exp2 (problem 3.3.7)

Percentage Accurate: 76.9% → 100.0%

Time: 7.9s

Alternatives: 8

Speedup: 20.5×

• 49.6% 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: 76.9% 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

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x)))))
   (if (<= t_0 1e-5)
     (+
      (* 0.002777777777777778 (pow x 6.0))
      (+ (* 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 <= 1e-5) {
		tmp = (0.002777777777777778 * pow(x, 6.0)) + ((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 <= 1d-5) then
        tmp = (0.002777777777777778d0 * (x ** 6.0d0)) + ((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 <= 1e-5) {
		tmp = (0.002777777777777778 * Math.pow(x, 6.0)) + ((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 <= 1e-5:
		tmp = (0.002777777777777778 * math.pow(x, 6.0)) + ((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 <= 1e-5)
		tmp = Float64(Float64(0.002777777777777778 * (x ^ 6.0)) + 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 <= 1e-5)
		tmp = (0.002777777777777778 * (x ^ 6.0)) + ((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, 1e-5], N[(N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[Power[x, 2.0], $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))) < 1.00000000000000008e-5

   1. Initial program 51.9%

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

      1. +-commutative 51.9%

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

      2. associate-+r- 51.7%

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

      3. +-commutative 51.7%

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

      4. associate-+r- 52.1%

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

      5. +-commutative 52.1%

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

      6. associate-+l- 51.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 1.00000000000000008e-5 < (+.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%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{x} - 2\right) + e^{-x}\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{-9}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x))))) (if (<= t_0 4e-9) (* x x) t_0)))

C

double code(double x) {
	double t_0 = (exp(x) - 2.0) + exp(-x);
	double tmp;
	if (t_0 <= 4e-9) {
		tmp = 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-9) then
        tmp = 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-9) {
		tmp = 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-9:
		tmp = 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-9)
		tmp = 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-9)
		tmp = 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-9], N[(x * x), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 4.00000000000000025e-9

   1. Initial program 51.7%

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

      1. +-commutative 51.7%

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

      2. associate-+r- 51.5%

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

      3. +-commutative 51.5%

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

      4. associate-+r- 51.9%

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

      5. +-commutative 51.9%

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

      6. associate-+l- 51.7%

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

   3. Simplified 51.7%

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

   4. Taylor expanded in x around 0 99.8%

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

      1. unpow2 99.8%

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

   6. Simplified 99.8%

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

   if 4.00000000000000025e-9 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 4 \cdot 10^{-9}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x} - 2\right) + e^{-x}\\ \end{array} \]

Alternative 3: 100.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x)))))
   (if (<= t_0 1e-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 <= 1e-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 <= 1e-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, 1e-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))) < 1.00000000000000008e-5

   1. Initial program 51.9%

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

      1. +-commutative 51.9%

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

      2. associate-+r- 51.7%

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

      3. +-commutative 51.7%

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

      4. associate-+r- 52.1%

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

      5. +-commutative 52.1%

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

      6. associate-+l- 51.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in x around 0 99.9%

      \[\leadsto \color{blue}{0.08333333333333333 \cdot {x}^{4} + {x}^{2}} \]
   5. 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%

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

   6. Simplified 99.9%

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

   if 1.00000000000000008e-5 < (+.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%

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

Alternative 4: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{6} \cdot 0.027777777777777776}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 6.0) (* x x) (sqrt (* (pow x 6.0) 0.027777777777777776))))

C

double code(double x) {
	double tmp;
	if (x <= 6.0) {
		tmp = x * x;
	} else {
		tmp = sqrt((pow(x, 6.0) * 0.027777777777777776));
	}
	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 = sqrt(((x ** 6.0d0) * 0.027777777777777776d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 6.0) {
		tmp = x * x;
	} else {
		tmp = Math.sqrt((Math.pow(x, 6.0) * 0.027777777777777776));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 6.0:
		tmp = x * x
	else:
		tmp = math.sqrt((math.pow(x, 6.0) * 0.027777777777777776))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 6.0)
		tmp = Float64(x * x);
	else
		tmp = sqrt(Float64((x ^ 6.0) * 0.027777777777777776));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 6.0)
		tmp = x * x;
	else
		tmp = sqrt(((x ^ 6.0) * 0.027777777777777776));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 6.0], N[(x * x), $MachinePrecision], N[Sqrt[N[(N[Power[x, 6.0], $MachinePrecision] * 0.027777777777777776), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6

   1. Initial program 68.0%

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

      1. +-commutative 68.0%

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

      2. associate-+r- 67.9%

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

      3. +-commutative 67.9%

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

      4. associate-+r- 68.2%

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

      5. +-commutative 68.2%

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

      6. associate-+l- 68.0%

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

   3. Simplified 68.0%

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

   4. Taylor expanded in x around 0 84.5%

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

      1. unpow2 84.5%

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

   6. Simplified 84.5%

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

   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 74.0%

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

      1. +-commutative 74.0%

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

      2. neg-mul-1 74.0%

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

      3. unsub-neg 74.0%

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

      4. cube-mult 74.0%

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

      5. unpow2 74.0%

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

      6. associate-*r* 74.0%

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

      7. distribute-rgt-out 74.0%

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

   6. Simplified 74.0%

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

   7. Taylor expanded in x around inf 74.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot {x}^{3}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 74.0%

         \[\leadsto \color{blue}{\sqrt{0.16666666666666666 \cdot {x}^{3}} \cdot \sqrt{0.16666666666666666 \cdot {x}^{3}}} \]

      2. sqrt-unprod 84.5%

         \[\leadsto \color{blue}{\sqrt{\left(0.16666666666666666 \cdot {x}^{3}\right) \cdot \left(0.16666666666666666 \cdot {x}^{3}\right)}} \]

      3. pow1/2 84.5%

         \[\leadsto \color{blue}{{\left(\left(0.16666666666666666 \cdot {x}^{3}\right) \cdot \left(0.16666666666666666 \cdot {x}^{3}\right)\right)}^{0.5}} \]

      4. swap-sqr 84.5%

         \[\leadsto {\color{blue}{\left(\left(0.16666666666666666 \cdot 0.16666666666666666\right) \cdot \left({x}^{3} \cdot {x}^{3}\right)\right)}}^{0.5} \]

      5. metadata-eval 84.5%

         \[\leadsto {\left(\color{blue}{0.027777777777777776} \cdot \left({x}^{3} \cdot {x}^{3}\right)\right)}^{0.5} \]

      6. pow-prod-up 84.5%

         \[\leadsto {\left(0.027777777777777776 \cdot \color{blue}{{x}^{\left(3 + 3\right)}}\right)}^{0.5} \]

      7. metadata-eval 84.5%

         \[\leadsto {\left(0.027777777777777776 \cdot {x}^{\color{blue}{6}}\right)}^{0.5} \]

   9. Applied egg-rr 84.5%

      \[\leadsto \color{blue}{{\left(0.027777777777777776 \cdot {x}^{6}\right)}^{0.5}} \]
   10. Step-by-step derivation

       1. unpow1/2 84.5%

          \[\leadsto \color{blue}{\sqrt{0.027777777777777776 \cdot {x}^{6}}} \]

   11. Simplified 84.5%

       \[\leadsto \color{blue}{\sqrt{0.027777777777777776 \cdot {x}^{6}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{6} \cdot 0.027777777777777776}\\ \end{array} \]

Alternative 5: 80.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.000175:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{-x} - \left(1 + \left(\left(x \cdot x\right) \cdot \left(-0.5 + x \cdot -0.16666666666666666\right) - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.000175)
   (* x x)
   (-
    (exp (- x))
    (+ 1.0 (- (* (* x x) (+ -0.5 (* x -0.16666666666666666))) x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.000175) {
		tmp = x * x;
	} else {
		tmp = exp(-x) - (1.0 + (((x * x) * (-0.5 + (x * -0.16666666666666666))) - x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.000175d0) then
        tmp = x * x
    else
        tmp = exp(-x) - (1.0d0 + (((x * x) * ((-0.5d0) + (x * (-0.16666666666666666d0)))) - x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.000175) {
		tmp = x * x;
	} else {
		tmp = Math.exp(-x) - (1.0 + (((x * x) * (-0.5 + (x * -0.16666666666666666))) - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.000175:
		tmp = x * x
	else:
		tmp = math.exp(-x) - (1.0 + (((x * x) * (-0.5 + (x * -0.16666666666666666))) - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.000175)
		tmp = Float64(x * x);
	else
		tmp = Float64(exp(Float64(-x)) - Float64(1.0 + Float64(Float64(Float64(x * x) * Float64(-0.5 + Float64(x * -0.16666666666666666))) - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.000175)
		tmp = x * x;
	else
		tmp = exp(-x) - (1.0 + (((x * x) * (-0.5 + (x * -0.16666666666666666))) - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.000175], N[(x * x), $MachinePrecision], N[(N[Exp[(-x)], $MachinePrecision] - N[(1.0 + N[(N[(N[(x * x), $MachinePrecision] * N[(-0.5 + N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.74999999999999998e-4

   1. Initial program 67.8%

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

      1. +-commutative 67.8%

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

      2. associate-+r- 67.7%

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

      3. +-commutative 67.7%

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

      4. associate-+r- 67.9%

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

      5. +-commutative 67.9%

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

      6. associate-+l- 67.8%

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

   3. Simplified 67.8%

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

   4. Taylor expanded in x around 0 85.0%

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

      1. unpow2 85.0%

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

   6. Simplified 85.0%

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

   if 1.74999999999999998e-4 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate-+r- 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r- 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+l- 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 72.8%

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

      1. +-commutative 72.8%

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

      2. neg-mul-1 72.8%

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

      3. unsub-neg 72.8%

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

      4. cube-mult 72.8%

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

      5. unpow2 72.8%

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

      6. associate-*r* 72.8%

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

      7. distribute-rgt-out 72.8%

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

   6. Simplified 72.8%

      \[\leadsto e^{-x} - \color{blue}{\left(1 + \left(\left(x \cdot x\right) \cdot \left(-0.5 + x \cdot -0.16666666666666666\right) - x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000175:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{-x} - \left(1 + \left(\left(x \cdot x\right) \cdot \left(-0.5 + x \cdot -0.16666666666666666\right) - x\right)\right)\\ \end{array} \]

Alternative 6: 80.3% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \left(--0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 6.0) {
		tmp = x * x;
	} else {
		tmp = (x * x) * (x * -(-0.16666666666666666));
	}
	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 = (x * x) * (x * -(-0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 6.0) {
		tmp = x * x;
	} else {
		tmp = (x * x) * (x * -(-0.16666666666666666));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 6.0:
		tmp = x * x
	else:
		tmp = (x * x) * (x * -(-0.16666666666666666))
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 6

   1. Initial program 68.0%

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

      1. +-commutative 68.0%

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

      2. associate-+r- 67.9%

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

      3. +-commutative 67.9%

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

      4. associate-+r- 68.2%

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

      5. +-commutative 68.2%

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

      6. associate-+l- 68.0%

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

   3. Simplified 68.0%

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

   4. Taylor expanded in x around 0 84.5%

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

      1. unpow2 84.5%

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

   6. Simplified 84.5%

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

   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 74.0%

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

      1. +-commutative 74.0%

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

      2. neg-mul-1 74.0%

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

      3. unsub-neg 74.0%

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

      4. cube-mult 74.0%

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

      5. unpow2 74.0%

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

      6. associate-*r* 74.0%

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

      7. distribute-rgt-out 74.0%

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

   6. Simplified 74.0%

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

   7. Taylor expanded in x around inf 74.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot {x}^{3} + 0.5 \cdot {x}^{2}} \]
   8. Step-by-step derivation

      1. metadata-eval 74.0%

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

      2. distribute-lft-neg-in 74.0%

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

      3. *-commutative 74.0%

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

      4. unpow3 74.0%

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

      5. associate-*l* 74.0%

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

      6. distribute-lft-neg-in 74.0%

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

      7. metadata-eval 74.0%

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

      8. unpow2 74.0%

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

      9. distribute-lft-neg-in 74.0%

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

      10. *-commutative 74.0%

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

      11. distribute-lft-neg-in 74.0%

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

      12. distribute-lft-in 74.0%

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

      13. distribute-lft-neg-in 74.0%

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

      14. fma-udef 74.0%

          \[\leadsto \left(\left(-x\right) \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, -0.16666666666666666, -0.5\right)} \]

      15. associate-*r* 74.0%

          \[\leadsto \color{blue}{\left(-x\right) \cdot \left(x \cdot \mathsf{fma}\left(x, -0.16666666666666666, -0.5\right)\right)} \]

   9. Simplified 74.0%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \left(x \cdot \mathsf{fma}\left(x, -0.16666666666666666, -0.5\right)\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-neg-out 74.0%

          \[\leadsto \color{blue}{-x \cdot \left(x \cdot \mathsf{fma}\left(x, -0.16666666666666666, -0.5\right)\right)} \]

       2. add-sqr-sqrt 74.0%

          \[\leadsto -\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left(x \cdot \mathsf{fma}\left(x, -0.16666666666666666, -0.5\right)\right) \]

       3. sqrt-unprod 74.0%

          \[\leadsto -\color{blue}{\sqrt{x \cdot x}} \cdot \left(x \cdot \mathsf{fma}\left(x, -0.16666666666666666, -0.5\right)\right) \]

       4. sqr-neg 74.0%

          \[\leadsto -\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}} \cdot \left(x \cdot \mathsf{fma}\left(x, -0.16666666666666666, -0.5\right)\right) \]

       5. sqrt-unprod 0.0%

          \[\leadsto -\color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} \cdot \left(x \cdot \mathsf{fma}\left(x, -0.16666666666666666, -0.5\right)\right) \]

       6. add-sqr-sqrt 0.1%

          \[\leadsto -\color{blue}{\left(-x\right)} \cdot \left(x \cdot \mathsf{fma}\left(x, -0.16666666666666666, -0.5\right)\right) \]

       7. associate-*r* 0.1%

          \[\leadsto -\color{blue}{\left(\left(-x\right) \cdot x\right) \cdot \mathsf{fma}\left(x, -0.16666666666666666, -0.5\right)} \]

       8. *-commutative 0.1%

          \[\leadsto -\color{blue}{\mathsf{fma}\left(x, -0.16666666666666666, -0.5\right) \cdot \left(\left(-x\right) \cdot x\right)} \]

       9. add-sqr-sqrt 0.0%

          \[\leadsto -\mathsf{fma}\left(x, -0.16666666666666666, -0.5\right) \cdot \left(\color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} \cdot x\right) \]

       10. sqrt-unprod 74.0%

           \[\leadsto -\mathsf{fma}\left(x, -0.16666666666666666, -0.5\right) \cdot \left(\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} \cdot x\right) \]

       11. sqr-neg 74.0%

           \[\leadsto -\mathsf{fma}\left(x, -0.16666666666666666, -0.5\right) \cdot \left(\sqrt{\color{blue}{x \cdot x}} \cdot x\right) \]

       12. sqrt-unprod 74.0%

           \[\leadsto -\mathsf{fma}\left(x, -0.16666666666666666, -0.5\right) \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot x\right) \]

       13. add-sqr-sqrt 74.0%

           \[\leadsto -\mathsf{fma}\left(x, -0.16666666666666666, -0.5\right) \cdot \left(\color{blue}{x} \cdot x\right) \]

   11. Applied egg-rr 74.0%

       \[\leadsto \color{blue}{-\mathsf{fma}\left(x, -0.16666666666666666, -0.5\right) \cdot \left(x \cdot x\right)} \]

   12. Taylor expanded in x around inf 74.0%

       \[\leadsto -\color{blue}{\left(-0.16666666666666666 \cdot x\right)} \cdot \left(x \cdot x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \left(--0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 7: 76.6% 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 75.8%

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

   1. +-commutative 75.8%

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

   2. associate-+r- 75.7%

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

   3. +-commutative 75.7%

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

   4. associate-+r- 75.9%

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

   5. +-commutative 75.9%

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

   6. associate-+l- 75.8%

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

3. Simplified 75.8%

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

4. Taylor expanded in x around 0 76.9%

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

   1. unpow2 76.9%

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

6. Simplified 76.9%

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

7. Final simplification 76.9%

   \[\leadsto x \cdot x \]

Alternative 8: 27.0% 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 75.8%

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

   1. +-commutative 75.8%

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

   2. associate-+r- 75.7%

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

   3. +-commutative 75.7%

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

   4. associate-+r- 75.9%

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

   5. +-commutative 75.9%

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

   6. associate-+l- 75.8%

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

3. Simplified 75.8%

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

5. Applied egg-rr 26.4%

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

6. Final simplification 26.4%

   \[\leadsto 0 \]

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 2023279 
(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))))