exp2 (problem 3.3.7)

Percentage Accurate: 54.2% → 99.9%

Time: 11.0s

Alternatives: 8

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: 54.2% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot 0.002777777777777778\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x + -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (- (exp x) 2.0) (exp (- x))) 4e-6)
   (*
    (pow x 2.0)
    (+
     1.0
     (*
      (pow x 2.0)
      (+ 0.08333333333333333 (* (pow x 2.0) 0.002777777777777778)))))
   (+ (* 2.0 (cosh x)) -2.0)))

C

double code(double x) {
	double tmp;
	if (((exp(x) - 2.0) + exp(-x)) <= 4e-6) {
		tmp = pow(x, 2.0) * (1.0 + (pow(x, 2.0) * (0.08333333333333333 + (pow(x, 2.0) * 0.002777777777777778))));
	} else {
		tmp = (2.0 * cosh(x)) + -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((exp(x) - 2.0d0) + exp(-x)) <= 4d-6) then
        tmp = (x ** 2.0d0) * (1.0d0 + ((x ** 2.0d0) * (0.08333333333333333d0 + ((x ** 2.0d0) * 0.002777777777777778d0))))
    else
        tmp = (2.0d0 * cosh(x)) + (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((Math.exp(x) - 2.0) + Math.exp(-x)) <= 4e-6) {
		tmp = Math.pow(x, 2.0) * (1.0 + (Math.pow(x, 2.0) * (0.08333333333333333 + (Math.pow(x, 2.0) * 0.002777777777777778))));
	} else {
		tmp = (2.0 * Math.cosh(x)) + -2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((math.exp(x) - 2.0) + math.exp(-x)) <= 4e-6:
		tmp = math.pow(x, 2.0) * (1.0 + (math.pow(x, 2.0) * (0.08333333333333333 + (math.pow(x, 2.0) * 0.002777777777777778))))
	else:
		tmp = (2.0 * math.cosh(x)) + -2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) - 2.0) + exp(Float64(-x))) <= 4e-6)
		tmp = Float64((x ^ 2.0) * Float64(1.0 + Float64((x ^ 2.0) * Float64(0.08333333333333333 + Float64((x ^ 2.0) * 0.002777777777777778)))));
	else
		tmp = Float64(Float64(2.0 * cosh(x)) + -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((exp(x) - 2.0) + exp(-x)) <= 4e-6)
		tmp = (x ^ 2.0) * (1.0 + ((x ^ 2.0) * (0.08333333333333333 + ((x ^ 2.0) * 0.002777777777777778))));
	else
		tmp = (2.0 * cosh(x)) + -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 4e-6], N[(N[Power[x, 2.0], $MachinePrecision] * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.08333333333333333 + N[(N[Power[x, 2.0], $MachinePrecision] * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 3.99999999999999982e-6

   1. Initial program 49.1%

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

      1. associate-+l- 49.1%

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

      2. sub-neg 49.1%

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

      3. sub-neg 49.1%

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

      4. distribute-neg-in 49.1%

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

      5. remove-double-neg 49.1%

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

      6. +-commutative 49.1%

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

      7. metadata-eval 49.1%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if 3.99999999999999982e-6 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

      2. sub-neg 99.8%

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

      3. sub-neg 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. remove-double-neg 99.8%

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

      6. +-commutative 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 99.8%

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

      1. sub-neg 99.8%

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

      2. *-rgt-identity 99.8%

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

      3. metadata-eval 99.8%

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

      4. associate-/l* 99.8%

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

      5. associate-*l/ 99.8%

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

      6. cosh-def 99.8%

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

      7. *-commutative 99.8%

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

      8. metadata-eval 99.8%

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

      9. fma-undefine 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, \cosh x, -2\right)} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \cosh x, -2\right)} \]
   8. Step-by-step derivation

      1. fma-undefine 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot 0.002777777777777778\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;{x}^{2} + 0.08333333333333333 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x + -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (- (exp x) 2.0) (exp (- x))) 4e-6)
   (+ (pow x 2.0) (* 0.08333333333333333 (pow x 4.0)))
   (+ (* 2.0 (cosh x)) -2.0)))

C

double code(double x) {
	double tmp;
	if (((exp(x) - 2.0) + exp(-x)) <= 4e-6) {
		tmp = pow(x, 2.0) + (0.08333333333333333 * pow(x, 4.0));
	} else {
		tmp = (2.0 * cosh(x)) + -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((exp(x) - 2.0d0) + exp(-x)) <= 4d-6) then
        tmp = (x ** 2.0d0) + (0.08333333333333333d0 * (x ** 4.0d0))
    else
        tmp = (2.0d0 * cosh(x)) + (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((Math.exp(x) - 2.0) + Math.exp(-x)) <= 4e-6) {
		tmp = Math.pow(x, 2.0) + (0.08333333333333333 * Math.pow(x, 4.0));
	} else {
		tmp = (2.0 * Math.cosh(x)) + -2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((math.exp(x) - 2.0) + math.exp(-x)) <= 4e-6:
		tmp = math.pow(x, 2.0) + (0.08333333333333333 * math.pow(x, 4.0))
	else:
		tmp = (2.0 * math.cosh(x)) + -2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) - 2.0) + exp(Float64(-x))) <= 4e-6)
		tmp = Float64((x ^ 2.0) + Float64(0.08333333333333333 * (x ^ 4.0)));
	else
		tmp = Float64(Float64(2.0 * cosh(x)) + -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((exp(x) - 2.0) + exp(-x)) <= 4e-6)
		tmp = (x ^ 2.0) + (0.08333333333333333 * (x ^ 4.0));
	else
		tmp = (2.0 * cosh(x)) + -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 4e-6], N[(N[Power[x, 2.0], $MachinePrecision] + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 3.99999999999999982e-6

   1. Initial program 49.1%

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

      1. associate-+l- 49.1%

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

      2. sub-neg 49.1%

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

      3. sub-neg 49.1%

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

      4. distribute-neg-in 49.1%

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

      5. remove-double-neg 49.1%

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

      6. +-commutative 49.1%

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

      7. metadata-eval 49.1%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. pow-prod-up 100.0%

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

      6. metadata-eval 100.0%

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

   9. Applied egg-rr 100.0%

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

   if 3.99999999999999982e-6 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

      2. sub-neg 99.8%

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

      3. sub-neg 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. remove-double-neg 99.8%

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

      6. +-commutative 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 99.8%

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

      1. sub-neg 99.8%

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

      2. *-rgt-identity 99.8%

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

      3. metadata-eval 99.8%

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

      4. associate-/l* 99.8%

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

      5. associate-*l/ 99.8%

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

      6. cosh-def 99.8%

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

      7. *-commutative 99.8%

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

      8. metadata-eval 99.8%

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

      9. fma-undefine 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, \cosh x, -2\right)} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \cosh x, -2\right)} \]
   8. Step-by-step derivation

      1. fma-undefine 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;{x}^{2} + 0.08333333333333333 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x + -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (- (exp x) 2.0) (exp (- x))) 4e-6)
   (fma x x (* 0.08333333333333333 (pow x 4.0)))
   (+ (* 2.0 (cosh x)) -2.0)))

C

double code(double x) {
	double tmp;
	if (((exp(x) - 2.0) + exp(-x)) <= 4e-6) {
		tmp = fma(x, x, (0.08333333333333333 * pow(x, 4.0)));
	} else {
		tmp = (2.0 * cosh(x)) + -2.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) - 2.0) + exp(Float64(-x))) <= 4e-6)
		tmp = fma(x, x, Float64(0.08333333333333333 * (x ^ 4.0)));
	else
		tmp = Float64(Float64(2.0 * cosh(x)) + -2.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 4e-6], N[(x * x + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 3.99999999999999982e-6

   1. Initial program 49.1%

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

      1. associate-+l- 49.1%

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

      2. sub-neg 49.1%

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

      3. sub-neg 49.1%

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

      4. distribute-neg-in 49.1%

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

      5. remove-double-neg 49.1%

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

      6. +-commutative 49.1%

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

      7. metadata-eval 49.1%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. unpow2 100.0%

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

      4. fma-define 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. pow-prod-up 100.0%

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

      8. metadata-eval 100.0%

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

   9. Applied egg-rr 100.0%

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

   if 3.99999999999999982e-6 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

      2. sub-neg 99.8%

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

      3. sub-neg 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. remove-double-neg 99.8%

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

      6. +-commutative 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 99.8%

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

      1. sub-neg 99.8%

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

      2. *-rgt-identity 99.8%

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

      3. metadata-eval 99.8%

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

      4. associate-/l* 99.8%

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

      5. associate-*l/ 99.8%

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

      6. cosh-def 99.8%

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

      7. *-commutative 99.8%

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

      8. metadata-eval 99.8%

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

      9. fma-undefine 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, \cosh x, -2\right)} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \cosh x, -2\right)} \]
   8. Step-by-step derivation

      1. fma-undefine 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (- (exp x) 2.0) (exp (- x))) 5e-10)
   (pow x 2.0)
   (+ (* 2.0 (cosh x)) -2.0)))

C

double code(double x) {
	double tmp;
	if (((exp(x) - 2.0) + exp(-x)) <= 5e-10) {
		tmp = pow(x, 2.0);
	} else {
		tmp = (2.0 * cosh(x)) + -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((exp(x) - 2.0d0) + exp(-x)) <= 5d-10) then
        tmp = x ** 2.0d0
    else
        tmp = (2.0d0 * cosh(x)) + (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((Math.exp(x) - 2.0) + Math.exp(-x)) <= 5e-10) {
		tmp = Math.pow(x, 2.0);
	} else {
		tmp = (2.0 * Math.cosh(x)) + -2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((math.exp(x) - 2.0) + math.exp(-x)) <= 5e-10:
		tmp = math.pow(x, 2.0)
	else:
		tmp = (2.0 * math.cosh(x)) + -2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) - 2.0) + exp(Float64(-x))) <= 5e-10)
		tmp = x ^ 2.0;
	else
		tmp = Float64(Float64(2.0 * cosh(x)) + -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((exp(x) - 2.0) + exp(-x)) <= 5e-10)
		tmp = x ^ 2.0;
	else
		tmp = (2.0 * cosh(x)) + -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 5e-10], N[Power[x, 2.0], $MachinePrecision], N[(N[(2.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 5.00000000000000031e-10

   1. Initial program 49.0%

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

      1. associate-+l- 49.0%

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

      2. sub-neg 49.0%

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

      3. sub-neg 49.0%

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

      4. distribute-neg-in 49.0%

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

      5. remove-double-neg 49.0%

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

      6. +-commutative 49.0%

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

      7. metadata-eval 49.0%

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

   3. Simplified 49.0%

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

   5. Taylor expanded in x around 0 99.9%

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

   if 5.00000000000000031e-10 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 96.8%

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

      1. associate-+l- 97.1%

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

      2. sub-neg 97.1%

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

      3. sub-neg 97.1%

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

      4. distribute-neg-in 97.1%

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

      5. remove-double-neg 97.1%

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

      6. +-commutative 97.1%

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

      7. metadata-eval 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around inf 97.1%

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

      1. sub-neg 97.1%

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

      2. *-rgt-identity 97.1%

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

      3. metadata-eval 97.1%

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

      4. associate-/l* 97.1%

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

      5. associate-*l/ 97.1%

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

      6. cosh-def 97.1%

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

      7. *-commutative 97.1%

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

      8. metadata-eval 97.1%

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

      9. fma-undefine 97.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, \cosh x, -2\right)} \]

   7. Simplified 97.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \cosh x, -2\right)} \]
   8. Step-by-step derivation

      1. fma-undefine 97.1%

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

   9. Applied egg-rr 97.1%

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

4. Final simplification 99.8%

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

Alternative 5: 98.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {x}^{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow x 2.0))

C

double code(double x) {
	return pow(x, 2.0);
}

Fortran

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

Java

public static double code(double x) {
	return Math.pow(x, 2.0);
}

Python

def code(x):
	return math.pow(x, 2.0)

Julia

function code(x)
	return x ^ 2.0
end

MATLAB

function tmp = code(x)
	tmp = x ^ 2.0;
end

Wolfram

code[x_] := N[Power[x, 2.0], $MachinePrecision]

Derivation

1. Initial program 50.6%

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

   1. associate-+l- 50.6%

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

   2. sub-neg 50.6%

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

   3. sub-neg 50.6%

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

   4. distribute-neg-in 50.6%

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

   5. remove-double-neg 50.6%

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

   6. +-commutative 50.6%

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

   7. metadata-eval 50.6%

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

3. Simplified 50.6%

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

5. Taylor expanded in x around 0 97.0%

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

6. Final simplification 97.0%

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

Alternative 6: 5.9% accurate, 41.2× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* x (+ x 2.0)))

C

double code(double x) {
	return x * (x + 2.0);
}

Fortran

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

Java

public static double code(double x) {
	return x * (x + 2.0);
}

Python

def code(x):
	return x * (x + 2.0)

Julia

function code(x)
	return Float64(x * Float64(x + 2.0))
end

MATLAB

function tmp = code(x)
	tmp = x * (x + 2.0);
end

Wolfram

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

Derivation

1. Initial program 50.6%

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

   1. associate-+l- 50.6%

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

   2. sub-neg 50.6%

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

   3. sub-neg 50.6%

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

   4. distribute-neg-in 50.6%

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

   5. remove-double-neg 50.6%

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

   6. +-commutative 50.6%

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

   7. metadata-eval 50.6%

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

3. Simplified 50.6%

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

   1. *-un-lft-identity 50.6%

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

   2. *-commutative 50.6%

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

   3. fma-define 50.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{x}, 1, e^{-x} + -2\right)} \]

   4. add-sqr-sqrt 25.7%

      \[\leadsto \mathsf{fma}\left(e^{x}, 1, e^{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}} + -2\right) \]

   5. sqrt-unprod 48.2%

      \[\leadsto \mathsf{fma}\left(e^{x}, 1, e^{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}} + -2\right) \]

   6. sqr-neg 48.2%

      \[\leadsto \mathsf{fma}\left(e^{x}, 1, e^{\sqrt{\color{blue}{x \cdot x}}} + -2\right) \]

   7. sqrt-unprod 22.4%

      \[\leadsto \mathsf{fma}\left(e^{x}, 1, e^{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} + -2\right) \]

   8. add-sqr-sqrt 47.6%

      \[\leadsto \mathsf{fma}\left(e^{x}, 1, e^{\color{blue}{x}} + -2\right) \]

6. Applied egg-rr 47.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(e^{x}, 1, e^{x} + -2\right)} \]
7. Step-by-step derivation

   1. fma-undefine 47.6%

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

   2. *-rgt-identity 47.6%

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

   3. associate-+r+ 47.6%

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

   4. count-2 47.6%

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

   5. *-commutative 47.6%

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

   6. metadata-eval 47.6%

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

   7. distribute-rgt-neg-in 47.6%

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

   8. +-commutative 47.6%

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

   9. *-commutative 47.6%

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

   10. distribute-lft-neg-in 47.6%

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

   11. metadata-eval 47.6%

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

8. Simplified 47.6%

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

9. Taylor expanded in x around 0 5.8%

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

    1. +-commutative 5.8%

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

11. Simplified 5.8%

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

12. Final simplification 5.8%

    \[\leadsto x \cdot \left(x + 2\right) \]
13. Add Preprocessing

Alternative 7: 5.9% accurate, 68.7× speedup

Math

\[\begin{array}{l} \\ x \cdot 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 2.0))

C

double code(double x) {
	return x * 2.0;
}

Fortran

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

Java

public static double code(double x) {
	return x * 2.0;
}

Python

def code(x):
	return x * 2.0

Julia

function code(x)
	return Float64(x * 2.0)
end

MATLAB

function tmp = code(x)
	tmp = x * 2.0;
end

Wolfram

code[x_] := N[(x * 2.0), $MachinePrecision]

Derivation

1. Initial program 50.6%

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

   1. associate-+l- 50.6%

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

   2. sub-neg 50.6%

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

   3. sub-neg 50.6%

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

   4. distribute-neg-in 50.6%

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

   5. remove-double-neg 50.6%

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

   6. +-commutative 50.6%

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

   7. metadata-eval 50.6%

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

3. Simplified 50.6%

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

   1. *-un-lft-identity 50.6%

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

   2. *-commutative 50.6%

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

   3. fma-define 50.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{x}, 1, e^{-x} + -2\right)} \]

   4. add-sqr-sqrt 25.7%

      \[\leadsto \mathsf{fma}\left(e^{x}, 1, e^{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}} + -2\right) \]

   5. sqrt-unprod 48.2%

      \[\leadsto \mathsf{fma}\left(e^{x}, 1, e^{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}} + -2\right) \]

   6. sqr-neg 48.2%

      \[\leadsto \mathsf{fma}\left(e^{x}, 1, e^{\sqrt{\color{blue}{x \cdot x}}} + -2\right) \]

   7. sqrt-unprod 22.4%

      \[\leadsto \mathsf{fma}\left(e^{x}, 1, e^{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} + -2\right) \]

   8. add-sqr-sqrt 47.6%

      \[\leadsto \mathsf{fma}\left(e^{x}, 1, e^{\color{blue}{x}} + -2\right) \]

6. Applied egg-rr 47.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(e^{x}, 1, e^{x} + -2\right)} \]
7. Step-by-step derivation

   1. fma-undefine 47.6%

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

   2. *-rgt-identity 47.6%

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

   3. associate-+r+ 47.6%

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

   4. count-2 47.6%

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

   5. *-commutative 47.6%

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

   6. metadata-eval 47.6%

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

   7. distribute-rgt-neg-in 47.6%

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

   8. +-commutative 47.6%

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

   9. *-commutative 47.6%

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

   10. distribute-lft-neg-in 47.6%

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

   11. metadata-eval 47.6%

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

8. Simplified 47.6%

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

9. Taylor expanded in x around 0 5.8%

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

    1. *-commutative 5.8%

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

11. Simplified 5.8%

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

12. Final simplification 5.8%

    \[\leadsto x \cdot 2 \]
13. Add Preprocessing

Alternative 8: 51.8% accurate, 206.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 50.6%

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

   1. associate-+l- 50.6%

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

   2. sub-neg 50.6%

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

   3. sub-neg 50.6%

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

   4. distribute-neg-in 50.6%

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

   5. remove-double-neg 50.6%

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

   6. +-commutative 50.6%

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

   7. metadata-eval 50.6%

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

3. Simplified 50.6%

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

5. Taylor expanded in x around inf 50.7%

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

   1. sub-neg 50.7%

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

   2. *-rgt-identity 50.7%

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

   3. metadata-eval 50.7%

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

   4. associate-/l* 50.7%

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

   5. associate-*l/ 50.7%

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

   6. cosh-def 50.7%

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

   7. *-commutative 50.7%

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

   8. metadata-eval 50.7%

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

   9. fma-undefine 50.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \cosh x, -2\right)} \]

7. Simplified 50.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, \cosh x, -2\right)} \]
8. Step-by-step derivation

   1. fma-undefine 50.7%

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

9. Applied egg-rr 50.7%

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

10. Taylor expanded in x around 0 47.1%

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

11. Final simplification 47.1%

    \[\leadsto 0 \]
12. Add Preprocessing

Developer target: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024055 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64
  :pre (<= (fabs x) 710.0)

  :alt
  (* 4.0 (* (sinh (/ x 2.0)) (sinh (/ x 2.0))))

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