exp2 (problem 3.3.7)

Percentage Accurate: 76.1% → 100.0%

Time: 9.8s

Alternatives: 8

Speedup: 68.7×

• 48.8% 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: 76.1% 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: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{x} - 2\right) + e^{-x}\\ \mathbf{if}\;t_0 \leq 0.005:\\ \;\;\;\;4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \left(0.08333333333333333 \cdot {x}^{4} + x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x)))))
   (if (<= t_0 0.005)
     (+
      (* 4.96031746031746e-5 (pow x 8.0))
      (+
       (* 0.002777777777777778 (pow x 6.0))
       (+ (* 0.08333333333333333 (pow x 4.0)) (* x x))))
     t_0)))

C

double code(double x) {
	double t_0 = (exp(x) - 2.0) + exp(-x);
	double tmp;
	if (t_0 <= 0.005) {
		tmp = (4.96031746031746e-5 * pow(x, 8.0)) + ((0.002777777777777778 * pow(x, 6.0)) + ((0.08333333333333333 * pow(x, 4.0)) + (x * x)));
	} 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 <= 0.005d0) then
        tmp = (4.96031746031746d-5 * (x ** 8.0d0)) + ((0.002777777777777778d0 * (x ** 6.0d0)) + ((0.08333333333333333d0 * (x ** 4.0d0)) + (x * x)))
    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 <= 0.005) {
		tmp = (4.96031746031746e-5 * Math.pow(x, 8.0)) + ((0.002777777777777778 * Math.pow(x, 6.0)) + ((0.08333333333333333 * Math.pow(x, 4.0)) + (x * x)));
	} 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 <= 0.005:
		tmp = (4.96031746031746e-5 * math.pow(x, 8.0)) + ((0.002777777777777778 * math.pow(x, 6.0)) + ((0.08333333333333333 * math.pow(x, 4.0)) + (x * x)))
	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 <= 0.005)
		tmp = Float64(Float64(4.96031746031746e-5 * (x ^ 8.0)) + Float64(Float64(0.002777777777777778 * (x ^ 6.0)) + Float64(Float64(0.08333333333333333 * (x ^ 4.0)) + Float64(x * x))));
	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 <= 0.005)
		tmp = (4.96031746031746e-5 * (x ^ 8.0)) + ((0.002777777777777778 * (x ^ 6.0)) + ((0.08333333333333333 * (x ^ 4.0)) + (x * x)));
	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, 0.005], 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[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x * x), $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))) < 0.0050000000000000001

   1. Initial program 47.2%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Applied egg-rr 100.0%

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

   if 0.0050000000000000001 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 0.005:\\ \;\;\;\;4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \left(0.08333333333333333 \cdot {x}^{4} + x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x} - 2\right) + e^{-x}\\ \end{array} \]

Alternative 2: 100.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x)))))
   (if (<= t_0 4e-5)
     (+
      (* 0.002777777777777778 (pow x 6.0))
      (+ (* 0.08333333333333333 (pow x 4.0)) (* x x)))
     t_0)))

C

double code(double x) {
	double t_0 = (exp(x) - 2.0) + exp(-x);
	double tmp;
	if (t_0 <= 4e-5) {
		tmp = (0.002777777777777778 * pow(x, 6.0)) + ((0.08333333333333333 * pow(x, 4.0)) + (x * x));
	} 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-5) then
        tmp = (0.002777777777777778d0 * (x ** 6.0d0)) + ((0.08333333333333333d0 * (x ** 4.0d0)) + (x * x))
    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-5) {
		tmp = (0.002777777777777778 * Math.pow(x, 6.0)) + ((0.08333333333333333 * Math.pow(x, 4.0)) + (x * x));
	} 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-5:
		tmp = (0.002777777777777778 * math.pow(x, 6.0)) + ((0.08333333333333333 * math.pow(x, 4.0)) + (x * x))
	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-5)
		tmp = Float64(Float64(0.002777777777777778 * (x ^ 6.0)) + Float64(Float64(0.08333333333333333 * (x ^ 4.0)) + Float64(x * x)));
	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-5)
		tmp = (0.002777777777777778 * (x ^ 6.0)) + ((0.08333333333333333 * (x ^ 4.0)) + (x * x));
	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-5], N[(N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x * x), $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))) < 4.00000000000000033e-5

   1. Initial program 46.8%

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

   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. unpow2 100.0%

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

   4. Applied egg-rr 100.0%

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

   if 4.00000000000000033e-5 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 99.9%

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

4. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x)))))
   (if (<= t_0 4e-5) (+ (* 0.08333333333333333 (pow x 4.0)) (* x x)) t_0)))

C

double code(double x) {
	double t_0 = (exp(x) - 2.0) + exp(-x);
	double tmp;
	if (t_0 <= 4e-5) {
		tmp = (0.08333333333333333 * pow(x, 4.0)) + (x * x);
	} 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-5) then
        tmp = (0.08333333333333333d0 * (x ** 4.0d0)) + (x * x)
    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-5) {
		tmp = (0.08333333333333333 * Math.pow(x, 4.0)) + (x * x);
	} 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-5:
		tmp = (0.08333333333333333 * math.pow(x, 4.0)) + (x * x)
	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-5)
		tmp = Float64(Float64(0.08333333333333333 * (x ^ 4.0)) + Float64(x * x));
	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-5)
		tmp = (0.08333333333333333 * (x ^ 4.0)) + (x * x);
	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-5], N[(N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x * x), $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))) < 4.00000000000000033e-5

   1. Initial program 46.8%

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

   2. Taylor expanded in x around 0 99.7%

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

      1. unpow2 100.0%

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

   4. Applied egg-rr 99.7%

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

   if 4.00000000000000033e-5 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 99.9%

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

4. Final simplification 99.8%

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

Alternative 4: 99.9% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x)))))
   (if (<= t_0 4e-5) (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 <= 4e-5) {
		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 <= 4e-5)
		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, 4e-5], 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))) < 4.00000000000000033e-5

   1. Initial program 46.8%

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

   2. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. fma-def 99.7%

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

   4. Applied egg-rr 99.7%

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

   if 4.00000000000000033e-5 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 99.9%

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

4. Final simplification 99.8%

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

Alternative 5: 93.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.2:\\ \;\;\;\;0.08333333333333333 \cdot {x}^{4} + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{x} \cdot 2 - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.2)
   (+ (* 0.08333333333333333 (pow x 4.0)) (* x x))
   (- (* (exp x) 2.0) 2.0)))

C

double code(double x) {
	double tmp;
	if (x <= 5.2) {
		tmp = (0.08333333333333333 * pow(x, 4.0)) + (x * x);
	} else {
		tmp = (exp(x) * 2.0) - 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5.2d0) then
        tmp = (0.08333333333333333d0 * (x ** 4.0d0)) + (x * x)
    else
        tmp = (exp(x) * 2.0d0) - 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.2) {
		tmp = (0.08333333333333333 * Math.pow(x, 4.0)) + (x * x);
	} else {
		tmp = (Math.exp(x) * 2.0) - 2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.2:
		tmp = (0.08333333333333333 * math.pow(x, 4.0)) + (x * x)
	else:
		tmp = (math.exp(x) * 2.0) - 2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.2)
		tmp = Float64(Float64(0.08333333333333333 * (x ^ 4.0)) + Float64(x * x));
	else
		tmp = Float64(Float64(exp(x) * 2.0) - 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.2)
		tmp = (0.08333333333333333 * (x ^ 4.0)) + (x * x);
	else
		tmp = (exp(x) * 2.0) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.2], N[(N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[x], $MachinePrecision] * 2.0), $MachinePrecision] - 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.20000000000000018

   1. Initial program 64.7%

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

   2. Taylor expanded in x around 0 93.6%

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

      1. unpow2 96.2%

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

   4. Applied egg-rr 93.6%

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

   if 5.20000000000000018 < 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. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 100.0%

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

      5. sqr-neg 100.0%

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

      6. sqrt-unprod 100.0%

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

      7. add-sqr-sqrt 100.0%

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

      8. count-2 100.0%

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

      9. *-commutative 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.2:\\ \;\;\;\;0.08333333333333333 \cdot {x}^{4} + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{x} \cdot 2 - 2\\ \end{array} \]

Alternative 6: 87.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{x} \cdot 2 - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.1) (* x x) (- (* (exp x) 2.0) 2.0)))

C

double code(double x) {
	double tmp;
	if (x <= 3.1) {
		tmp = x * x;
	} else {
		tmp = (exp(x) * 2.0) - 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.1d0) then
        tmp = x * x
    else
        tmp = (exp(x) * 2.0d0) - 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.1) {
		tmp = x * x;
	} else {
		tmp = (Math.exp(x) * 2.0) - 2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.1:
		tmp = x * x
	else:
		tmp = (math.exp(x) * 2.0) - 2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.1)
		tmp = Float64(x * x);
	else
		tmp = Float64(Float64(exp(x) * 2.0) - 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.1)
		tmp = x * x;
	else
		tmp = (exp(x) * 2.0) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.1], N[(x * x), $MachinePrecision], N[(N[(N[Exp[x], $MachinePrecision] * 2.0), $MachinePrecision] - 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.10000000000000009

   1. Initial program 64.7%

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

   2. Taylor expanded in x around 0 84.4%

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

      1. unpow2 96.2%

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

   4. Applied egg-rr 84.4%

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

   if 3.10000000000000009 < 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. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 100.0%

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

      5. sqr-neg 100.0%

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

      6. sqrt-unprod 100.0%

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

      7. add-sqr-sqrt 100.0%

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

      8. count-2 100.0%

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

      9. *-commutative 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{x} \cdot 2 - 2\\ \end{array} \]

Alternative 7: 84.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.3:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.002777777777777778 \cdot {x}^{6}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.3) (* x x) (* 0.002777777777777778 (pow x 6.0))))

C

double code(double x) {
	double tmp;
	if (x <= 4.3) {
		tmp = x * x;
	} 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 <= 4.3d0) then
        tmp = x * x
    else
        tmp = 0.002777777777777778d0 * (x ** 6.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.3) {
		tmp = x * x;
	} else {
		tmp = 0.002777777777777778 * Math.pow(x, 6.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.3:
		tmp = x * x
	else:
		tmp = 0.002777777777777778 * math.pow(x, 6.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.3)
		tmp = Float64(x * x);
	else
		tmp = Float64(0.002777777777777778 * (x ^ 6.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.3)
		tmp = x * x;
	else
		tmp = 0.002777777777777778 * (x ^ 6.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.3], N[(x * x), $MachinePrecision], N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.29999999999999982

   1. Initial program 64.7%

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

   2. Taylor expanded in x around 0 84.4%

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

      1. unpow2 96.2%

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

   4. Applied egg-rr 84.4%

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

   if 4.29999999999999982 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 89.1%

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

   3. Taylor expanded in x around inf 89.1%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.3:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.002777777777777778 \cdot {x}^{6}\\ \end{array} \]

Alternative 8: 76.9% 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 72.9%

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

2. Taylor expanded in x around 0 77.7%

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

   1. unpow2 96.0%

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

4. Applied egg-rr 77.7%

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

5. Final simplification 77.7%

   \[\leadsto x \cdot x \]

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 2023326 
(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))))