exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 9.2s

Alternatives: 17

Speedup: 1.0×

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return Math.exp(-(1.0 - (x * x)));
}

Python

def code(x):
	return math.exp(-(1.0 - (x * x)))

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return Math.exp(-(1.0 - (x * x)));
}

Python

def code(x):
	return math.exp(-(1.0 - (x * x)))

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{x \cdot x + -1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (exp (+ (* x x) -1.0)))

C

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

Fortran

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

Java

public static double code(double x) {
	return Math.exp(((x * x) + -1.0));
}

Python

def code(x):
	return math.exp(((x * x) + -1.0))

Julia

function code(x)
	return exp(Float64(Float64(x * x) + -1.0))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Step-by-step derivation

   1. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

   3. associate--r- N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

   7. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 74.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-13}:\\ \;\;\;\;e^{-1}\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 2e-13) (exp -1.0) (exp x)))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 2e-13) {
		tmp = exp(-1.0);
	} else {
		tmp = exp(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * x) <= 2d-13) then
        tmp = exp((-1.0d0))
    else
        tmp = exp(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 2e-13) {
		tmp = Math.exp(-1.0);
	} else {
		tmp = Math.exp(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 2e-13:
		tmp = math.exp(-1.0)
	else:
		tmp = math.exp(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 2e-13)
		tmp = exp(-1.0);
	else
		tmp = exp(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 2e-13)
		tmp = exp(-1.0);
	else
		tmp = exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-13], N[Exp[-1.0], $MachinePrecision], N[Exp[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.0000000000000001e-13

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. exp-lowering-exp.f64 99.8%

         \[\leadsto \mathsf{exp.f64}\left(-1\right) \]

   7. Simplified 99.8%

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

   if 2.0000000000000001e-13 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

   7. Simplified 100.0%

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

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot \left(x \cdot 1\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{1}{2}\right)\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{1}{2}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{1}{2}\right)\right)\right)\right) \]

      5. distribute-lft-out N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2} + x \cdot \frac{1}{2}\right)\right)\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot \left(\frac{x}{2} + x \cdot \frac{1}{2}\right)\right)\right) \]

      7. div-inv N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot \left(\frac{x}{2} + \frac{x}{2}\right)\right)\right) \]

      8. flip-+ N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot \frac{\frac{x}{2} \cdot \frac{x}{2} - \frac{x}{2} \cdot \frac{x}{2}}{\frac{x}{2} - \frac{x}{2}}\right)\right) \]

      9. +-inverses N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot \frac{0}{\frac{x}{2} - \frac{x}{2}}\right)\right) \]

      10. +-inverses N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot \frac{0}{0}\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\frac{x \cdot 0}{0}\right)\right) \]

      12. *-rgt-identity N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\frac{\left(x \cdot 1\right) \cdot 0}{0}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\frac{\left(x \cdot \left(\frac{1}{2} + \frac{1}{2}\right)\right) \cdot 0}{0}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\frac{\left(x \cdot \left(\frac{1}{2} + \frac{1}{2}\right)\right) \cdot 0}{0}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\frac{\left(x \cdot \left(\frac{1}{2} + \frac{1}{2}\right)\right) \cdot 0}{0}\right)\right) \]

      16. distribute-lft-out N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\frac{\left(x \cdot \frac{1}{2} + x \cdot \frac{1}{2}\right) \cdot 0}{0}\right)\right) \]

      17. div-inv N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\frac{\left(\frac{x}{2} + x \cdot \frac{1}{2}\right) \cdot 0}{0}\right)\right) \]

      18. div-inv N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\frac{\left(\frac{x}{2} + \frac{x}{2}\right) \cdot 0}{0}\right)\right) \]

      19. +-inverses N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\frac{\left(\frac{x}{2} + \frac{x}{2}\right) \cdot \left(\frac{x}{2} - \frac{x}{2}\right)}{0}\right)\right) \]

      20. difference-of-squares N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\frac{\frac{x}{2} \cdot \frac{x}{2} - \frac{x}{2} \cdot \frac{x}{2}}{0}\right)\right) \]

      21. +-inverses N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\frac{\frac{x}{2} \cdot \frac{x}{2} - \frac{x}{2} \cdot \frac{x}{2}}{\frac{x}{2} - \frac{x}{2}}\right)\right) \]

      22. flip-+ N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\frac{x}{2} + \frac{x}{2}\right)\right) \]

      23. count-2 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(2 \cdot \frac{x}{2}\right)\right) \]

      24. associate-*r/ N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\frac{2 \cdot x}{2}\right)\right) \]

   9. Applied egg-rr 49.6%

      \[\leadsto e^{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot 0.16666666666666666\\ t_1 := x \cdot \left(x \cdot x\right)\\ t_2 := \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-13}:\\ \;\;\;\;e^{-1}\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{+60}:\\ \;\;\;\;\frac{1 - t\_2 \cdot t\_2}{\left(1 - t\_2\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(x \cdot x + -1\right)\right)}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\frac{\left(x \cdot t\_1\right) \cdot \left(0.125 + t\_1 \cdot \left(t\_1 \cdot 0.004629629629629629\right)\right)}{0.25 + t\_0 \cdot \left(t\_0 - 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t\_1 \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) 0.16666666666666666))
        (t_1 (* x (* x x)))
        (t_2 (* (* x x) (* (* x x) (* x x)))))
   (if (<= (* x x) 2e-13)
     (exp -1.0)
     (if (<= (* x x) 4e+60)
       (/
        (- 1.0 (* t_2 t_2))
        (* (- 1.0 t_2) (+ 1.0 (* (* x x) (+ (* x x) -1.0)))))
       (if (<= (* x x) 5e+152)
         (/
          (* (* x t_1) (+ 0.125 (* t_1 (* t_1 0.004629629629629629))))
          (+ 0.25 (* t_0 (- t_0 0.5))))
         (* x (* t_1 0.5)))))))

C

double code(double x) {
	double t_0 = (x * x) * 0.16666666666666666;
	double t_1 = x * (x * x);
	double t_2 = (x * x) * ((x * x) * (x * x));
	double tmp;
	if ((x * x) <= 2e-13) {
		tmp = exp(-1.0);
	} else if ((x * x) <= 4e+60) {
		tmp = (1.0 - (t_2 * t_2)) / ((1.0 - t_2) * (1.0 + ((x * x) * ((x * x) + -1.0))));
	} else if ((x * x) <= 5e+152) {
		tmp = ((x * t_1) * (0.125 + (t_1 * (t_1 * 0.004629629629629629)))) / (0.25 + (t_0 * (t_0 - 0.5)));
	} else {
		tmp = x * (t_1 * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (x * x) * 0.16666666666666666d0
    t_1 = x * (x * x)
    t_2 = (x * x) * ((x * x) * (x * x))
    if ((x * x) <= 2d-13) then
        tmp = exp((-1.0d0))
    else if ((x * x) <= 4d+60) then
        tmp = (1.0d0 - (t_2 * t_2)) / ((1.0d0 - t_2) * (1.0d0 + ((x * x) * ((x * x) + (-1.0d0)))))
    else if ((x * x) <= 5d+152) then
        tmp = ((x * t_1) * (0.125d0 + (t_1 * (t_1 * 0.004629629629629629d0)))) / (0.25d0 + (t_0 * (t_0 - 0.5d0)))
    else
        tmp = x * (t_1 * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * 0.16666666666666666;
	double t_1 = x * (x * x);
	double t_2 = (x * x) * ((x * x) * (x * x));
	double tmp;
	if ((x * x) <= 2e-13) {
		tmp = Math.exp(-1.0);
	} else if ((x * x) <= 4e+60) {
		tmp = (1.0 - (t_2 * t_2)) / ((1.0 - t_2) * (1.0 + ((x * x) * ((x * x) + -1.0))));
	} else if ((x * x) <= 5e+152) {
		tmp = ((x * t_1) * (0.125 + (t_1 * (t_1 * 0.004629629629629629)))) / (0.25 + (t_0 * (t_0 - 0.5)));
	} else {
		tmp = x * (t_1 * 0.5);
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) * 0.16666666666666666
	t_1 = x * (x * x)
	t_2 = (x * x) * ((x * x) * (x * x))
	tmp = 0
	if (x * x) <= 2e-13:
		tmp = math.exp(-1.0)
	elif (x * x) <= 4e+60:
		tmp = (1.0 - (t_2 * t_2)) / ((1.0 - t_2) * (1.0 + ((x * x) * ((x * x) + -1.0))))
	elif (x * x) <= 5e+152:
		tmp = ((x * t_1) * (0.125 + (t_1 * (t_1 * 0.004629629629629629)))) / (0.25 + (t_0 * (t_0 - 0.5)))
	else:
		tmp = x * (t_1 * 0.5)
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * 0.16666666666666666)
	t_1 = Float64(x * Float64(x * x))
	t_2 = Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * x)))
	tmp = 0.0
	if (Float64(x * x) <= 2e-13)
		tmp = exp(-1.0);
	elseif (Float64(x * x) <= 4e+60)
		tmp = Float64(Float64(1.0 - Float64(t_2 * t_2)) / Float64(Float64(1.0 - t_2) * Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(x * x) + -1.0)))));
	elseif (Float64(x * x) <= 5e+152)
		tmp = Float64(Float64(Float64(x * t_1) * Float64(0.125 + Float64(t_1 * Float64(t_1 * 0.004629629629629629)))) / Float64(0.25 + Float64(t_0 * Float64(t_0 - 0.5))));
	else
		tmp = Float64(x * Float64(t_1 * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) * 0.16666666666666666;
	t_1 = x * (x * x);
	t_2 = (x * x) * ((x * x) * (x * x));
	tmp = 0.0;
	if ((x * x) <= 2e-13)
		tmp = exp(-1.0);
	elseif ((x * x) <= 4e+60)
		tmp = (1.0 - (t_2 * t_2)) / ((1.0 - t_2) * (1.0 + ((x * x) * ((x * x) + -1.0))));
	elseif ((x * x) <= 5e+152)
		tmp = ((x * t_1) * (0.125 + (t_1 * (t_1 * 0.004629629629629629)))) / (0.25 + (t_0 * (t_0 - 0.5)));
	else
		tmp = x * (t_1 * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 2e-13], N[Exp[-1.0], $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 4e+60], N[(N[(1.0 - N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - t$95$2), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 5e+152], N[(N[(N[(x * t$95$1), $MachinePrecision] * N[(0.125 + N[(t$95$1 * N[(t$95$1 * 0.004629629629629629), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.25 + N[(t$95$0 * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(t$95$1 * 0.5), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x x) < 2.0000000000000001e-13

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. exp-lowering-exp.f64 99.8%

         \[\leadsto \mathsf{exp.f64}\left(-1\right) \]

   7. Simplified 99.8%

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

   if 2.0000000000000001e-13 < (*.f64 x x) < 3.9999999999999998e60

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2}} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 3.4%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   10. Simplified 3.4%

       \[\leadsto \color{blue}{1 + x \cdot x} \]
   11. Step-by-step derivation

       1. flip3-+ N/A

          \[\leadsto \frac{{1}^{3} + {\left(x \cdot x\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot \left(x \cdot x\right)\right)}} \]

       2. div-inv N/A

          \[\leadsto \left({1}^{3} + {\left(x \cdot x\right)}^{3}\right) \cdot \color{blue}{\frac{1}{1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot \left(x \cdot x\right)\right)}} \]

       3. metadata-eval N/A

          \[\leadsto \left(1 + {\left(x \cdot x\right)}^{3}\right) \cdot \frac{1}{1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot \left(x \cdot x\right)\right)} \]

       4. flip-+ N/A

          \[\leadsto \frac{1 \cdot 1 - {\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{3}}{1 - {\left(x \cdot x\right)}^{3}} \cdot \frac{\color{blue}{1}}{1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot \left(x \cdot x\right)\right)} \]

       5. metadata-eval N/A

          \[\leadsto \frac{1 \cdot 1 - {\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{3}}{{1}^{3} - {\left(x \cdot x\right)}^{3}} \cdot \frac{1}{1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot \left(x \cdot x\right)\right)} \]

       6. frac-times N/A

          \[\leadsto \frac{\left(1 \cdot 1 - {\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 1}{\color{blue}{\left({1}^{3} - {\left(x \cdot x\right)}^{3}\right) \cdot \left(1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot \left(x \cdot x\right)\right)\right)}} \]

       7. metadata-eval N/A

          \[\leadsto \frac{\left(1 - {\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 1}{\left({\color{blue}{1}}^{3} - {\left(x \cdot x\right)}^{3}\right) \cdot \left(1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot \left(x \cdot x\right)\right)\right)} \]

       8. metadata-eval N/A

          \[\leadsto \frac{\left({1}^{3} - {\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 1}{\left({\color{blue}{1}}^{3} - {\left(x \cdot x\right)}^{3}\right) \cdot \left(1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot \left(x \cdot x\right)\right)\right)} \]

       9. unpow-prod-down N/A

          \[\leadsto \frac{\left({1}^{3} - {\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}^{3}\right) \cdot 1}{\left({1}^{\color{blue}{3}} - {\left(x \cdot x\right)}^{3}\right) \cdot \left(1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot \left(x \cdot x\right)\right)\right)} \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\left({1}^{3} - {\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}^{3}\right) \cdot 1\right), \color{blue}{\left(\left({1}^{3} - {\left(x \cdot x\right)}^{3}\right) \cdot \left(1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot \left(x \cdot x\right)\right)\right)\right)}\right) \]

   12. Applied egg-rr 39.6%

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

   if 3.9999999999999998e60 < (*.f64 x x) < 5e152

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 45.4%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 45.4%

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

   11. Taylor expanded in x around inf

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

   13. Simplified 45.4%

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

   15. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.125 + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.004629629629629629\right)\right)}{0.25 + \left(\left(x \cdot x\right) \cdot 0.16666666666666666\right) \cdot \left(\left(x \cdot x\right) \cdot 0.16666666666666666 - 0.5\right)}} \]

   if 5e152 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot x\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

   10. Simplified 100.0%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{4}} \]
   12. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot {x}^{\left(2 \cdot \color{blue}{2}\right)} \]

       2. pow-sqr N/A

          \[\leadsto \frac{1}{2} \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

       4. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

       5. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

       6. *-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot x\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot x\right)}\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot x\right)}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right) \]

       10. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot {x}^{\color{blue}{3}}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

       12. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

       16. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{x \cdot \left(0.5 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-13}:\\ \;\;\;\;e^{-1}\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{+60}:\\ \;\;\;\;\frac{1 - \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}{\left(1 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(x \cdot x + -1\right)\right)}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\frac{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.125 + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.004629629629629629\right)\right)}{0.25 + \left(\left(x \cdot x\right) \cdot 0.16666666666666666\right) \cdot \left(\left(x \cdot x\right) \cdot 0.16666666666666666 - 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot 0.16666666666666666\\ t_1 := x \cdot \left(x \cdot x\right)\\ t_2 := \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{+60}:\\ \;\;\;\;\frac{1 - t\_2 \cdot t\_2}{\left(1 - t\_2\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(x \cdot x + -1\right)\right)}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\frac{\left(x \cdot t\_1\right) \cdot \left(0.125 + t\_1 \cdot \left(t\_1 \cdot 0.004629629629629629\right)\right)}{0.25 + t\_0 \cdot \left(t\_0 - 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t\_1 \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) 0.16666666666666666))
        (t_1 (* x (* x x)))
        (t_2 (* (* x x) (* (* x x) (* x x)))))
   (if (<= (* x x) 4e+60)
     (/
      (- 1.0 (* t_2 t_2))
      (* (- 1.0 t_2) (+ 1.0 (* (* x x) (+ (* x x) -1.0)))))
     (if (<= (* x x) 5e+152)
       (/
        (* (* x t_1) (+ 0.125 (* t_1 (* t_1 0.004629629629629629))))
        (+ 0.25 (* t_0 (- t_0 0.5))))
       (* x (* t_1 0.5))))))

C

double code(double x) {
	double t_0 = (x * x) * 0.16666666666666666;
	double t_1 = x * (x * x);
	double t_2 = (x * x) * ((x * x) * (x * x));
	double tmp;
	if ((x * x) <= 4e+60) {
		tmp = (1.0 - (t_2 * t_2)) / ((1.0 - t_2) * (1.0 + ((x * x) * ((x * x) + -1.0))));
	} else if ((x * x) <= 5e+152) {
		tmp = ((x * t_1) * (0.125 + (t_1 * (t_1 * 0.004629629629629629)))) / (0.25 + (t_0 * (t_0 - 0.5)));
	} else {
		tmp = x * (t_1 * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (x * x) * 0.16666666666666666d0
    t_1 = x * (x * x)
    t_2 = (x * x) * ((x * x) * (x * x))
    if ((x * x) <= 4d+60) then
        tmp = (1.0d0 - (t_2 * t_2)) / ((1.0d0 - t_2) * (1.0d0 + ((x * x) * ((x * x) + (-1.0d0)))))
    else if ((x * x) <= 5d+152) then
        tmp = ((x * t_1) * (0.125d0 + (t_1 * (t_1 * 0.004629629629629629d0)))) / (0.25d0 + (t_0 * (t_0 - 0.5d0)))
    else
        tmp = x * (t_1 * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * 0.16666666666666666;
	double t_1 = x * (x * x);
	double t_2 = (x * x) * ((x * x) * (x * x));
	double tmp;
	if ((x * x) <= 4e+60) {
		tmp = (1.0 - (t_2 * t_2)) / ((1.0 - t_2) * (1.0 + ((x * x) * ((x * x) + -1.0))));
	} else if ((x * x) <= 5e+152) {
		tmp = ((x * t_1) * (0.125 + (t_1 * (t_1 * 0.004629629629629629)))) / (0.25 + (t_0 * (t_0 - 0.5)));
	} else {
		tmp = x * (t_1 * 0.5);
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) * 0.16666666666666666
	t_1 = x * (x * x)
	t_2 = (x * x) * ((x * x) * (x * x))
	tmp = 0
	if (x * x) <= 4e+60:
		tmp = (1.0 - (t_2 * t_2)) / ((1.0 - t_2) * (1.0 + ((x * x) * ((x * x) + -1.0))))
	elif (x * x) <= 5e+152:
		tmp = ((x * t_1) * (0.125 + (t_1 * (t_1 * 0.004629629629629629)))) / (0.25 + (t_0 * (t_0 - 0.5)))
	else:
		tmp = x * (t_1 * 0.5)
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * 0.16666666666666666)
	t_1 = Float64(x * Float64(x * x))
	t_2 = Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * x)))
	tmp = 0.0
	if (Float64(x * x) <= 4e+60)
		tmp = Float64(Float64(1.0 - Float64(t_2 * t_2)) / Float64(Float64(1.0 - t_2) * Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(x * x) + -1.0)))));
	elseif (Float64(x * x) <= 5e+152)
		tmp = Float64(Float64(Float64(x * t_1) * Float64(0.125 + Float64(t_1 * Float64(t_1 * 0.004629629629629629)))) / Float64(0.25 + Float64(t_0 * Float64(t_0 - 0.5))));
	else
		tmp = Float64(x * Float64(t_1 * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) * 0.16666666666666666;
	t_1 = x * (x * x);
	t_2 = (x * x) * ((x * x) * (x * x));
	tmp = 0.0;
	if ((x * x) <= 4e+60)
		tmp = (1.0 - (t_2 * t_2)) / ((1.0 - t_2) * (1.0 + ((x * x) * ((x * x) + -1.0))));
	elseif ((x * x) <= 5e+152)
		tmp = ((x * t_1) * (0.125 + (t_1 * (t_1 * 0.004629629629629629)))) / (0.25 + (t_0 * (t_0 - 0.5)));
	else
		tmp = x * (t_1 * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 4e+60], N[(N[(1.0 - N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - t$95$2), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 5e+152], N[(N[(N[(x * t$95$1), $MachinePrecision] * N[(0.125 + N[(t$95$1 * N[(t$95$1 * 0.004629629629629629), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.25 + N[(t$95$0 * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(t$95$1 * 0.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 3.9999999999999998e60

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

      2. *-lowering-*.f64 27.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

   7. Simplified 27.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2}} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 16.2%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   10. Simplified 16.2%

       \[\leadsto \color{blue}{1 + x \cdot x} \]
   11. Step-by-step derivation

       1. flip3-+ N/A

          \[\leadsto \frac{{1}^{3} + {\left(x \cdot x\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot \left(x \cdot x\right)\right)}} \]

       2. div-inv N/A

          \[\leadsto \left({1}^{3} + {\left(x \cdot x\right)}^{3}\right) \cdot \color{blue}{\frac{1}{1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot \left(x \cdot x\right)\right)}} \]

       3. metadata-eval N/A

          \[\leadsto \left(1 + {\left(x \cdot x\right)}^{3}\right) \cdot \frac{1}{1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot \left(x \cdot x\right)\right)} \]

       4. flip-+ N/A

          \[\leadsto \frac{1 \cdot 1 - {\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{3}}{1 - {\left(x \cdot x\right)}^{3}} \cdot \frac{\color{blue}{1}}{1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot \left(x \cdot x\right)\right)} \]

       5. metadata-eval N/A

          \[\leadsto \frac{1 \cdot 1 - {\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{3}}{{1}^{3} - {\left(x \cdot x\right)}^{3}} \cdot \frac{1}{1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot \left(x \cdot x\right)\right)} \]

       6. frac-times N/A

          \[\leadsto \frac{\left(1 \cdot 1 - {\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 1}{\color{blue}{\left({1}^{3} - {\left(x \cdot x\right)}^{3}\right) \cdot \left(1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot \left(x \cdot x\right)\right)\right)}} \]

       7. metadata-eval N/A

          \[\leadsto \frac{\left(1 - {\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 1}{\left({\color{blue}{1}}^{3} - {\left(x \cdot x\right)}^{3}\right) \cdot \left(1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot \left(x \cdot x\right)\right)\right)} \]

       8. metadata-eval N/A

          \[\leadsto \frac{\left({1}^{3} - {\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{3}\right) \cdot 1}{\left({\color{blue}{1}}^{3} - {\left(x \cdot x\right)}^{3}\right) \cdot \left(1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot \left(x \cdot x\right)\right)\right)} \]

       9. unpow-prod-down N/A

          \[\leadsto \frac{\left({1}^{3} - {\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}^{3}\right) \cdot 1}{\left({1}^{\color{blue}{3}} - {\left(x \cdot x\right)}^{3}\right) \cdot \left(1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot \left(x \cdot x\right)\right)\right)} \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\left({1}^{3} - {\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}^{3}\right) \cdot 1\right), \color{blue}{\left(\left({1}^{3} - {\left(x \cdot x\right)}^{3}\right) \cdot \left(1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot \left(x \cdot x\right)\right)\right)\right)}\right) \]

   12. Applied egg-rr 20.2%

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

   if 3.9999999999999998e60 < (*.f64 x x) < 5e152

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 45.4%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 45.4%

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

   11. Taylor expanded in x around inf

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

   13. Simplified 45.4%

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

   15. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.125 + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.004629629629629629\right)\right)}{0.25 + \left(\left(x \cdot x\right) \cdot 0.16666666666666666\right) \cdot \left(\left(x \cdot x\right) \cdot 0.16666666666666666 - 0.5\right)}} \]

   if 5e152 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot x\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

   10. Simplified 100.0%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{4}} \]
   12. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot {x}^{\left(2 \cdot \color{blue}{2}\right)} \]

       2. pow-sqr N/A

          \[\leadsto \frac{1}{2} \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

       4. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

       5. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

       6. *-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot x\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot x\right)}\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot x\right)}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right) \]

       10. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot {x}^{\color{blue}{3}}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

       12. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

       16. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{x \cdot \left(0.5 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{+60}:\\ \;\;\;\;\frac{1 - \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}{\left(1 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(x \cdot x + -1\right)\right)}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\frac{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.125 + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.004629629629629629\right)\right)}{0.25 + \left(\left(x \cdot x\right) \cdot 0.16666666666666666\right) \cdot \left(\left(x \cdot x\right) \cdot 0.16666666666666666 - 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+152}:\\ \;\;\;\;1 + \frac{\left(x \cdot x\right) \cdot \left(1 - t\_0 \cdot t\_0\right)}{1 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) (+ 0.5 (* x (* x 0.16666666666666666))))))
   (if (<= (* x x) 5e+152)
     (+ 1.0 (/ (* (* x x) (- 1.0 (* t_0 t_0))) (- 1.0 t_0)))
     (* x (* (* x (* x x)) 0.5)))))

C

double code(double x) {
	double t_0 = (x * x) * (0.5 + (x * (x * 0.16666666666666666)));
	double tmp;
	if ((x * x) <= 5e+152) {
		tmp = 1.0 + (((x * x) * (1.0 - (t_0 * t_0))) / (1.0 - t_0));
	} else {
		tmp = x * ((x * (x * x)) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * x) * (0.5d0 + (x * (x * 0.16666666666666666d0)))
    if ((x * x) <= 5d+152) then
        tmp = 1.0d0 + (((x * x) * (1.0d0 - (t_0 * t_0))) / (1.0d0 - t_0))
    else
        tmp = x * ((x * (x * x)) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * (0.5 + (x * (x * 0.16666666666666666)));
	double tmp;
	if ((x * x) <= 5e+152) {
		tmp = 1.0 + (((x * x) * (1.0 - (t_0 * t_0))) / (1.0 - t_0));
	} else {
		tmp = x * ((x * (x * x)) * 0.5);
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) * (0.5 + (x * (x * 0.16666666666666666)))
	tmp = 0
	if (x * x) <= 5e+152:
		tmp = 1.0 + (((x * x) * (1.0 - (t_0 * t_0))) / (1.0 - t_0))
	else:
		tmp = x * ((x * (x * x)) * 0.5)
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(x * 0.16666666666666666))))
	tmp = 0.0
	if (Float64(x * x) <= 5e+152)
		tmp = Float64(1.0 + Float64(Float64(Float64(x * x) * Float64(1.0 - Float64(t_0 * t_0))) / Float64(1.0 - t_0)));
	else
		tmp = Float64(x * Float64(Float64(x * Float64(x * x)) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) * (0.5 + (x * (x * 0.16666666666666666)));
	tmp = 0.0;
	if ((x * x) <= 5e+152)
		tmp = 1.0 + (((x * x) * (1.0 - (t_0 * t_0))) / (1.0 - t_0));
	else
		tmp = x * ((x * (x * x)) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e+152], N[(1.0 + N[(N[(N[(x * x), $MachinePrecision] * N[(1.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5e152

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

      2. *-lowering-*.f64 34.7%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

   7. Simplified 34.7%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 19.4%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 19.4%

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

       1. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

       2. flip-+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1 \cdot 1 - \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)}{1 - x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)} \cdot \left(\color{blue}{x} \cdot x\right)\right)\right) \]

       3. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\left(1 \cdot 1 - \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)\right) \cdot \left(x \cdot x\right)}{\color{blue}{1 - x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)}}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(1 \cdot 1 - \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right), \color{blue}{\left(1 - x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)}\right)\right) \]

   12. Applied egg-rr 25.2%

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

   if 5e152 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot x\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

   10. Simplified 100.0%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{4}} \]
   12. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot {x}^{\left(2 \cdot \color{blue}{2}\right)} \]

       2. pow-sqr N/A

          \[\leadsto \frac{1}{2} \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

       4. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

       5. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

       6. *-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot x\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot x\right)}\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot x\right)}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right) \]

       10. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot {x}^{\color{blue}{3}}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

       12. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

       16. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{x \cdot \left(0.5 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+152}:\\ \;\;\;\;1 + \frac{\left(x \cdot x\right) \cdot \left(1 - \left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)\right)}{1 - \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot 0.16666666666666666\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+152}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(1 + \frac{\left(x \cdot x\right) \cdot \left(0.125 + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 0.004629629629629629\right)}{0.25 + t\_0 \cdot \left(t\_0 - 0.5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x 0.16666666666666666))))
   (if (<= (* x x) 5e+152)
     (+
      1.0
      (*
       (* x x)
       (+
        1.0
        (/
         (*
          (* x x)
          (+ 0.125 (* (* (* x x) (* (* x x) (* x x))) 0.004629629629629629)))
         (+ 0.25 (* t_0 (- t_0 0.5)))))))
     (* x (* (* x (* x x)) 0.5)))))

C

double code(double x) {
	double t_0 = x * (x * 0.16666666666666666);
	double tmp;
	if ((x * x) <= 5e+152) {
		tmp = 1.0 + ((x * x) * (1.0 + (((x * x) * (0.125 + (((x * x) * ((x * x) * (x * x))) * 0.004629629629629629))) / (0.25 + (t_0 * (t_0 - 0.5))))));
	} else {
		tmp = x * ((x * (x * x)) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * 0.16666666666666666d0)
    if ((x * x) <= 5d+152) then
        tmp = 1.0d0 + ((x * x) * (1.0d0 + (((x * x) * (0.125d0 + (((x * x) * ((x * x) * (x * x))) * 0.004629629629629629d0))) / (0.25d0 + (t_0 * (t_0 - 0.5d0))))))
    else
        tmp = x * ((x * (x * x)) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (x * 0.16666666666666666);
	double tmp;
	if ((x * x) <= 5e+152) {
		tmp = 1.0 + ((x * x) * (1.0 + (((x * x) * (0.125 + (((x * x) * ((x * x) * (x * x))) * 0.004629629629629629))) / (0.25 + (t_0 * (t_0 - 0.5))))));
	} else {
		tmp = x * ((x * (x * x)) * 0.5);
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * 0.16666666666666666)
	tmp = 0
	if (x * x) <= 5e+152:
		tmp = 1.0 + ((x * x) * (1.0 + (((x * x) * (0.125 + (((x * x) * ((x * x) * (x * x))) * 0.004629629629629629))) / (0.25 + (t_0 * (t_0 - 0.5))))))
	else:
		tmp = x * ((x * (x * x)) * 0.5)
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * 0.16666666666666666))
	tmp = 0.0
	if (Float64(x * x) <= 5e+152)
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(1.0 + Float64(Float64(Float64(x * x) * Float64(0.125 + Float64(Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * x))) * 0.004629629629629629))) / Float64(0.25 + Float64(t_0 * Float64(t_0 - 0.5)))))));
	else
		tmp = Float64(x * Float64(Float64(x * Float64(x * x)) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * 0.16666666666666666);
	tmp = 0.0;
	if ((x * x) <= 5e+152)
		tmp = 1.0 + ((x * x) * (1.0 + (((x * x) * (0.125 + (((x * x) * ((x * x) * (x * x))) * 0.004629629629629629))) / (0.25 + (t_0 * (t_0 - 0.5))))));
	else
		tmp = x * ((x * (x * x)) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e+152], N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(N[(N[(x * x), $MachinePrecision] * N[(0.125 + N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.004629629629629629), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.25 + N[(t$95$0 * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5e152

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

      2. *-lowering-*.f64 34.7%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

   7. Simplified 34.7%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 19.4%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 19.4%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{6}\right)}\right)\right)\right)\right) \]

       2. flip3-+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{{\frac{1}{2}}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)}^{3}}{\color{blue}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{6}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)}}\right)\right)\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\frac{\left(x \cdot x\right) \cdot \left({\frac{1}{2}}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)}^{3}\right)}{\color{blue}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{6}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)}}\right)\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(x \cdot x\right) \cdot \left({\frac{1}{2}}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)}^{3}\right)\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{2} + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{6}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)\right)\right)}\right)\right)\right)\right) \]

   12. Applied egg-rr 22.8%

       \[\leadsto 1 + \left(x \cdot x\right) \cdot \left(1 + \color{blue}{\frac{\left(x \cdot x\right) \cdot \left(0.125 + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 0.004629629629629629\right)}{0.25 + \left(x \cdot \left(x \cdot 0.16666666666666666\right)\right) \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right) - 0.5\right)}}\right) \]

   if 5e152 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot x\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

   10. Simplified 100.0%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{4}} \]
   12. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot {x}^{\left(2 \cdot \color{blue}{2}\right)} \]

       2. pow-sqr N/A

          \[\leadsto \frac{1}{2} \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

       4. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

       5. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

       6. *-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot x\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot x\right)}\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot x\right)}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right) \]

       10. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot {x}^{\color{blue}{3}}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

       12. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

       16. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{x \cdot \left(0.5 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+152}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(1 + \frac{\left(x \cdot x\right) \cdot \left(0.125 + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 0.004629629629629629\right)}{0.25 + \left(x \cdot \left(x \cdot 0.16666666666666666\right)\right) \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right) - 0.5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+102}:\\ \;\;\;\;\frac{1 - t\_0 \cdot t\_0}{\left(1 + t\_0\right) \cdot \left(1 - x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) (* x x))))
   (if (<= (* x x) 2e+102)
     (/ (- 1.0 (* t_0 t_0)) (* (+ 1.0 t_0) (- 1.0 (* x x))))
     (* x (* x (* (* x (* x (* x x))) 0.16666666666666666))))))

C

double code(double x) {
	double t_0 = (x * x) * (x * x);
	double tmp;
	if ((x * x) <= 2e+102) {
		tmp = (1.0 - (t_0 * t_0)) / ((1.0 + t_0) * (1.0 - (x * x)));
	} else {
		tmp = x * (x * ((x * (x * (x * x))) * 0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * x) * (x * x)
    if ((x * x) <= 2d+102) then
        tmp = (1.0d0 - (t_0 * t_0)) / ((1.0d0 + t_0) * (1.0d0 - (x * x)))
    else
        tmp = x * (x * ((x * (x * (x * x))) * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * (x * x);
	double tmp;
	if ((x * x) <= 2e+102) {
		tmp = (1.0 - (t_0 * t_0)) / ((1.0 + t_0) * (1.0 - (x * x)));
	} else {
		tmp = x * (x * ((x * (x * (x * x))) * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) * (x * x)
	tmp = 0
	if (x * x) <= 2e+102:
		tmp = (1.0 - (t_0 * t_0)) / ((1.0 + t_0) * (1.0 - (x * x)))
	else:
		tmp = x * (x * ((x * (x * (x * x))) * 0.16666666666666666))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(x * x))
	tmp = 0.0
	if (Float64(x * x) <= 2e+102)
		tmp = Float64(Float64(1.0 - Float64(t_0 * t_0)) / Float64(Float64(1.0 + t_0) * Float64(1.0 - Float64(x * x))));
	else
		tmp = Float64(x * Float64(x * Float64(Float64(x * Float64(x * Float64(x * x))) * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) * (x * x);
	tmp = 0.0;
	if ((x * x) <= 2e+102)
		tmp = (1.0 - (t_0 * t_0)) / ((1.0 + t_0) * (1.0 - (x * x)));
	else
		tmp = x * (x * ((x * (x * (x * x))) * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 2e+102], N[(N[(1.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + t$95$0), $MachinePrecision] * N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.99999999999999995e102

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

      2. *-lowering-*.f64 31.8%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

   7. Simplified 31.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2}} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 15.4%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   10. Simplified 15.4%

       \[\leadsto \color{blue}{1 + x \cdot x} \]
   11. Step-by-step derivation

       1. flip-+ N/A

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

       2. div-inv N/A

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

       3. metadata-eval N/A

          \[\leadsto \left(1 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{1 - x \cdot x} \]

       4. flip-- N/A

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

       5. frac-times N/A

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

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(1 \cdot 1 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 1\right), \color{blue}{\left(\left(1 + \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(1 - x \cdot x\right)\right)}\right) \]

   12. Applied egg-rr 19.1%

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

   if 1.99999999999999995e102 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 100.0%

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

   11. Taylor expanded in x around inf

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

       1. metadata-eval N/A

          \[\leadsto \frac{1}{6} \cdot {x}^{\left(2 \cdot \color{blue}{3}\right)} \]

       2. pow-sqr N/A

          \[\leadsto \frac{1}{6} \cdot \left({x}^{3} \cdot \color{blue}{{x}^{3}}\right) \]

       3. cube-prod N/A

          \[\leadsto \frac{1}{6} \cdot {\left(x \cdot x\right)}^{\color{blue}{3}} \]

       4. unpow2 N/A

          \[\leadsto \frac{1}{6} \cdot {\left({x}^{2}\right)}^{3} \]

       5. unpow3 N/A

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

       6. pow-sqr N/A

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

       7. metadata-eval N/A

          \[\leadsto \frac{1}{6} \cdot \left({x}^{4} \cdot {x}^{2}\right) \]

       8. associate-*l* N/A

          \[\leadsto \left(\frac{1}{6} \cdot {x}^{4}\right) \cdot \color{blue}{{x}^{2}} \]

       9. metadata-eval N/A

          \[\leadsto \left(\frac{1}{6} \cdot {x}^{\left(2 \cdot 2\right)}\right) \cdot {x}^{2} \]

       10. pow-sqr N/A

           \[\leadsto \left(\frac{1}{6} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot {x}^{2} \]

       11. associate-*l* N/A

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

       12. unpow2 N/A

           \[\leadsto \left(\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \left(x \cdot x\right)\right) \cdot {x}^{2} \]

       13. associate-*r* N/A

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

       14. associate-*r* N/A

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

       15. unpow2 N/A

           \[\leadsto \left(\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot {x}^{2}\right) \cdot {x}^{2} \]

       16. *-commutative N/A

           \[\leadsto \left(\left({x}^{2} \cdot \frac{1}{6}\right) \cdot {x}^{2}\right) \cdot {x}^{2} \]

       17. associate-*r* N/A

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

       18. *-commutative N/A

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

       19. unpow2 N/A

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

       20. associate-*l* N/A

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

       21. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot {x}^{2}\right)\right)\right)}\right) \]

   13. Simplified 100.0%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+102}:\\ \;\;\;\;\frac{1 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{\left(1 + \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(1 - x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 2e-13)
   1.0
   (* (* x x) (* (* x x) (+ 0.5 (* x (* x 0.16666666666666666)))))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 2e-13) {
		tmp = 1.0;
	} else {
		tmp = (x * x) * ((x * x) * (0.5 + (x * (x * 0.16666666666666666))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * x) <= 2d-13) then
        tmp = 1.0d0
    else
        tmp = (x * x) * ((x * x) * (0.5d0 + (x * (x * 0.16666666666666666d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 2e-13) {
		tmp = 1.0;
	} else {
		tmp = (x * x) * ((x * x) * (0.5 + (x * (x * 0.16666666666666666))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 2e-13:
		tmp = 1.0
	else:
		tmp = (x * x) * ((x * x) * (0.5 + (x * (x * 0.16666666666666666))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 2e-13)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(x * 0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 2e-13)
		tmp = 1.0;
	else
		tmp = (x * x) * ((x * x) * (0.5 + (x * (x * 0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-13], 1.0, N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.0000000000000001e-13

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

      2. *-lowering-*.f64 17.8%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

   7. Simplified 17.8%

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

   9. Applied egg-rr 17.8%

      \[\leadsto \color{blue}{1} \]

   if 2.0000000000000001e-13 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 80.9%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 80.9%

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

   11. Taylor expanded in x around inf

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

   13. Simplified 80.9%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 51.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 2e-13)
   1.0
   (* x (* x (* (* x (* x (* x x))) 0.16666666666666666)))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 2e-13) {
		tmp = 1.0;
	} else {
		tmp = x * (x * ((x * (x * (x * x))) * 0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * x) <= 2d-13) then
        tmp = 1.0d0
    else
        tmp = x * (x * ((x * (x * (x * x))) * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 2e-13) {
		tmp = 1.0;
	} else {
		tmp = x * (x * ((x * (x * (x * x))) * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 2e-13:
		tmp = 1.0
	else:
		tmp = x * (x * ((x * (x * (x * x))) * 0.16666666666666666))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 2e-13)
		tmp = 1.0;
	else
		tmp = Float64(x * Float64(x * Float64(Float64(x * Float64(x * Float64(x * x))) * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 2e-13)
		tmp = 1.0;
	else
		tmp = x * (x * ((x * (x * (x * x))) * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-13], 1.0, N[(x * N[(x * N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.0000000000000001e-13

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

      2. *-lowering-*.f64 17.8%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

   7. Simplified 17.8%

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

   9. Applied egg-rr 17.8%

      \[\leadsto \color{blue}{1} \]

   if 2.0000000000000001e-13 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 80.9%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 80.9%

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

   11. Taylor expanded in x around inf

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

       1. metadata-eval N/A

          \[\leadsto \frac{1}{6} \cdot {x}^{\left(2 \cdot \color{blue}{3}\right)} \]

       2. pow-sqr N/A

          \[\leadsto \frac{1}{6} \cdot \left({x}^{3} \cdot \color{blue}{{x}^{3}}\right) \]

       3. cube-prod N/A

          \[\leadsto \frac{1}{6} \cdot {\left(x \cdot x\right)}^{\color{blue}{3}} \]

       4. unpow2 N/A

          \[\leadsto \frac{1}{6} \cdot {\left({x}^{2}\right)}^{3} \]

       5. unpow3 N/A

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

       6. pow-sqr N/A

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

       7. metadata-eval N/A

          \[\leadsto \frac{1}{6} \cdot \left({x}^{4} \cdot {x}^{2}\right) \]

       8. associate-*l* N/A

          \[\leadsto \left(\frac{1}{6} \cdot {x}^{4}\right) \cdot \color{blue}{{x}^{2}} \]

       9. metadata-eval N/A

          \[\leadsto \left(\frac{1}{6} \cdot {x}^{\left(2 \cdot 2\right)}\right) \cdot {x}^{2} \]

       10. pow-sqr N/A

           \[\leadsto \left(\frac{1}{6} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot {x}^{2} \]

       11. associate-*l* N/A

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

       12. unpow2 N/A

           \[\leadsto \left(\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \left(x \cdot x\right)\right) \cdot {x}^{2} \]

       13. associate-*r* N/A

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

       14. associate-*r* N/A

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

       15. unpow2 N/A

           \[\leadsto \left(\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot {x}^{2}\right) \cdot {x}^{2} \]

       16. *-commutative N/A

           \[\leadsto \left(\left({x}^{2} \cdot \frac{1}{6}\right) \cdot {x}^{2}\right) \cdot {x}^{2} \]

       17. associate-*r* N/A

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

       18. *-commutative N/A

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

       19. unpow2 N/A

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

       20. associate-*l* N/A

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

       21. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} \cdot {x}^{2}\right)\right)\right)}\right) \]

   13. Simplified 80.9%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ 1 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.16666666666666666 + 0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  1.0
  (* (* x x) (+ 1.0 (* x (* x (+ (* (* x x) 0.16666666666666666) 0.5)))))))

C

double code(double x) {
	return 1.0 + ((x * x) * (1.0 + (x * (x * (((x * x) * 0.16666666666666666) + 0.5)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + ((x * x) * (1.0d0 + (x * (x * (((x * x) * 0.16666666666666666d0) + 0.5d0)))))
end function

Java

public static double code(double x) {
	return 1.0 + ((x * x) * (1.0 + (x * (x * (((x * x) * 0.16666666666666666) + 0.5)))));
}

Python

def code(x):
	return 1.0 + ((x * x) * (1.0 + (x * (x * (((x * x) * 0.16666666666666666) + 0.5)))))

Julia

function code(x)
	return Float64(1.0 + Float64(Float64(x * x) * Float64(1.0 + Float64(x * Float64(x * Float64(Float64(Float64(x * x) * 0.16666666666666666) + 0.5))))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + ((x * x) * (1.0 + (x * (x * (((x * x) * 0.16666666666666666) + 0.5)))));
end

Wolfram

code[x_] := N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(x * N[(x * N[(N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Step-by-step derivation

   1. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

   3. associate--r- N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

   7. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

   2. *-lowering-*.f64 59.2%

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

7. Simplified 59.2%

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

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)} \]
9. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

   16. *-lowering-*.f64 49.6%

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

10. Simplified 49.6%

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

11. Final simplification 49.6%

    \[\leadsto 1 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.16666666666666666 + 0.5\right)\right)\right) \]
12. Add Preprocessing

Alternative 11: 47.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 2e-13) 1.0 (* x (* x (+ 1.0 (* x (* x 0.5)))))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 2e-13) {
		tmp = 1.0;
	} else {
		tmp = x * (x * (1.0 + (x * (x * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * x) <= 2d-13) then
        tmp = 1.0d0
    else
        tmp = x * (x * (1.0d0 + (x * (x * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 2e-13) {
		tmp = 1.0;
	} else {
		tmp = x * (x * (1.0 + (x * (x * 0.5))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 2e-13:
		tmp = 1.0
	else:
		tmp = x * (x * (1.0 + (x * (x * 0.5))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 2e-13)
		tmp = 1.0;
	else
		tmp = Float64(x * Float64(x * Float64(1.0 + Float64(x * Float64(x * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 2e-13)
		tmp = 1.0;
	else
		tmp = x * (x * (1.0 + (x * (x * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-13], 1.0, N[(x * N[(x * N[(1.0 + N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.0000000000000001e-13

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

      2. *-lowering-*.f64 17.8%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

   7. Simplified 17.8%

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

   9. Applied egg-rr 17.8%

      \[\leadsto \color{blue}{1} \]

   if 2.0000000000000001e-13 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot x\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 75.7%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

   10. Simplified 75.7%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{{x}^{4} \cdot \left(\frac{1}{2} + \frac{1}{{x}^{2}}\right)} \]
   12. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto {x}^{\left(2 \cdot 2\right)} \cdot \left(\frac{1}{2} + \frac{1}{{x}^{2}}\right) \]

       2. pow-sqr N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. distribute-rgt-in N/A

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

       6. lft-mult-inverse N/A

          \[\leadsto \left(x \cdot x\right) \cdot \left(\frac{1}{2} \cdot {x}^{2} + 1\right) \]

       7. +-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot x\right)}\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)}\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)}\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)}\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

       15. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot x\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right)\right)\right) \]

       18. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

       19. *-lowering-*.f64 75.7%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

   13. Simplified 75.7%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot 0.5\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 47.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 2e-13) 1.0 (* x (* (* x (* x x)) 0.5))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 2e-13) {
		tmp = 1.0;
	} else {
		tmp = x * ((x * (x * x)) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * x) <= 2d-13) then
        tmp = 1.0d0
    else
        tmp = x * ((x * (x * x)) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 2e-13) {
		tmp = 1.0;
	} else {
		tmp = x * ((x * (x * x)) * 0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 2e-13:
		tmp = 1.0
	else:
		tmp = x * ((x * (x * x)) * 0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 2e-13)
		tmp = 1.0;
	else
		tmp = Float64(x * Float64(Float64(x * Float64(x * x)) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 2e-13)
		tmp = 1.0;
	else
		tmp = x * ((x * (x * x)) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-13], 1.0, N[(x * N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.0000000000000001e-13

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

      2. *-lowering-*.f64 17.8%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

   7. Simplified 17.8%

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

   9. Applied egg-rr 17.8%

      \[\leadsto \color{blue}{1} \]

   if 2.0000000000000001e-13 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot x\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 75.7%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

   10. Simplified 75.7%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{4}} \]
   12. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot {x}^{\left(2 \cdot \color{blue}{2}\right)} \]

       2. pow-sqr N/A

          \[\leadsto \frac{1}{2} \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

       4. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

       5. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

       6. *-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot x\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot x\right)}\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot x\right)}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right) \]

       10. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot {x}^{\color{blue}{3}}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

       12. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

       16. *-lowering-*.f64 75.7%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   13. Simplified 75.7%

       \[\leadsto \color{blue}{x \cdot \left(0.5 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.3% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.16666666666666666\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ 1.0 (* (* x x) (* (* x (* x (* x x))) 0.16666666666666666))))

C

double code(double x) {
	return 1.0 + ((x * x) * ((x * (x * (x * x))) * 0.16666666666666666));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + ((x * x) * ((x * (x * (x * x))) * 0.16666666666666666d0))
end function

Java

public static double code(double x) {
	return 1.0 + ((x * x) * ((x * (x * (x * x))) * 0.16666666666666666));
}

Python

def code(x):
	return 1.0 + ((x * x) * ((x * (x * (x * x))) * 0.16666666666666666))

Julia

function code(x)
	return Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(x * Float64(x * Float64(x * x))) * 0.16666666666666666)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + ((x * x) * ((x * (x * (x * x))) * 0.16666666666666666));
end

Wolfram

code[x_] := N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Step-by-step derivation

   1. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

   3. associate--r- N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

   7. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

   2. *-lowering-*.f64 59.2%

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

7. Simplified 59.2%

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

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)} \]
9. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

   16. *-lowering-*.f64 49.6%

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

10. Simplified 49.6%

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

11. Taylor expanded in x around inf

    \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \color{blue}{\left(\frac{1}{6} \cdot {x}^{4}\right)}\right)\right) \]
12. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{4}\right)}\right)\right)\right) \]

    2. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\frac{1}{6}, \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right)\right)\right) \]

    3. pow-plus N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\frac{1}{6}, \left({x}^{3} \cdot \color{blue}{x}\right)\right)\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{{x}^{3}}\right)\right)\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right)\right)\right)\right) \]

    6. cube-mult N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right)\right) \]

    7. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right)\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right)\right) \]

    9. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

    10. *-lowering-*.f64 49.6%

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

13. Simplified 49.6%

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

14. Final simplification 49.6%

    \[\leadsto 1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.16666666666666666\right) \]
15. Add Preprocessing

Alternative 14: 47.4% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ 1 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.5\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 1.0 (* (* x x) (* x (* x 0.5)))))

C

double code(double x) {
	return 1.0 + ((x * x) * (x * (x * 0.5)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + ((x * x) * (x * (x * 0.5d0)))
end function

Java

public static double code(double x) {
	return 1.0 + ((x * x) * (x * (x * 0.5)));
}

Python

def code(x):
	return 1.0 + ((x * x) * (x * (x * 0.5)))

Julia

function code(x)
	return Float64(1.0 + Float64(Float64(x * x) * Float64(x * Float64(x * 0.5))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + ((x * x) * (x * (x * 0.5)));
end

Wolfram

code[x_] := N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Step-by-step derivation

   1. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

   3. associate--r- N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

   7. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

   2. *-lowering-*.f64 59.2%

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

7. Simplified 59.2%

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

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \]
9. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot x\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 47.0%

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

10. Simplified 47.0%

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

11. Taylor expanded in x around inf

    \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)}\right)\right) \]
12. Step-by-step derivation

    1. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} \cdot \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

    2. associate-*r* N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(\frac{1}{2} \cdot x\right) \cdot \color{blue}{x}\right)\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right)\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

    6. *-lowering-*.f64 47.0%

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

13. Simplified 47.0%

    \[\leadsto 1 + \left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot 0.5\right)\right)} \]
14. Add Preprocessing

Alternative 15: 35.7% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 0.2) 1.0 (* x x)))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.2) {
		tmp = 1.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * x) <= 0.2d0) then
        tmp = 1.0d0
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 0.2) {
		tmp = 1.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.2:
		tmp = 1.0
	else:
		tmp = x * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.2)
		tmp = 1.0;
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 0.2)
		tmp = 1.0;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.2], 1.0, N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 0.20000000000000001

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

      2. *-lowering-*.f64 17.8%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

   7. Simplified 17.8%

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

   9. Applied egg-rr 17.8%

      \[\leadsto \color{blue}{1} \]

   if 0.20000000000000001 < (*.f64 x x)

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

      3. associate--r- N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2}} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 55.2%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   10. Simplified 55.2%

       \[\leadsto \color{blue}{1 + x \cdot x} \]

   11. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. *-lowering-*.f64 55.2%

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   13. Simplified 55.2%

       \[\leadsto \color{blue}{x \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 35.7% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ x \cdot x + 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (* x x) 1.0))

C

double code(double x) {
	return (x * x) + 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return (x * x) + 1.0

Julia

function code(x)
	return Float64(Float64(x * x) + 1.0)
end

MATLAB

function tmp = code(x)
	tmp = (x * x) + 1.0;
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Step-by-step derivation

   1. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

   3. associate--r- N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

   7. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

   2. *-lowering-*.f64 59.2%

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

7. Simplified 59.2%

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

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + {x}^{2}} \]
9. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{x}\right)\right) \]

   3. *-lowering-*.f64 36.6%

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

10. Simplified 36.6%

    \[\leadsto \color{blue}{1 + x \cdot x} \]

11. Final simplification 36.6%

    \[\leadsto x \cdot x + 1 \]
12. Add Preprocessing

Alternative 17: 10.5% accurate, 106.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Step-by-step derivation

   1. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(1 - x \cdot x\right)\right)\right) \]

   3. associate--r- N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\left(0 - 1\right) + x \cdot x\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(-1 + x \cdot x\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x + -1\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), -1\right)\right) \]

   7. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -1\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({x}^{2}\right)}\right) \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(x \cdot x\right)\right) \]

   2. *-lowering-*.f64 59.2%

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, x\right)\right) \]

7. Simplified 59.2%

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

9. Applied egg-rr 10.4%

   \[\leadsto \color{blue}{1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024161 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))