exp2 (problem 3.3.7)

Percentage Accurate: 52.9% → 99.9%

Time: 11.3s

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: 52.9% 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: 99.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.005:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot \left(1 + \left(x\_m \cdot x\_m\right) \cdot 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x\_m - 2\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.005)
   (* (* x_m x_m) (+ 1.0 (* (* x_m x_m) 0.08333333333333333)))
   (- (* 2.0 (cosh x_m)) 2.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.005) {
		tmp = (x_m * x_m) * (1.0 + ((x_m * x_m) * 0.08333333333333333));
	} 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 <= 0.005d0) then
        tmp = (x_m * x_m) * (1.0d0 + ((x_m * x_m) * 0.08333333333333333d0))
    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 <= 0.005) {
		tmp = (x_m * x_m) * (1.0 + ((x_m * x_m) * 0.08333333333333333));
	} 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 <= 0.005:
		tmp = (x_m * x_m) * (1.0 + ((x_m * x_m) * 0.08333333333333333))
	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 <= 0.005)
		tmp = Float64(Float64(x_m * x_m) * Float64(1.0 + Float64(Float64(x_m * x_m) * 0.08333333333333333)));
	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 <= 0.005)
		tmp = (x_m * x_m) * (1.0 + ((x_m * x_m) * 0.08333333333333333));
	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, 0.005], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $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 < 0.0050000000000000001

   1. Initial program 48.8%

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

      1. associate-+l- 48.8%

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

      2. sub-neg 48.8%

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

      3. sub-neg 48.8%

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

      4. distribute-neg-in 48.8%

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

      5. remove-double-neg 48.8%

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

      6. +-commutative 48.8%

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

      7. metadata-eval 48.8%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in x around 0 98.7%

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

      1. *-commutative 98.7%

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

   7. Simplified 98.7%

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

      1. unpow2 98.8%

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

   9. Applied egg-rr 98.7%

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

       1. unpow2 98.8%

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

   11. Applied egg-rr 98.7%

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

   if 0.0050000000000000001 < x

   1. Initial program 90.2%

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

      1. associate-+l- 90.2%

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

      2. sub-neg 90.2%

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

      3. sub-neg 90.2%

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

      4. distribute-neg-in 90.2%

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

      5. remove-double-neg 90.2%

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

      6. +-commutative 90.2%

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

      7. metadata-eval 90.2%

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

   3. Simplified 90.2%

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

      1. +-commutative 90.2%

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

      2. associate-+r+ 90.2%

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

      3. metadata-eval 90.2%

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

      4. sub-neg 90.2%

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

      5. +-commutative 90.2%

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

      6. associate-+r- 89.0%

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

      7. +-commutative 89.0%

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

      8. cosh-undef 89.2%

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

   6. Applied egg-rr 89.2%

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

Alternative 2: 99.1% accurate, 0.6× speedup

Math

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

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 (* (* x_m x_m) 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 + ((x_m * x_m) * 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 * x_m) * 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 + ((x_m * x_m) * 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 + ((x_m * x_m) * 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(Float64(x_m * x_m) * 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 * x_m) * 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[(x$95$m * x$95$m), $MachinePrecision] * 3.1001984126984127e-6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]

Derivation

1. Initial program 49.5%

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

   1. associate-+l- 49.5%

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

   2. sub-neg 49.5%

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

   3. sub-neg 49.5%

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

   4. distribute-neg-in 49.5%

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

   5. remove-double-neg 49.5%

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

   6. +-commutative 49.5%

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

   7. metadata-eval 49.5%

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

3. Simplified 49.5%

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

   1. +-commutative 49.5%

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

   2. associate-+r+ 49.5%

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

   3. metadata-eval 49.5%

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

   4. sub-neg 49.5%

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

   5. add-sqr-sqrt 49.4%

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

   6. pow2 49.4%

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

   7. +-commutative 49.4%

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

   8. sub-neg 49.4%

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

   9. metadata-eval 49.4%

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

   10. associate-+r+ 49.4%

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

   11. +-commutative 49.4%

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

   12. +-commutative 49.4%

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

   13. cosh-undef 49.4%

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

6. Applied egg-rr 49.4%

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

7. Taylor expanded in x around 0 98.5%

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

   1. *-commutative 98.5%

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

9. Simplified 98.5%

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

    1. unpow2 98.5%

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

11. Applied egg-rr 98.5%

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

Alternative 3: 99.0% accurate, 1.7× speedup

Math

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

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 (* (* x_m x_m) 4.96031746031746e-5))))))))

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 + ((x_m * x_m) * 4.96031746031746e-5))))));
}

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 + ((x_m * x_m) * 4.96031746031746d-5))))))
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 + ((x_m * x_m) * 4.96031746031746e-5))))));
}

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 + ((x_m * x_m) * 4.96031746031746e-5))))))

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(Float64(x_m * x_m) * 4.96031746031746e-5)))))))
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 + ((x_m * x_m) * 4.96031746031746e-5))))));
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[(N[(x$95$m * x$95$m), $MachinePrecision] * 4.96031746031746e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.5%

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

   1. associate-+l- 49.5%

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

   2. sub-neg 49.5%

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

   3. sub-neg 49.5%

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

   4. distribute-neg-in 49.5%

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

   5. remove-double-neg 49.5%

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

   6. +-commutative 49.5%

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

   7. metadata-eval 49.5%

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

3. Simplified 49.5%

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

5. Taylor expanded in x around 0 98.4%

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

   1. *-commutative 98.4%

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

7. Simplified 98.4%

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

   1. unpow2 98.5%

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

9. Applied egg-rr 98.4%

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

    1. unpow2 98.5%

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

11. Applied egg-rr 98.4%

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

    1. unpow2 98.5%

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

13. Applied egg-rr 98.4%

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

Alternative 4: 98.7% accurate, 18.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \left(x\_m \cdot x\_m\right) \cdot \left(1 + \left(x\_m \cdot x\_m\right) \cdot 0.08333333333333333\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (* (* x_m x_m) (+ 1.0 (* (* x_m x_m) 0.08333333333333333))))

C

x_m = fabs(x);
double code(double x_m) {
	return (x_m * x_m) * (1.0 + ((x_m * x_m) * 0.08333333333333333));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (x_m * x_m) * (1.0d0 + ((x_m * x_m) * 0.08333333333333333d0))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (x_m * x_m) * (1.0 + ((x_m * x_m) * 0.08333333333333333));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (x_m * x_m) * (1.0 + ((x_m * x_m) * 0.08333333333333333))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(x_m * x_m) * Float64(1.0 + Float64(Float64(x_m * x_m) * 0.08333333333333333)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (x_m * x_m) * (1.0 + ((x_m * x_m) * 0.08333333333333333));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.5%

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

   1. associate-+l- 49.5%

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

   2. sub-neg 49.5%

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

   3. sub-neg 49.5%

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

   4. distribute-neg-in 49.5%

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

   5. remove-double-neg 49.5%

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

   6. +-commutative 49.5%

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

   7. metadata-eval 49.5%

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

3. Simplified 49.5%

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

5. Taylor expanded in x around 0 97.9%

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

   1. *-commutative 97.9%

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

7. Simplified 97.9%

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

   1. unpow2 98.5%

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

9. Applied egg-rr 97.9%

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

    1. unpow2 98.5%

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

11. Applied egg-rr 97.9%

    \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + \left(x \cdot x\right) \cdot 0.08333333333333333\right) \]
12. Add Preprocessing

Alternative 5: 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 49.5%

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

   1. associate-+l- 49.5%

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

   2. sub-neg 49.5%

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

   3. sub-neg 49.5%

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

   4. distribute-neg-in 49.5%

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

   5. remove-double-neg 49.5%

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

   6. +-commutative 49.5%

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

   7. metadata-eval 49.5%

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

3. Simplified 49.5%

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

5. Taylor expanded in x around 0 97.2%

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

   1. unpow2 98.5%

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

7. Applied egg-rr 97.2%

   \[\leadsto \color{blue}{x \cdot x} \]
8. 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 2024121 
(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))))