exp2 (problem 3.3.7)

Percentage Accurate: 53.5% → 99.9%

Time: 10.7s

Alternatives: 6

Speedup: 68.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 0.001:\\ \;\;\;\;{x}^{2} \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot \left(0.002777777777777778 + \left(x \cdot x\right) \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (- (exp x) 2.0) (exp (- x))) 0.001)
   (*
    (pow x 2.0)
    (+
     1.0
     (*
      (* x x)
      (+
       0.08333333333333333
       (* (* x x) (+ 0.002777777777777778 (* (* x x) 4.96031746031746e-5)))))))
   (- (* 2.0 (cosh x)) 2.0)))

C

double code(double x) {
	double tmp;
	if (((exp(x) - 2.0) + exp(-x)) <= 0.001) {
		tmp = pow(x, 2.0) * (1.0 + ((x * x) * (0.08333333333333333 + ((x * x) * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5))))));
	} 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)) <= 0.001d0) then
        tmp = (x ** 2.0d0) * (1.0d0 + ((x * x) * (0.08333333333333333d0 + ((x * x) * (0.002777777777777778d0 + ((x * x) * 4.96031746031746d-5))))))
    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)) <= 0.001) {
		tmp = Math.pow(x, 2.0) * (1.0 + ((x * x) * (0.08333333333333333 + ((x * x) * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5))))));
	} 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)) <= 0.001:
		tmp = math.pow(x, 2.0) * (1.0 + ((x * x) * (0.08333333333333333 + ((x * x) * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5))))))
	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))) <= 0.001)
		tmp = Float64((x ^ 2.0) * Float64(1.0 + Float64(Float64(x * x) * Float64(0.08333333333333333 + Float64(Float64(x * x) * Float64(0.002777777777777778 + Float64(Float64(x * x) * 4.96031746031746e-5)))))));
	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)) <= 0.001)
		tmp = (x ^ 2.0) * (1.0 + ((x * x) * (0.08333333333333333 + ((x * x) * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5))))));
	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], 0.001], N[(N[Power[x, 2.0], $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.08333333333333333 + N[(N[(x * x), $MachinePrecision] * N[(0.002777777777777778 + N[(N[(x * x), $MachinePrecision] * 4.96031746031746e-5), $MachinePrecision]), $MachinePrecision]), $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) #s(literal 2 binary64)) (exp.f64 (neg.f64 x))) < 1e-3

   1. Initial program 49.7%

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

      1. associate-+l- 49.7%

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

      2. sub-neg 49.7%

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

      3. sub-neg 49.7%

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

      4. distribute-neg-in 49.7%

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

      5. remove-double-neg 49.7%

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

      6. +-commutative 49.7%

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

      7. metadata-eval 49.7%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. unpow2 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. unpow2 100.0%

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

   11. Applied egg-rr 100.0%

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

       1. unpow2 100.0%

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

   13. Applied egg-rr 100.0%

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

   if 1e-3 < (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x)))

   1. Initial program 98.4%

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

      1. associate-+l- 98.4%

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

      2. sub-neg 98.4%

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

      3. sub-neg 98.4%

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

      4. distribute-neg-in 98.4%

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

      5. remove-double-neg 98.4%

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

      6. +-commutative 98.4%

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

      7. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

      1. +-commutative 98.4%

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

      2. associate-+r+ 98.4%

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

      3. metadata-eval 98.4%

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

      4. sub-neg 98.4%

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

      5. +-commutative 98.4%

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

      6. associate-+r- 98.4%

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

      7. +-commutative 98.4%

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

      8. cosh-undef 98.4%

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

   6. Applied egg-rr 98.4%

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

Alternative 2: 99.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.035:\\ \;\;\;\;{x}^{2} \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot 0.002777777777777778\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.035)
   (*
    (pow x 2.0)
    (+
     1.0
     (* (* x x) (+ 0.08333333333333333 (* (* x x) 0.002777777777777778)))))
   (- (* 2.0 (cosh x)) 2.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.035) {
		tmp = pow(x, 2.0) * (1.0 + ((x * x) * (0.08333333333333333 + ((x * x) * 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 (x <= 0.035d0) then
        tmp = (x ** 2.0d0) * (1.0d0 + ((x * x) * (0.08333333333333333d0 + ((x * x) * 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 (x <= 0.035) {
		tmp = Math.pow(x, 2.0) * (1.0 + ((x * x) * (0.08333333333333333 + ((x * x) * 0.002777777777777778))));
	} else {
		tmp = (2.0 * Math.cosh(x)) - 2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.035:
		tmp = math.pow(x, 2.0) * (1.0 + ((x * x) * (0.08333333333333333 + ((x * x) * 0.002777777777777778))))
	else:
		tmp = (2.0 * math.cosh(x)) - 2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.035)
		tmp = Float64((x ^ 2.0) * Float64(1.0 + Float64(Float64(x * x) * Float64(0.08333333333333333 + Float64(Float64(x * x) * 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 (x <= 0.035)
		tmp = (x ^ 2.0) * (1.0 + ((x * x) * (0.08333333333333333 + ((x * x) * 0.002777777777777778))));
	else
		tmp = (2.0 * cosh(x)) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.035], N[(N[Power[x, 2.0], $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.08333333333333333 + N[(N[(x * x), $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 x < 0.035000000000000003

   1. Initial program 50.3%

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

      1. associate-+l- 50.3%

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

      2. sub-neg 50.3%

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

      3. sub-neg 50.3%

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

      4. distribute-neg-in 50.3%

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

      5. remove-double-neg 50.3%

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

      6. +-commutative 50.3%

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

      7. metadata-eval 50.3%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in x around 0 99.0%

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

      1. *-commutative 99.0%

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

   7. Simplified 99.0%

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

      1. unpow2 99.1%

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

   9. Applied egg-rr 99.0%

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

       1. unpow2 99.1%

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

   11. Applied egg-rr 99.0%

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

   if 0.035000000000000003 < 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 3: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0062:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0062)
   (+ (* x x) (* 0.08333333333333333 (pow x 4.0)))
   (- (* 2.0 (cosh x)) 2.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0062) {
		tmp = (x * x) + (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 (x <= 0.0062d0) then
        tmp = (x * x) + (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 (x <= 0.0062) {
		tmp = (x * x) + (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 x <= 0.0062:
		tmp = (x * x) + (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 (x <= 0.0062)
		tmp = Float64(Float64(x * x) + 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 (x <= 0.0062)
		tmp = (x * x) + (0.08333333333333333 * (x ^ 4.0));
	else
		tmp = (2.0 * cosh(x)) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0062], N[(N[(x * x), $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 x < 0.00619999999999999978

   1. Initial program 50.3%

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

      1. associate-+l- 50.3%

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

      2. sub-neg 50.3%

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

      3. sub-neg 50.3%

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

      4. distribute-neg-in 50.3%

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

      5. remove-double-neg 50.3%

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

      6. +-commutative 50.3%

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

      7. metadata-eval 50.3%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in x around 0 98.8%

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

      1. distribute-rgt-in 98.8%

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

      2. *-lft-identity 98.8%

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

      3. associate-*l* 98.8%

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

      4. pow-sqr 98.8%

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

      5. metadata-eval 98.8%

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

   7. Simplified 98.8%

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

      1. unpow2 99.1%

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

   9. Applied egg-rr 98.8%

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

   if 0.00619999999999999978 < 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 4: 98.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.000145:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.000145) (* x x) (- (* 2.0 (cosh x)) 2.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.000145) {
		tmp = x * x;
	} 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 (x <= 0.000145d0) then
        tmp = x * x
    else
        tmp = (2.0d0 * cosh(x)) - 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.000145) {
		tmp = x * x;
	} else {
		tmp = (2.0 * Math.cosh(x)) - 2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.000145:
		tmp = x * x
	else:
		tmp = (2.0 * math.cosh(x)) - 2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.000145)
		tmp = Float64(x * x);
	else
		tmp = Float64(Float64(2.0 * cosh(x)) - 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.000145)
		tmp = x * x;
	else
		tmp = (2.0 * cosh(x)) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.000145], N[(x * x), $MachinePrecision], N[(N[(2.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.45e-4

   1. Initial program 50.3%

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

      1. associate-+l- 50.3%

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

      2. sub-neg 50.3%

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

      3. sub-neg 50.3%

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

      4. distribute-neg-in 50.3%

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

      5. remove-double-neg 50.3%

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

      6. +-commutative 50.3%

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

      7. metadata-eval 50.3%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in x around 0 97.8%

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

      1. unpow2 99.1%

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

   7. Applied egg-rr 97.8%

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

   if 1.45e-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 5: 98.2% accurate, 68.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.8%

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

   1. associate-+l- 50.8%

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

   2. sub-neg 50.8%

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

   3. sub-neg 50.8%

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

   4. distribute-neg-in 50.8%

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

   5. remove-double-neg 50.8%

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

   6. +-commutative 50.8%

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

   7. metadata-eval 50.8%

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

3. Simplified 50.8%

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

5. Taylor expanded in x around 0 96.9%

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

   1. unpow2 98.4%

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

7. Applied egg-rr 96.9%

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

Alternative 6: 5.9% accurate, 206.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 50.8%

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

   1. associate-+l- 50.8%

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

   2. sub-neg 50.8%

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

   3. sub-neg 50.8%

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

   4. distribute-neg-in 50.8%

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

   5. remove-double-neg 50.8%

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

   6. +-commutative 50.8%

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

   7. metadata-eval 50.8%

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

3. Simplified 50.8%

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

5. Taylor expanded in x around 0 47.3%

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

6. Taylor expanded in x around 0 5.8%

   \[\leadsto \color{blue}{x} \]
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 2024144 
(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))))