exp2 (problem 3.3.7)

Percentage Accurate: 77.1% → 99.9%

Time: 8.6s

Alternatives: 7

Speedup: 18.7×

• 49.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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: 77.1% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;0.002777777777777778 \cdot {x}^{6} + \left(x \cdot x\right) \cdot \left(0.08333333333333333 \cdot \left(x \cdot x\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (- (exp x) 2.0) (exp (- x))) 4e-7)
   (+
    (* 0.002777777777777778 (pow x 6.0))
    (* (* x x) (+ (* 0.08333333333333333 (* x x)) 1.0)))
   (- (* 2.0 (cosh x)) 2.0)))

C

double code(double x) {
	double tmp;
	if (((exp(x) - 2.0) + exp(-x)) <= 4e-7) {
		tmp = (0.002777777777777778 * pow(x, 6.0)) + ((x * x) * ((0.08333333333333333 * (x * x)) + 1.0));
	} 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-7) then
        tmp = (0.002777777777777778d0 * (x ** 6.0d0)) + ((x * x) * ((0.08333333333333333d0 * (x * x)) + 1.0d0))
    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-7) {
		tmp = (0.002777777777777778 * Math.pow(x, 6.0)) + ((x * x) * ((0.08333333333333333 * (x * x)) + 1.0));
	} 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-7:
		tmp = (0.002777777777777778 * math.pow(x, 6.0)) + ((x * x) * ((0.08333333333333333 * (x * x)) + 1.0))
	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-7)
		tmp = Float64(Float64(0.002777777777777778 * (x ^ 6.0)) + Float64(Float64(x * x) * Float64(Float64(0.08333333333333333 * Float64(x * x)) + 1.0)));
	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-7)
		tmp = (0.002777777777777778 * (x ^ 6.0)) + ((x * x) * ((0.08333333333333333 * (x * x)) + 1.0));
	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-7], N[(N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(N[(0.08333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $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))) < 3.9999999999999998e-7

   1. Initial program 64.0%

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

      1. +-commutative 64.0%

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

      2. sub-neg 64.0%

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

      3. +-commutative 64.0%

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

      4. associate-+l+ 64.0%

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

      5. sub-neg 64.0%

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

      6. remove-double-neg 64.0%

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

      7. sub-neg 64.0%

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

      8. neg-sub0 64.0%

         \[\leadsto \left(e^{-x} - 2\right) - \color{blue}{\left(0 - e^{x}\right)} \]

      9. associate--r- 64.0%

         \[\leadsto \color{blue}{\left(\left(e^{-x} - 2\right) - 0\right) + e^{x}} \]

      10. +-commutative 64.0%

          \[\leadsto \color{blue}{e^{x} + \left(\left(e^{-x} - 2\right) - 0\right)} \]

      11. --rgt-identity 64.0%

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

      12. sub-neg 64.0%

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

      13. metadata-eval 64.0%

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

   3. Simplified 64.0%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. metadata-eval 100.0%

         \[\leadsto 0.002777777777777778 \cdot {x}^{6} + \left(0.08333333333333333 \cdot {x}^{\color{blue}{\left(2 + 2\right)}} + {x}^{2}\right) \]

      2. pow-prod-up 100.0%

         \[\leadsto 0.002777777777777778 \cdot {x}^{6} + \left(0.08333333333333333 \cdot \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} + {x}^{2}\right) \]

      3. associate-*r* 100.0%

         \[\leadsto 0.002777777777777778 \cdot {x}^{6} + \left(\color{blue}{\left(0.08333333333333333 \cdot {x}^{2}\right) \cdot {x}^{2}} + {x}^{2}\right) \]

      4. distribute-lft1-in 100.0%

         \[\leadsto 0.002777777777777778 \cdot {x}^{6} + \color{blue}{\left(0.08333333333333333 \cdot {x}^{2} + 1\right) \cdot {x}^{2}} \]

      5. unpow2 100.0%

         \[\leadsto 0.002777777777777778 \cdot {x}^{6} + \left(0.08333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} + 1\right) \cdot {x}^{2} \]

      6. unpow2 100.0%

         \[\leadsto 0.002777777777777778 \cdot {x}^{6} + \left(0.08333333333333333 \cdot \left(x \cdot x\right) + 1\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   6. Applied egg-rr 100.0%

      \[\leadsto 0.002777777777777778 \cdot {x}^{6} + \color{blue}{\left(0.08333333333333333 \cdot \left(x \cdot x\right) + 1\right) \cdot \left(x \cdot x\right)} \]

   if 3.9999999999999998e-7 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+l+ 100.0%

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

      5. sub-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. neg-sub0 100.0%

         \[\leadsto \left(e^{-x} - 2\right) - \color{blue}{\left(0 - e^{x}\right)} \]

      9. associate--r- 100.0%

         \[\leadsto \color{blue}{\left(\left(e^{-x} - 2\right) - 0\right) + e^{x}} \]

      10. +-commutative 100.0%

          \[\leadsto \color{blue}{e^{x} + \left(\left(e^{-x} - 2\right) - 0\right)} \]

      11. --rgt-identity 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. associate-+r+ 100.0%

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

      2. metadata-eval 100.0%

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

      3. sub-neg 100.0%

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

      4. cosh-undef 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;0.002777777777777778 \cdot {x}^{6} + \left(x \cdot x\right) \cdot \left(0.08333333333333333 \cdot \left(x \cdot x\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \]

Alternative 2: 94.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.006:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.08333333333333333 \cdot \left(x \cdot x\right) - -1\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.006)
   (* (* x x) (- (* 0.08333333333333333 (* x x)) -1.0))
   (- (* 2.0 (cosh x)) 2.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.006) {
		tmp = (x * x) * ((0.08333333333333333 * (x * x)) - -1.0);
	} 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 <= 0.006d0) then
        tmp = (x * x) * ((0.08333333333333333d0 * (x * x)) - (-1.0d0))
    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 <= 0.006) {
		tmp = (x * x) * ((0.08333333333333333 * (x * x)) - -1.0);
	} else {
		tmp = (2.0 * Math.cosh(x)) - 2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.006:
		tmp = (x * x) * ((0.08333333333333333 * (x * x)) - -1.0)
	else:
		tmp = (2.0 * math.cosh(x)) - 2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.006)
		tmp = Float64(Float64(x * x) * Float64(Float64(0.08333333333333333 * Float64(x * x)) - -1.0));
	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 <= 0.006)
		tmp = (x * x) * ((0.08333333333333333 * (x * x)) - -1.0);
	else
		tmp = (2.0 * cosh(x)) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.006], N[(N[(x * x), $MachinePrecision] * N[(N[(0.08333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0060000000000000001

   1. Initial program 76.8%

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

      1. +-commutative 76.8%

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

      2. sub-neg 76.8%

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

      3. +-commutative 76.8%

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

      4. associate-+l+ 76.8%

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

      5. sub-neg 76.8%

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

      6. remove-double-neg 76.8%

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

      7. sub-neg 76.8%

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

      8. neg-sub0 76.8%

         \[\leadsto \left(e^{-x} - 2\right) - \color{blue}{\left(0 - e^{x}\right)} \]

      9. associate--r- 76.8%

         \[\leadsto \color{blue}{\left(\left(e^{-x} - 2\right) - 0\right) + e^{x}} \]

      10. +-commutative 76.8%

          \[\leadsto \color{blue}{e^{x} + \left(\left(e^{-x} - 2\right) - 0\right)} \]

      11. --rgt-identity 76.8%

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

      12. sub-neg 76.8%

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

      13. metadata-eval 76.8%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in x around 0 92.2%

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

      1. fma-def 92.2%

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

      2. unpow2 92.2%

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

   6. Simplified 92.2%

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

      1. fma-udef 92.2%

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

      2. unpow2 92.2%

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

      3. +-commutative 92.2%

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

      4. unpow2 92.2%

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

   8. Applied egg-rr 92.2%

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

      1. metadata-eval 92.2%

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

      2. pow-sqr 92.2%

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

      3. pow2 92.2%

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

      4. pow2 92.2%

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

      5. associate-*r* 92.2%

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

      6. distribute-rgt1-in 92.2%

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

      7. fma-def 92.2%

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

   10. Applied egg-rr 92.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right) \cdot \left(x \cdot x\right)} \]
   11. Step-by-step derivation

       1. metadata-eval 92.2%

          \[\leadsto \mathsf{fma}\left(0.08333333333333333, x \cdot x, \color{blue}{--1}\right) \cdot \left(x \cdot x\right) \]

       2. metadata-eval 92.2%

          \[\leadsto \mathsf{fma}\left(0.08333333333333333, x \cdot x, -\color{blue}{\left(-1\right)}\right) \cdot \left(x \cdot x\right) \]

       3. fma-neg 92.2%

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

       4. metadata-eval 92.2%

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

   12. Applied egg-rr 92.2%

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

   if 0.0060000000000000001 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+l+ 100.0%

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

      5. sub-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. neg-sub0 100.0%

         \[\leadsto \left(e^{-x} - 2\right) - \color{blue}{\left(0 - e^{x}\right)} \]

      9. associate--r- 100.0%

         \[\leadsto \color{blue}{\left(\left(e^{-x} - 2\right) - 0\right) + e^{x}} \]

      10. +-commutative 100.0%

          \[\leadsto \color{blue}{e^{x} + \left(\left(e^{-x} - 2\right) - 0\right)} \]

      11. --rgt-identity 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. associate-+r+ 100.0%

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

      2. metadata-eval 100.0%

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

      3. sub-neg 100.0%

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

      4. cosh-undef 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.006:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.08333333333333333 \cdot \left(x \cdot x\right) - -1\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \]

Alternative 3: 93.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.08333333333333333 \cdot \left(x \cdot x\right) - -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.6)
   (* (* x x) (- (* 0.08333333333333333 (* x x)) -1.0))
   (expm1 x)))

C

double code(double x) {
	double tmp;
	if (x <= 2.6) {
		tmp = (x * x) * ((0.08333333333333333 * (x * x)) - -1.0);
	} else {
		tmp = expm1(x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.6) {
		tmp = (x * x) * ((0.08333333333333333 * (x * x)) - -1.0);
	} else {
		tmp = Math.expm1(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.6:
		tmp = (x * x) * ((0.08333333333333333 * (x * x)) - -1.0)
	else:
		tmp = math.expm1(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.6)
		tmp = Float64(Float64(x * x) * Float64(Float64(0.08333333333333333 * Float64(x * x)) - -1.0));
	else
		tmp = expm1(x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.6], N[(N[(x * x), $MachinePrecision] * N[(N[(0.08333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(Exp[x] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.60000000000000009

   1. Initial program 76.8%

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

      1. +-commutative 76.8%

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

      2. sub-neg 76.8%

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

      3. +-commutative 76.8%

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

      4. associate-+l+ 76.8%

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

      5. sub-neg 76.8%

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

      6. remove-double-neg 76.8%

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

      7. sub-neg 76.8%

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

      8. neg-sub0 76.8%

         \[\leadsto \left(e^{-x} - 2\right) - \color{blue}{\left(0 - e^{x}\right)} \]

      9. associate--r- 76.8%

         \[\leadsto \color{blue}{\left(\left(e^{-x} - 2\right) - 0\right) + e^{x}} \]

      10. +-commutative 76.8%

          \[\leadsto \color{blue}{e^{x} + \left(\left(e^{-x} - 2\right) - 0\right)} \]

      11. --rgt-identity 76.8%

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

      12. sub-neg 76.8%

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

      13. metadata-eval 76.8%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in x around 0 92.2%

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

      1. fma-def 92.2%

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

      2. unpow2 92.2%

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

   6. Simplified 92.2%

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

      1. fma-udef 92.2%

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

      2. unpow2 92.2%

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

      3. +-commutative 92.2%

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

      4. unpow2 92.2%

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

   8. Applied egg-rr 92.2%

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

      1. metadata-eval 92.2%

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

      2. pow-sqr 92.2%

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

      3. pow2 92.2%

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

      4. pow2 92.2%

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

      5. associate-*r* 92.2%

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

      6. distribute-rgt1-in 92.2%

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

      7. fma-def 92.2%

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

   10. Applied egg-rr 92.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right) \cdot \left(x \cdot x\right)} \]
   11. Step-by-step derivation

       1. metadata-eval 92.2%

          \[\leadsto \mathsf{fma}\left(0.08333333333333333, x \cdot x, \color{blue}{--1}\right) \cdot \left(x \cdot x\right) \]

       2. metadata-eval 92.2%

          \[\leadsto \mathsf{fma}\left(0.08333333333333333, x \cdot x, -\color{blue}{\left(-1\right)}\right) \cdot \left(x \cdot x\right) \]

       3. fma-neg 92.2%

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

       4. metadata-eval 92.2%

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

   12. Applied egg-rr 92.2%

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

   if 2.60000000000000009 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+l+ 100.0%

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

      5. sub-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. neg-sub0 100.0%

         \[\leadsto \left(e^{-x} - 2\right) - \color{blue}{\left(0 - e^{x}\right)} \]

      9. associate--r- 100.0%

         \[\leadsto \color{blue}{\left(\left(e^{-x} - 2\right) - 0\right) + e^{x}} \]

      10. +-commutative 100.0%

          \[\leadsto \color{blue}{e^{x} + \left(\left(e^{-x} - 2\right) - 0\right)} \]

      11. --rgt-identity 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.1%

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

   5. Taylor expanded in x around inf 99.1%

      \[\leadsto \color{blue}{e^{x} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 99.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right)} \]

   7. Simplified 99.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.08333333333333333 \cdot \left(x \cdot x\right) - -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 4: 81.3% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.08333333333333333 \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.5) (* x x) (* (* x x) (* 0.08333333333333333 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= 3.5) {
		tmp = x * x;
	} else {
		tmp = (x * x) * (0.08333333333333333 * (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.5d0) then
        tmp = x * x
    else
        tmp = (x * x) * (0.08333333333333333d0 * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.5) {
		tmp = x * x;
	} else {
		tmp = (x * x) * (0.08333333333333333 * (x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.5:
		tmp = x * x
	else:
		tmp = (x * x) * (0.08333333333333333 * (x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.5)
		tmp = Float64(x * x);
	else
		tmp = Float64(Float64(x * x) * Float64(0.08333333333333333 * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.5)
		tmp = x * x;
	else
		tmp = (x * x) * (0.08333333333333333 * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.5], N[(x * x), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(0.08333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5

   1. Initial program 76.8%

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

      1. +-commutative 76.8%

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

      2. sub-neg 76.8%

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

      3. +-commutative 76.8%

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

      4. associate-+l+ 76.8%

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

      5. sub-neg 76.8%

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

      6. remove-double-neg 76.8%

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

      7. sub-neg 76.8%

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

      8. neg-sub0 76.8%

         \[\leadsto \left(e^{-x} - 2\right) - \color{blue}{\left(0 - e^{x}\right)} \]

      9. associate--r- 76.8%

         \[\leadsto \color{blue}{\left(\left(e^{-x} - 2\right) - 0\right) + e^{x}} \]

      10. +-commutative 76.8%

          \[\leadsto \color{blue}{e^{x} + \left(\left(e^{-x} - 2\right) - 0\right)} \]

      11. --rgt-identity 76.8%

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

      12. sub-neg 76.8%

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

      13. metadata-eval 76.8%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in x around 0 81.8%

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

      1. unpow2 81.8%

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

   6. Simplified 81.8%

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

   if 3.5 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+l+ 100.0%

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

      5. sub-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. neg-sub0 100.0%

         \[\leadsto \left(e^{-x} - 2\right) - \color{blue}{\left(0 - e^{x}\right)} \]

      9. associate--r- 100.0%

         \[\leadsto \color{blue}{\left(\left(e^{-x} - 2\right) - 0\right) + e^{x}} \]

      10. +-commutative 100.0%

          \[\leadsto \color{blue}{e^{x} + \left(\left(e^{-x} - 2\right) - 0\right)} \]

      11. --rgt-identity 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 67.6%

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

      1. fma-def 67.6%

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

      2. unpow2 67.6%

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

   6. Simplified 67.6%

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

      1. fma-udef 67.6%

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

      2. unpow2 67.6%

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

      3. +-commutative 67.6%

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

      4. unpow2 67.6%

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

   8. Applied egg-rr 67.6%

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

      1. metadata-eval 67.6%

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

      2. pow-sqr 67.6%

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

      3. pow2 67.6%

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

      4. pow2 67.6%

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

      5. associate-*r* 67.6%

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

      6. distribute-rgt1-in 67.6%

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

      7. fma-def 67.6%

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

   10. Applied egg-rr 67.6%

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

   11. Taylor expanded in x around inf 67.6%

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

       1. unpow2 67.6%

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

   13. Simplified 67.6%

       \[\leadsto \color{blue}{\left(0.08333333333333333 \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.08333333333333333 \cdot \left(x \cdot x\right)\right)\\ \end{array} \]

Alternative 5: 87.8% accurate, 18.7× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot x\right) \cdot \left(0.08333333333333333 \cdot \left(x \cdot x\right) - -1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (* x x) (- (* 0.08333333333333333 (* x x)) -1.0)))

C

double code(double x) {
	return (x * x) * ((0.08333333333333333 * (x * x)) - -1.0);
}

Fortran

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

Java

public static double code(double x) {
	return (x * x) * ((0.08333333333333333 * (x * x)) - -1.0);
}

Python

def code(x):
	return (x * x) * ((0.08333333333333333 * (x * x)) - -1.0)

Julia

function code(x)
	return Float64(Float64(x * x) * Float64(Float64(0.08333333333333333 * Float64(x * x)) - -1.0))
end

MATLAB

function tmp = code(x)
	tmp = (x * x) * ((0.08333333333333333 * (x * x)) - -1.0);
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * N[(N[(0.08333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.4%

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

   1. +-commutative 83.4%

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

   2. sub-neg 83.4%

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

   3. +-commutative 83.4%

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

   4. associate-+l+ 83.4%

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

   5. sub-neg 83.4%

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

   6. remove-double-neg 83.4%

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

   7. sub-neg 83.4%

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

   8. neg-sub0 83.4%

      \[\leadsto \left(e^{-x} - 2\right) - \color{blue}{\left(0 - e^{x}\right)} \]

   9. associate--r- 83.4%

      \[\leadsto \color{blue}{\left(\left(e^{-x} - 2\right) - 0\right) + e^{x}} \]

   10. +-commutative 83.4%

       \[\leadsto \color{blue}{e^{x} + \left(\left(e^{-x} - 2\right) - 0\right)} \]

   11. --rgt-identity 83.4%

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

   12. sub-neg 83.4%

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

   13. metadata-eval 83.4%

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

3. Simplified 83.4%

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

4. Taylor expanded in x around 0 85.2%

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

   1. fma-def 85.2%

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

   2. unpow2 85.2%

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

6. Simplified 85.2%

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

   1. fma-udef 85.2%

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

   2. unpow2 85.2%

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

   3. +-commutative 85.2%

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

   4. unpow2 85.2%

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

8. Applied egg-rr 85.2%

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

   1. metadata-eval 85.2%

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

   2. pow-sqr 85.2%

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

   3. pow2 85.2%

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

   4. pow2 85.2%

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

   5. associate-*r* 85.2%

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

   6. distribute-rgt1-in 85.2%

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

   7. fma-def 85.2%

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

10. Applied egg-rr 85.2%

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right) \cdot \left(x \cdot x\right)} \]
11. Step-by-step derivation

    1. metadata-eval 85.2%

       \[\leadsto \mathsf{fma}\left(0.08333333333333333, x \cdot x, \color{blue}{--1}\right) \cdot \left(x \cdot x\right) \]

    2. metadata-eval 85.2%

       \[\leadsto \mathsf{fma}\left(0.08333333333333333, x \cdot x, -\color{blue}{\left(-1\right)}\right) \cdot \left(x \cdot x\right) \]

    3. fma-neg 85.2%

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

    4. metadata-eval 85.2%

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

12. Applied egg-rr 85.2%

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

13. Final simplification 85.2%

    \[\leadsto \left(x \cdot x\right) \cdot \left(0.08333333333333333 \cdot \left(x \cdot x\right) - -1\right) \]

Alternative 6: 75.4% 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 83.4%

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

   1. +-commutative 83.4%

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

   2. sub-neg 83.4%

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

   3. +-commutative 83.4%

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

   4. associate-+l+ 83.4%

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

   5. sub-neg 83.4%

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

   6. remove-double-neg 83.4%

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

   7. sub-neg 83.4%

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

   8. neg-sub0 83.4%

      \[\leadsto \left(e^{-x} - 2\right) - \color{blue}{\left(0 - e^{x}\right)} \]

   9. associate--r- 83.4%

      \[\leadsto \color{blue}{\left(\left(e^{-x} - 2\right) - 0\right) + e^{x}} \]

   10. +-commutative 83.4%

       \[\leadsto \color{blue}{e^{x} + \left(\left(e^{-x} - 2\right) - 0\right)} \]

   11. --rgt-identity 83.4%

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

   12. sub-neg 83.4%

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

   13. metadata-eval 83.4%

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

3. Simplified 83.4%

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

4. Taylor expanded in x around 0 73.3%

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

   1. unpow2 73.3%

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

6. Simplified 73.3%

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

7. Final simplification 73.3%

   \[\leadsto x \cdot x \]

Alternative 7: 4.3% 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 83.4%

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

   1. +-commutative 83.4%

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

   2. sub-neg 83.4%

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

   3. +-commutative 83.4%

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

   4. associate-+l+ 83.4%

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

   5. sub-neg 83.4%

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

   6. remove-double-neg 83.4%

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

   7. sub-neg 83.4%

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

   8. neg-sub0 83.4%

      \[\leadsto \left(e^{-x} - 2\right) - \color{blue}{\left(0 - e^{x}\right)} \]

   9. associate--r- 83.4%

      \[\leadsto \color{blue}{\left(\left(e^{-x} - 2\right) - 0\right) + e^{x}} \]

   10. +-commutative 83.4%

       \[\leadsto \color{blue}{e^{x} + \left(\left(e^{-x} - 2\right) - 0\right)} \]

   11. --rgt-identity 83.4%

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

   12. sub-neg 83.4%

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

   13. metadata-eval 83.4%

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

3. Simplified 83.4%

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

4. Taylor expanded in x around 0 57.7%

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

5. Taylor expanded in x around 0 4.5%

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

6. Final simplification 4.5%

   \[\leadsto x \]

Developer target: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 4 \cdot {\sinh \left(\frac{x}{2}\right)}^{2} \end{array} \]

FPCore

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

C

double code(double x) {
	return 4.0 * pow(sinh((x / 2.0)), 2.0);
}

Fortran

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

Java

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

Python

def code(x):
	return 4.0 * math.pow(math.sinh((x / 2.0)), 2.0)

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(4.0 * N[Power[N[Sinh[N[(x / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023297 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

  :herbie-target
  (* 4.0 (pow (sinh (/ x 2.0)) 2.0))

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