exp2 (problem 3.3.7)

Percentage Accurate: 76.5% → 100.0%

Time: 7.8s

Alternatives: 7

Speedup: 1.9×

• 49.5% 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: 76.5% 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: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 0.002:\\ \;\;\;\;4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \left(x \cdot x + {x}^{4} \cdot 0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (- (exp x) 2.0) (exp (- x))) 0.002)
   (+
    (* 4.96031746031746e-5 (pow x 8.0))
    (+
     (* 0.002777777777777778 (pow x 6.0))
     (+ (* x x) (* (pow x 4.0) 0.08333333333333333))))
   (- (* 2.0 (cosh x)) 2.0)))

C

double code(double x) {
	double tmp;
	if (((exp(x) - 2.0) + exp(-x)) <= 0.002) {
		tmp = (4.96031746031746e-5 * pow(x, 8.0)) + ((0.002777777777777778 * pow(x, 6.0)) + ((x * x) + (pow(x, 4.0) * 0.08333333333333333)));
	} 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)) <= 0.002d0) then
        tmp = (4.96031746031746d-5 * (x ** 8.0d0)) + ((0.002777777777777778d0 * (x ** 6.0d0)) + ((x * x) + ((x ** 4.0d0) * 0.08333333333333333d0)))
    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)) <= 0.002) {
		tmp = (4.96031746031746e-5 * Math.pow(x, 8.0)) + ((0.002777777777777778 * Math.pow(x, 6.0)) + ((x * x) + (Math.pow(x, 4.0) * 0.08333333333333333)));
	} 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)) <= 0.002:
		tmp = (4.96031746031746e-5 * math.pow(x, 8.0)) + ((0.002777777777777778 * math.pow(x, 6.0)) + ((x * x) + (math.pow(x, 4.0) * 0.08333333333333333)))
	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))) <= 0.002)
		tmp = Float64(Float64(4.96031746031746e-5 * (x ^ 8.0)) + Float64(Float64(0.002777777777777778 * (x ^ 6.0)) + Float64(Float64(x * x) + Float64((x ^ 4.0) * 0.08333333333333333))));
	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)) <= 0.002)
		tmp = (4.96031746031746e-5 * (x ^ 8.0)) + ((0.002777777777777778 * (x ^ 6.0)) + ((x * x) + ((x ^ 4.0) * 0.08333333333333333)));
	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], 0.002], N[(N[(4.96031746031746e-5 * N[Power[x, 8.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * 0.08333333333333333), $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))) < 2e-3

   1. Initial program 42.9%

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

      1. +-commutative 42.9%

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

      2. associate-+r- 42.8%

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

      3. +-commutative 42.8%

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

      4. associate-+r- 42.8%

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

      5. +-commutative 42.8%

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

      6. associate-+l- 42.9%

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

   3. Simplified 42.9%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \left(0.08333333333333333 \cdot {x}^{4} + {x}^{2}\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \left(\color{blue}{{x}^{4} \cdot 0.08333333333333333} + {x}^{2}\right)\right) \]

      2. pow2 100.0%

         \[\leadsto 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \left({x}^{4} \cdot 0.08333333333333333 + \color{blue}{x \cdot x}\right)\right) \]

      3. fma-def 100.0%

         \[\leadsto 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \color{blue}{\mathsf{fma}\left({x}^{4}, 0.08333333333333333, x \cdot x\right)}\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \color{blue}{\mathsf{fma}\left({x}^{4}, 0.08333333333333333, x \cdot x\right)}\right) \]
   7. Step-by-step derivation

      1. fma-udef 100.0%

         \[\leadsto 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \color{blue}{\left({x}^{4} \cdot 0.08333333333333333 + x \cdot x\right)}\right) \]

   8. Applied egg-rr 100.0%

      \[\leadsto 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \color{blue}{\left({x}^{4} \cdot 0.08333333333333333 + x \cdot x\right)}\right) \]

   if 2e-3 < (+.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. associate-+r- 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+r- 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{x} + e^{-x}\right)\right)} - 2 \]

      2. expm1-udef 100.0%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(e^{x} + e^{-x}\right)} - 1\right)} - 2 \]

      3. cosh-undef 100.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \cosh x}\right)} - 1\right) - 2 \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \cosh x\right)} - 1\right)} - 2 \]
   7. Step-by-step derivation

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \cosh x\right)\right)} - 2 \]

      2. expm1-log1p 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 0.002:\\ \;\;\;\;4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \left(x \cdot x + {x}^{4} \cdot 0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(x, x, {x}^{4} \cdot 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (- (exp x) 2.0) (exp (- x))) 1e-8)
   (fma x x (* (pow x 4.0) 0.08333333333333333))
   (- (* 2.0 (cosh x)) 2.0)))

C

double code(double x) {
	double tmp;
	if (((exp(x) - 2.0) + exp(-x)) <= 1e-8) {
		tmp = fma(x, x, (pow(x, 4.0) * 0.08333333333333333));
	} 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))) <= 1e-8)
		tmp = fma(x, x, Float64((x ^ 4.0) * 0.08333333333333333));
	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], 1e-8], N[(x * x + N[(N[Power[x, 4.0], $MachinePrecision] * 0.08333333333333333), $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))) < 1e-8

   1. Initial program 42.5%

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

      1. +-commutative 42.5%

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

      2. associate-+r- 42.4%

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

      3. +-commutative 42.4%

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

      4. associate-+r- 42.5%

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

      5. +-commutative 42.5%

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

      6. associate-+l- 42.5%

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

   3. Simplified 42.5%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. fma-def 100.0%

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

   6. Simplified 100.0%

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

   if 1e-8 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+r- 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r- 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+l- 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 99.9%

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

      1. expm1-log1p-u 99.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{x} + e^{-x}\right)\right)} - 2 \]

      2. expm1-udef 99.8%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(e^{x} + e^{-x}\right)} - 1\right)} - 2 \]

      3. cosh-undef 99.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \cosh x}\right)} - 1\right) - 2 \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \cosh x\right)} - 1\right)} - 2 \]
   7. Step-by-step derivation

      1. expm1-def 99.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \cosh x\right)\right)} - 2 \]

      2. expm1-log1p 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(x, x, {x}^{4} \cdot 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \]

Alternative 3: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 10^{-8}:\\ \;\;\;\;x \cdot x + {x}^{4} \cdot 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (- (exp x) 2.0) (exp (- x))) 1e-8)
   (+ (* x x) (* (pow x 4.0) 0.08333333333333333))
   (- (* 2.0 (cosh x)) 2.0)))

C

double code(double x) {
	double tmp;
	if (((exp(x) - 2.0) + exp(-x)) <= 1e-8) {
		tmp = (x * x) + (pow(x, 4.0) * 0.08333333333333333);
	} 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)) <= 1d-8) then
        tmp = (x * x) + ((x ** 4.0d0) * 0.08333333333333333d0)
    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)) <= 1e-8) {
		tmp = (x * x) + (Math.pow(x, 4.0) * 0.08333333333333333);
	} 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)) <= 1e-8:
		tmp = (x * x) + (math.pow(x, 4.0) * 0.08333333333333333)
	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))) <= 1e-8)
		tmp = Float64(Float64(x * x) + Float64((x ^ 4.0) * 0.08333333333333333));
	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)) <= 1e-8)
		tmp = (x * x) + ((x ^ 4.0) * 0.08333333333333333);
	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], 1e-8], N[(N[(x * x), $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * 0.08333333333333333), $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))) < 1e-8

   1. Initial program 42.5%

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

      1. +-commutative 42.5%

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

      2. associate-+r- 42.4%

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

      3. +-commutative 42.4%

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

      4. associate-+r- 42.5%

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

      5. +-commutative 42.5%

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

      6. associate-+l- 42.5%

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

   3. Simplified 42.5%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. fma-def 100.0%

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

   6. Simplified 100.0%

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

      1. fma-udef 100.0%

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

   8. Applied egg-rr 100.0%

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

   if 1e-8 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+r- 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r- 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+l- 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 99.9%

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

      1. expm1-log1p-u 99.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{x} + e^{-x}\right)\right)} - 2 \]

      2. expm1-udef 99.8%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(e^{x} + e^{-x}\right)} - 1\right)} - 2 \]

      3. cosh-undef 99.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \cosh x}\right)} - 1\right) - 2 \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \cosh x\right)} - 1\right)} - 2 \]
   7. Step-by-step derivation

      1. expm1-def 99.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \cosh x\right)\right)} - 2 \]

      2. expm1-log1p 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 10^{-8}:\\ \;\;\;\;x \cdot x + {x}^{4} \cdot 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \]

Alternative 4: 88.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00016:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.00016) (* x x) (- (* 2.0 (cosh x)) 2.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.00016) {
		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 <= 0.00016d0) 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 <= 0.00016) {
		tmp = x * x;
	} else {
		tmp = (2.0 * Math.cosh(x)) - 2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.00016:
		tmp = x * x
	else:
		tmp = (2.0 * math.cosh(x)) - 2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.00016)
		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 <= 0.00016)
		tmp = x * x;
	else
		tmp = (2.0 * cosh(x)) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.00016], 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 < 1.60000000000000013e-4

   1. Initial program 64.1%

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

      1. +-commutative 64.1%

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

      2. associate-+r- 64.1%

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

      3. +-commutative 64.1%

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

      4. associate-+r- 64.1%

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

      5. +-commutative 64.1%

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

      6. associate-+l- 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in x around 0 79.0%

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

      1. unpow2 79.0%

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

   6. Simplified 79.0%

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

   if 1.60000000000000013e-4 < x

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-+r- 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+l- 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 99.8%

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

      1. expm1-log1p-u 99.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{x} + e^{-x}\right)\right)} - 2 \]

      2. expm1-udef 99.7%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(e^{x} + e^{-x}\right)} - 1\right)} - 2 \]

      3. cosh-undef 99.7%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \cosh x}\right)} - 1\right) - 2 \]

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \cosh x\right)} - 1\right)} - 2 \]
   7. Step-by-step derivation

      1. expm1-def 99.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \cosh x\right)\right)} - 2 \]

      2. expm1-log1p 99.8%

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

   8. Simplified 99.8%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00016:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \]

Alternative 5: 80.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 6.0) (* x x) (* 0.16666666666666666 (pow x 3.0))))

C

double code(double x) {
	double tmp;
	if (x <= 6.0) {
		tmp = x * x;
	} else {
		tmp = 0.16666666666666666 * pow(x, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 6.0d0) then
        tmp = x * x
    else
        tmp = 0.16666666666666666d0 * (x ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 6.0) {
		tmp = x * x;
	} else {
		tmp = 0.16666666666666666 * Math.pow(x, 3.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 6.0:
		tmp = x * x
	else:
		tmp = 0.16666666666666666 * math.pow(x, 3.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 6.0)
		tmp = Float64(x * x);
	else
		tmp = Float64(0.16666666666666666 * (x ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 6.0)
		tmp = x * x;
	else
		tmp = 0.16666666666666666 * (x ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 6.0], N[(x * x), $MachinePrecision], N[(0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6

   1. Initial program 64.2%

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

      1. +-commutative 64.2%

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

      2. associate-+r- 64.2%

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

      3. +-commutative 64.2%

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

      4. associate-+r- 64.2%

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

      5. +-commutative 64.2%

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

      6. associate-+l- 64.2%

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

   3. Simplified 64.2%

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

   4. Taylor expanded in x around 0 78.8%

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

      1. unpow2 78.8%

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

   6. Simplified 78.8%

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

   if 6 < 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. associate-+r- 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+r- 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 68.0%

      \[\leadsto e^{-x} - \color{blue}{\left(1 + \left(-1 \cdot x + \left(-0.5 \cdot {x}^{2} + -0.16666666666666666 \cdot {x}^{3}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 68.0%

         \[\leadsto e^{-x} - \left(1 + \color{blue}{\left(\left(-0.5 \cdot {x}^{2} + -0.16666666666666666 \cdot {x}^{3}\right) + -1 \cdot x\right)}\right) \]

      2. neg-mul-1 68.0%

         \[\leadsto e^{-x} - \left(1 + \left(\left(-0.5 \cdot {x}^{2} + -0.16666666666666666 \cdot {x}^{3}\right) + \color{blue}{\left(-x\right)}\right)\right) \]

      3. unsub-neg 68.0%

         \[\leadsto e^{-x} - \left(1 + \color{blue}{\left(\left(-0.5 \cdot {x}^{2} + -0.16666666666666666 \cdot {x}^{3}\right) - x\right)}\right) \]

      4. cube-mult 68.0%

         \[\leadsto e^{-x} - \left(1 + \left(\left(-0.5 \cdot {x}^{2} + -0.16666666666666666 \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right) - x\right)\right) \]

      5. unpow2 68.0%

         \[\leadsto e^{-x} - \left(1 + \left(\left(-0.5 \cdot {x}^{2} + -0.16666666666666666 \cdot \left(x \cdot \color{blue}{{x}^{2}}\right)\right) - x\right)\right) \]

      6. associate-*r* 68.0%

         \[\leadsto e^{-x} - \left(1 + \left(\left(-0.5 \cdot {x}^{2} + \color{blue}{\left(-0.16666666666666666 \cdot x\right) \cdot {x}^{2}}\right) - x\right)\right) \]

      7. distribute-rgt-out 68.0%

         \[\leadsto e^{-x} - \left(1 + \left(\color{blue}{{x}^{2} \cdot \left(-0.5 + -0.16666666666666666 \cdot x\right)} - x\right)\right) \]

   6. Simplified 68.0%

      \[\leadsto e^{-x} - \color{blue}{\left(1 + \left(\left(x \cdot x\right) \cdot \left(-0.5 + x \cdot -0.16666666666666666\right) - x\right)\right)} \]

   7. Taylor expanded in x around inf 68.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot {x}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot {x}^{3}\\ \end{array} \]

Alternative 6: 76.4% 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 71.6%

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

   1. +-commutative 71.6%

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

   2. associate-+r- 71.6%

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

   3. +-commutative 71.6%

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

   4. associate-+r- 71.6%

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

   5. +-commutative 71.6%

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

   6. associate-+l- 71.6%

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

3. Simplified 71.6%

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

4. Taylor expanded in x around 0 73.2%

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

   1. unpow2 73.2%

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

6. Simplified 73.2%

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

7. Final simplification 73.2%

   \[\leadsto x \cdot x \]

Alternative 7: 26.8% 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 71.6%

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

   1. +-commutative 71.6%

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

   2. associate-+r- 71.6%

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

   3. +-commutative 71.6%

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

   4. associate-+r- 71.6%

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

   5. +-commutative 71.6%

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

   6. associate-+l- 71.6%

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

3. Simplified 71.6%

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

5. Applied egg-rr 20.8%

   \[\leadsto \color{blue}{0} \]

6. Final simplification 20.8%

   \[\leadsto 0 \]

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 2023275 
(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))))