((x_m * x_m) * (0.08333333333333333 + ((x_m * x_m) * (0.002777777777777778 + (4.96031746031746e-5 * (x_m * x_m)))))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.08333333333333333 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.002777777777777778 + N[(4.96031746031746e-5 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.0%

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

   1. associate-+l- 58.0%

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

   2. sub-neg 58.0%

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

   3. sub-neg 58.0%

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

   4. distribute-neg-in 58.0%

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

   5. remove-double-neg 58.0%

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

   6. +-commutative 58.0%

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

   7. metadata-eval 58.0%

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

3. Simplified 58.0%

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

5. Taylor expanded in x around 0 98.9%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + 4.96031746031746 \cdot 10^{-5} \cdot {x}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 98.9%

      \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + \color{blue}{{x}^{2} \cdot 4.96031746031746 \cdot 10^{-5}}\right)\right)\right) \]

7. Simplified 98.9%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + {x}^{2} \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right)} \]
8. Step-by-step derivation

   1. unpow2 98.9%

      \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + \color{blue}{\left(x \cdot x\right)} \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right) \]

9. Applied egg-rr 98.9%

   \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + \color{blue}{\left(x \cdot x\right)} \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right) \]
10. Step-by-step derivation

    1. unpow2 98.9%

       \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + \color{blue}{\left(x \cdot x\right)} \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right) \]

11. Applied egg-rr 98.9%

    \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.002777777777777778 + \left(x \cdot x\right) \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right) \]
12. Step-by-step derivation

    1. unpow2 98.9%

       \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + \color{blue}{\left(x \cdot x\right)} \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right) \]

13. Applied egg-rr 98.9%

    \[\leadsto {x}^{2} \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot \left(0.002777777777777778 + \left(x \cdot x\right) \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right) \]

14. Final simplification 98.9%

    \[\leadsto {x}^{2} \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot \left(0.002777777777777778 + 4.96031746031746 \cdot 10^{-5} \cdot \left(x \cdot x\right)\right)\right)\right) \]
15. Add Preprocessing

Alternative 4: 98.3% 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 58.0%

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

   1. associate-+l- 58.0%

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

   2. sub-neg 58.0%

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

   3. sub-neg 58.0%

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

   4. distribute-neg-in 58.0%

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

   5. remove-double-neg 58.0%

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

   6. +-commutative 58.0%

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

   7. metadata-eval 58.0%

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

3. Simplified 58.0%

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

5. Taylor expanded in x around 0 98.1%

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

   1. unpow2 98.9%

      \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + \color{blue}{\left(x \cdot x\right)} \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right) \]

7. Applied egg-rr 98.1%

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

Alternative 5: 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 58.0%

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

   1. associate-+l- 58.0%

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

   2. sub-neg 58.0%

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

   3. sub-neg 58.0%

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

   4. distribute-neg-in 58.0%

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

   5. remove-double-neg 58.0%

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

   6. +-commutative 58.0%

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

   7. metadata-eval 58.0%

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

3. Simplified 58.0%

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

5. Taylor expanded in x around 0 56.3%

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

6. Taylor expanded in x around 0 6.0%

   \[\leadsto \color{blue}{x} \]
7. 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 2024136 
(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))))