exp2 (problem 3.3.7)

Percentage Accurate: 76.4% → 100.0%

Time: 5.2s

Alternatives: 9

Speedup: 1.9×

• 50.1% 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: 76.4% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{x} - 2\right) + e^{-x}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4} + 0.002777777777777778 \cdot {x}^{6}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x)))))
   (if (<= t_0 5e-6)
     (fma
      x
      x
      (+
       (* 0.08333333333333333 (pow x 4.0))
       (* 0.002777777777777778 (pow x 6.0))))
     t_0)))

C

double code(double x) {
	double t_0 = (exp(x) - 2.0) + exp(-x);
	double tmp;
	if (t_0 <= 5e-6) {
		tmp = fma(x, x, ((0.08333333333333333 * pow(x, 4.0)) + (0.002777777777777778 * pow(x, 6.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(exp(x) - 2.0) + exp(Float64(-x)))
	tmp = 0.0
	if (t_0 <= 5e-6)
		tmp = fma(x, x, Float64(Float64(0.08333333333333333 * (x ^ 4.0)) + Float64(0.002777777777777778 * (x ^ 6.0))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-6], N[(x * x + N[(N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 5.00000000000000041e-6

   1. Initial program 55.1%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. unpow2 100.0%

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

      4. fma-def 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-def 100.0%

         \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\mathsf{fma}\left(0.08333333333333333, {x}^{4}, 0.002777777777777778 \cdot {x}^{6}\right)}\right) \]

   4. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

   if 5.00000000000000041e-6 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

      \[\left(e^{x} - 2\right) + e^{-x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{x} - 2\right) + e^{-x}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;0.002777777777777778 \cdot {x}^{6} + \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x)))))
   (if (<= t_0 5e-6)
     (+
      (* 0.002777777777777778 (pow x 6.0))
      (fma x x (* 0.08333333333333333 (pow x 4.0))))
     t_0)))

C

double code(double x) {
	double t_0 = (exp(x) - 2.0) + exp(-x);
	double tmp;
	if (t_0 <= 5e-6) {
		tmp = (0.002777777777777778 * pow(x, 6.0)) + fma(x, x, (0.08333333333333333 * pow(x, 4.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(exp(x) - 2.0) + exp(Float64(-x)))
	tmp = 0.0
	if (t_0 <= 5e-6)
		tmp = Float64(Float64(0.002777777777777778 * (x ^ 6.0)) + fma(x, x, Float64(0.08333333333333333 * (x ^ 4.0))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-6], N[(N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(x * x + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 5.00000000000000041e-6

   1. Initial program 55.1%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 99.9%

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

      2. unpow2 99.9%

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

      3. fma-def 99.9%

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

   4. Applied egg-rr 100.0%

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

   if 5.00000000000000041e-6 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

      \[\left(e^{x} - 2\right) + e^{-x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

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

Alternative 3: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{x} - 2\right) + e^{-x}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;0.08333333333333333 \cdot {x}^{4} + {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x)))))
   (if (<= t_0 5e-6) (+ (* 0.08333333333333333 (pow x 4.0)) (pow x 2.0)) t_0)))

C

double code(double x) {
	double t_0 = (exp(x) - 2.0) + exp(-x);
	double tmp;
	if (t_0 <= 5e-6) {
		tmp = (0.08333333333333333 * pow(x, 4.0)) + pow(x, 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (exp(x) - 2.0d0) + exp(-x)
    if (t_0 <= 5d-6) then
        tmp = (0.08333333333333333d0 * (x ** 4.0d0)) + (x ** 2.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (Math.exp(x) - 2.0) + Math.exp(-x);
	double tmp;
	if (t_0 <= 5e-6) {
		tmp = (0.08333333333333333 * Math.pow(x, 4.0)) + Math.pow(x, 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (math.exp(x) - 2.0) + math.exp(-x)
	tmp = 0
	if t_0 <= 5e-6:
		tmp = (0.08333333333333333 * math.pow(x, 4.0)) + math.pow(x, 2.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(exp(x) - 2.0) + exp(Float64(-x)))
	tmp = 0.0
	if (t_0 <= 5e-6)
		tmp = Float64(Float64(0.08333333333333333 * (x ^ 4.0)) + (x ^ 2.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (exp(x) - 2.0) + exp(-x);
	tmp = 0.0;
	if (t_0 <= 5e-6)
		tmp = (0.08333333333333333 * (x ^ 4.0)) + (x ^ 2.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-6], N[(N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 5.00000000000000041e-6

   1. Initial program 55.1%

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

   2. Taylor expanded in x around 0 99.9%

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

   if 5.00000000000000041e-6 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

      \[\left(e^{x} - 2\right) + e^{-x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

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

Alternative 4: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{x} - 2\right) + e^{-x}\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{-16}:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x)))))
   (if (<= t_0 4e-16) (pow x 2.0) t_0)))

C

double code(double x) {
	double t_0 = (exp(x) - 2.0) + exp(-x);
	double tmp;
	if (t_0 <= 4e-16) {
		tmp = pow(x, 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (exp(x) - 2.0d0) + exp(-x)
    if (t_0 <= 4d-16) then
        tmp = x ** 2.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (Math.exp(x) - 2.0) + Math.exp(-x);
	double tmp;
	if (t_0 <= 4e-16) {
		tmp = Math.pow(x, 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (math.exp(x) - 2.0) + math.exp(-x)
	tmp = 0
	if t_0 <= 4e-16:
		tmp = math.pow(x, 2.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(exp(x) - 2.0) + exp(Float64(-x)))
	tmp = 0.0
	if (t_0 <= 4e-16)
		tmp = x ^ 2.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (exp(x) - 2.0) + exp(-x);
	tmp = 0.0;
	if (t_0 <= 4e-16)
		tmp = x ^ 2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-16], N[Power[x, 2.0], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 3.9999999999999999e-16

   1. Initial program 54.7%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 3.9999999999999999e-16 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 99.4%

      \[\left(e^{x} - 2\right) + e^{-x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 4 \cdot 10^{-16}:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x} - 2\right) + e^{-x}\\ \end{array} \]

Alternative 5: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{x} - 2\right) + e^{-x}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x)))))
   (if (<= t_0 5e-6) (fma x x (* 0.08333333333333333 (pow x 4.0))) t_0)))

C

double code(double x) {
	double t_0 = (exp(x) - 2.0) + exp(-x);
	double tmp;
	if (t_0 <= 5e-6) {
		tmp = fma(x, x, (0.08333333333333333 * pow(x, 4.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(exp(x) - 2.0) + exp(Float64(-x)))
	tmp = 0.0
	if (t_0 <= 5e-6)
		tmp = fma(x, x, Float64(0.08333333333333333 * (x ^ 4.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-6], N[(x * x + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 5.00000000000000041e-6

   1. Initial program 55.1%

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

   2. Taylor expanded in x around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. unpow2 99.9%

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

      3. fma-def 99.9%

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

   4. Applied egg-rr 99.9%

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

   if 5.00000000000000041e-6 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

      \[\left(e^{x} - 2\right) + e^{-x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

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

Alternative 6: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{x}^{18} \cdot 2.1433470507544582 \cdot 10^{-8}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.5) (pow x 2.0) (cbrt (* (pow x 18.0) 2.1433470507544582e-8))))

C

double code(double x) {
	double tmp;
	if (x <= 4.5) {
		tmp = pow(x, 2.0);
	} else {
		tmp = cbrt((pow(x, 18.0) * 2.1433470507544582e-8));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.5) {
		tmp = Math.pow(x, 2.0);
	} else {
		tmp = Math.cbrt((Math.pow(x, 18.0) * 2.1433470507544582e-8));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.5)
		tmp = x ^ 2.0;
	else
		tmp = cbrt(Float64((x ^ 18.0) * 2.1433470507544582e-8));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4.5], N[Power[x, 2.0], $MachinePrecision], N[Power[N[(N[Power[x, 18.0], $MachinePrecision] * 2.1433470507544582e-8), $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.5

   1. Initial program 69.5%

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

   2. Taylor expanded in x around 0 83.2%

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

   if 4.5 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 83.4%

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

   3. Taylor expanded in x around inf 83.4%

      \[\leadsto \color{blue}{0.002777777777777778 \cdot {x}^{6}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 83.4%

         \[\leadsto \color{blue}{\sqrt{0.002777777777777778 \cdot {x}^{6}} \cdot \sqrt{0.002777777777777778 \cdot {x}^{6}}} \]

      2. sqrt-unprod 92.4%

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

      3. *-commutative 92.4%

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

      4. *-commutative 92.4%

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

      5. swap-sqr 92.4%

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

      6. pow-prod-up 92.4%

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

      7. metadata-eval 92.4%

         \[\leadsto \sqrt{{x}^{\color{blue}{12}} \cdot \left(0.002777777777777778 \cdot 0.002777777777777778\right)} \]

      8. metadata-eval 92.4%

         \[\leadsto \sqrt{{x}^{12} \cdot \color{blue}{7.71604938271605 \cdot 10^{-6}}} \]

   5. Applied egg-rr 92.4%

      \[\leadsto \color{blue}{\sqrt{{x}^{12} \cdot 7.71604938271605 \cdot 10^{-6}}} \]
   6. Step-by-step derivation

      1. add-cbrt-cube 95.4%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{{x}^{12} \cdot 7.71604938271605 \cdot 10^{-6}} \cdot \sqrt{{x}^{12} \cdot 7.71604938271605 \cdot 10^{-6}}\right) \cdot \sqrt{{x}^{12} \cdot 7.71604938271605 \cdot 10^{-6}}}} \]

      2. pow1/3 95.4%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{{x}^{12} \cdot 7.71604938271605 \cdot 10^{-6}} \cdot \sqrt{{x}^{12} \cdot 7.71604938271605 \cdot 10^{-6}}\right) \cdot \sqrt{{x}^{12} \cdot 7.71604938271605 \cdot 10^{-6}}\right)}^{0.3333333333333333}} \]

      3. pow3 95.4%

         \[\leadsto {\color{blue}{\left({\left(\sqrt{{x}^{12} \cdot 7.71604938271605 \cdot 10^{-6}}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. *-commutative 95.4%

         \[\leadsto {\left({\left(\sqrt{\color{blue}{7.71604938271605 \cdot 10^{-6} \cdot {x}^{12}}}\right)}^{3}\right)}^{0.3333333333333333} \]

      5. sqrt-prod 95.4%

         \[\leadsto {\left({\color{blue}{\left(\sqrt{7.71604938271605 \cdot 10^{-6}} \cdot \sqrt{{x}^{12}}\right)}}^{3}\right)}^{0.3333333333333333} \]

      6. metadata-eval 95.4%

         \[\leadsto {\left({\left(\color{blue}{0.002777777777777778} \cdot \sqrt{{x}^{12}}\right)}^{3}\right)}^{0.3333333333333333} \]

      7. sqrt-pow1 95.4%

         \[\leadsto {\left({\left(0.002777777777777778 \cdot \color{blue}{{x}^{\left(\frac{12}{2}\right)}}\right)}^{3}\right)}^{0.3333333333333333} \]

      8. metadata-eval 95.4%

         \[\leadsto {\left({\left(0.002777777777777778 \cdot {x}^{\color{blue}{6}}\right)}^{3}\right)}^{0.3333333333333333} \]

      9. *-commutative 95.4%

         \[\leadsto {\left({\color{blue}{\left({x}^{6} \cdot 0.002777777777777778\right)}}^{3}\right)}^{0.3333333333333333} \]

      10. unpow-prod-down 95.4%

          \[\leadsto {\color{blue}{\left({\left({x}^{6}\right)}^{3} \cdot {0.002777777777777778}^{3}\right)}}^{0.3333333333333333} \]

      11. pow3 95.4%

          \[\leadsto {\left(\color{blue}{\left(\left({x}^{6} \cdot {x}^{6}\right) \cdot {x}^{6}\right)} \cdot {0.002777777777777778}^{3}\right)}^{0.3333333333333333} \]

      12. pow-prod-up 95.4%

          \[\leadsto {\left(\left(\color{blue}{{x}^{\left(6 + 6\right)}} \cdot {x}^{6}\right) \cdot {0.002777777777777778}^{3}\right)}^{0.3333333333333333} \]

      13. metadata-eval 95.4%

          \[\leadsto {\left(\left({x}^{\color{blue}{12}} \cdot {x}^{6}\right) \cdot {0.002777777777777778}^{3}\right)}^{0.3333333333333333} \]

      14. pow-prod-up 95.4%

          \[\leadsto {\left(\color{blue}{{x}^{\left(12 + 6\right)}} \cdot {0.002777777777777778}^{3}\right)}^{0.3333333333333333} \]

      15. metadata-eval 95.4%

          \[\leadsto {\left({x}^{\color{blue}{18}} \cdot {0.002777777777777778}^{3}\right)}^{0.3333333333333333} \]

      16. metadata-eval 95.4%

          \[\leadsto {\left({x}^{18} \cdot \color{blue}{2.1433470507544582 \cdot 10^{-8}}\right)}^{0.3333333333333333} \]

   7. Applied egg-rr 95.4%

      \[\leadsto \color{blue}{{\left({x}^{18} \cdot 2.1433470507544582 \cdot 10^{-8}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. unpow1/3 95.4%

         \[\leadsto \color{blue}{\sqrt[3]{{x}^{18} \cdot 2.1433470507544582 \cdot 10^{-8}}} \]

   9. Simplified 95.4%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{18} \cdot 2.1433470507544582 \cdot 10^{-8}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{x}^{18} \cdot 2.1433470507544582 \cdot 10^{-8}}\\ \end{array} \]

Alternative 7: 83.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.002777777777777778 \cdot {x}^{6}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.5) (pow x 2.0) (* 0.002777777777777778 (pow x 6.0))))

C

double code(double x) {
	double tmp;
	if (x <= 4.5) {
		tmp = pow(x, 2.0);
	} else {
		tmp = 0.002777777777777778 * pow(x, 6.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4.5d0) then
        tmp = x ** 2.0d0
    else
        tmp = 0.002777777777777778d0 * (x ** 6.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.5) {
		tmp = Math.pow(x, 2.0);
	} else {
		tmp = 0.002777777777777778 * Math.pow(x, 6.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.5:
		tmp = math.pow(x, 2.0)
	else:
		tmp = 0.002777777777777778 * math.pow(x, 6.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.5)
		tmp = x ^ 2.0;
	else
		tmp = Float64(0.002777777777777778 * (x ^ 6.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.5)
		tmp = x ^ 2.0;
	else
		tmp = 0.002777777777777778 * (x ^ 6.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.5], N[Power[x, 2.0], $MachinePrecision], N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.5

   1. Initial program 69.5%

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

   2. Taylor expanded in x around 0 83.2%

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

   if 4.5 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 83.4%

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

   3. Taylor expanded in x around inf 83.4%

      \[\leadsto \color{blue}{0.002777777777777778 \cdot {x}^{6}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.002777777777777778 \cdot {x}^{6}\\ \end{array} \]

Alternative 8: 75.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return x ^ 2.0
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

2. Taylor expanded in x around 0 78.0%

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

3. Final simplification 78.0%

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

Alternative 9: 25.7% 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 77.0%

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

3. Applied egg-rr 28.1%

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

4. Final simplification 28.1%

   \[\leadsto 0 \]

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 2023322 
(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))))