exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 9.8s

Alternatives: 14

Speedup: 1.0×

• 49.6% 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: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.5:\\ \;\;\;\;e^{-1}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 0.5

   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 98.9%

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

   7. Simplified 98.9%

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

   if 0.5 < (*.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%

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

Alternative 3: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\\ \mathbf{if}\;x \cdot x \leq 0.5:\\ \;\;\;\;e^{-1}\\ \mathbf{elif}\;x \cdot x \leq 10^{+150}:\\ \;\;\;\;1 + \frac{x \cdot \left(x \cdot \left(1 - x \cdot \left(\left(x \cdot t\_0\right) \cdot \left(\left(x \cdot x\right) \cdot t\_0\right)\right)\right)\right)}{1 - x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* x (* x 0.16666666666666666)))))
   (if (<= (* x x) 0.5)
     (exp -1.0)
     (if (<= (* x x) 1e+150)
       (+
        1.0
        (/
         (* x (* x (- 1.0 (* x (* (* x t_0) (* (* x x) t_0))))))
         (- 1.0 (* x (* x (+ 0.5 (* (* x x) 0.16666666666666666)))))))
       (* x (* x (* (* x x) 0.5)))))))

C

double code(double x) {
	double t_0 = 0.5 + (x * (x * 0.16666666666666666));
	double tmp;
	if ((x * x) <= 0.5) {
		tmp = exp(-1.0);
	} else if ((x * x) <= 1e+150) {
		tmp = 1.0 + ((x * (x * (1.0 - (x * ((x * t_0) * ((x * x) * t_0)))))) / (1.0 - (x * (x * (0.5 + ((x * x) * 0.16666666666666666))))));
	} 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 = 0.5d0 + (x * (x * 0.16666666666666666d0))
    if ((x * x) <= 0.5d0) then
        tmp = exp((-1.0d0))
    else if ((x * x) <= 1d+150) then
        tmp = 1.0d0 + ((x * (x * (1.0d0 - (x * ((x * t_0) * ((x * x) * t_0)))))) / (1.0d0 - (x * (x * (0.5d0 + ((x * x) * 0.16666666666666666d0))))))
    else
        tmp = x * (x * ((x * x) * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 0.5 + (x * (x * 0.16666666666666666));
	double tmp;
	if ((x * x) <= 0.5) {
		tmp = Math.exp(-1.0);
	} else if ((x * x) <= 1e+150) {
		tmp = 1.0 + ((x * (x * (1.0 - (x * ((x * t_0) * ((x * x) * t_0)))))) / (1.0 - (x * (x * (0.5 + ((x * x) * 0.16666666666666666))))));
	} else {
		tmp = x * (x * ((x * x) * 0.5));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 + (x * (x * 0.16666666666666666))
	tmp = 0
	if (x * x) <= 0.5:
		tmp = math.exp(-1.0)
	elif (x * x) <= 1e+150:
		tmp = 1.0 + ((x * (x * (1.0 - (x * ((x * t_0) * ((x * x) * t_0)))))) / (1.0 - (x * (x * (0.5 + ((x * x) * 0.16666666666666666))))))
	else:
		tmp = x * (x * ((x * x) * 0.5))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 + Float64(x * Float64(x * 0.16666666666666666)))
	tmp = 0.0
	if (Float64(x * x) <= 0.5)
		tmp = exp(-1.0);
	elseif (Float64(x * x) <= 1e+150)
		tmp = Float64(1.0 + Float64(Float64(x * Float64(x * Float64(1.0 - Float64(x * Float64(Float64(x * t_0) * Float64(Float64(x * x) * t_0)))))) / Float64(1.0 - Float64(x * Float64(x * Float64(0.5 + Float64(Float64(x * x) * 0.16666666666666666)))))));
	else
		tmp = Float64(x * Float64(x * Float64(Float64(x * x) * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 + (x * (x * 0.16666666666666666));
	tmp = 0.0;
	if ((x * x) <= 0.5)
		tmp = exp(-1.0);
	elseif ((x * x) <= 1e+150)
		tmp = 1.0 + ((x * (x * (1.0 - (x * ((x * t_0) * ((x * x) * t_0)))))) / (1.0 - (x * (x * (0.5 + ((x * x) * 0.16666666666666666))))));
	else
		tmp = x * (x * ((x * x) * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 + N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 0.5], N[Exp[-1.0], $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 1e+150], N[(1.0 + N[(N[(x * N[(x * N[(1.0 - N[(x * N[(N[(x * t$95$0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(x * N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 0.5

   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 98.9%

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

   7. Simplified 98.9%

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

   if 0.5 < (*.f64 x x) < 9.99999999999999981e149

   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. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 40.3%

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

   10. Simplified 40.3%

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

       1. *-commutative N/A

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

       2. flip-+ N/A

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

       3. associate-*l/ N/A

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

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

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

   12. Applied egg-rr 64.7%

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

       1. associate-*r* N/A

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

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

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

   14. Applied egg-rr 64.7%

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

   if 9.99999999999999981e149 < (*.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. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

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

   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

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

       2. pow-plus N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow3 N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

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

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

       11. *-commutative N/A

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.5:\\ \;\;\;\;e^{-1}\\ \mathbf{elif}\;x \cdot x \leq 10^{+150}:\\ \;\;\;\;1 + \frac{x \cdot \left(x \cdot \left(1 - x \cdot \left(\left(x \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)\right)\right)\right)}{1 - x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\\ \mathbf{if}\;x \cdot x \leq 10^{+150}:\\ \;\;\;\;1 + \frac{x \cdot \left(x \cdot \left(1 - x \cdot \left(\left(x \cdot t\_0\right) \cdot \left(\left(x \cdot x\right) \cdot t\_0\right)\right)\right)\right)}{1 - x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* x (* x 0.16666666666666666)))))
   (if (<= (* x x) 1e+150)
     (+
      1.0
      (/
       (* x (* x (- 1.0 (* x (* (* x t_0) (* (* x x) t_0))))))
       (- 1.0 (* x (* x (+ 0.5 (* (* x x) 0.16666666666666666)))))))
     (* x (* x (* (* x x) 0.5))))))

C

double code(double x) {
	double t_0 = 0.5 + (x * (x * 0.16666666666666666));
	double tmp;
	if ((x * x) <= 1e+150) {
		tmp = 1.0 + ((x * (x * (1.0 - (x * ((x * t_0) * ((x * x) * t_0)))))) / (1.0 - (x * (x * (0.5 + ((x * x) * 0.16666666666666666))))));
	} 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 = 0.5d0 + (x * (x * 0.16666666666666666d0))
    if ((x * x) <= 1d+150) then
        tmp = 1.0d0 + ((x * (x * (1.0d0 - (x * ((x * t_0) * ((x * x) * t_0)))))) / (1.0d0 - (x * (x * (0.5d0 + ((x * x) * 0.16666666666666666d0))))))
    else
        tmp = x * (x * ((x * x) * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 0.5 + (x * (x * 0.16666666666666666));
	double tmp;
	if ((x * x) <= 1e+150) {
		tmp = 1.0 + ((x * (x * (1.0 - (x * ((x * t_0) * ((x * x) * t_0)))))) / (1.0 - (x * (x * (0.5 + ((x * x) * 0.16666666666666666))))));
	} else {
		tmp = x * (x * ((x * x) * 0.5));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 + (x * (x * 0.16666666666666666))
	tmp = 0
	if (x * x) <= 1e+150:
		tmp = 1.0 + ((x * (x * (1.0 - (x * ((x * t_0) * ((x * x) * t_0)))))) / (1.0 - (x * (x * (0.5 + ((x * x) * 0.16666666666666666))))))
	else:
		tmp = x * (x * ((x * x) * 0.5))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 + Float64(x * Float64(x * 0.16666666666666666)))
	tmp = 0.0
	if (Float64(x * x) <= 1e+150)
		tmp = Float64(1.0 + Float64(Float64(x * Float64(x * Float64(1.0 - Float64(x * Float64(Float64(x * t_0) * Float64(Float64(x * x) * t_0)))))) / Float64(1.0 - Float64(x * Float64(x * Float64(0.5 + Float64(Float64(x * x) * 0.16666666666666666)))))));
	else
		tmp = Float64(x * Float64(x * Float64(Float64(x * x) * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 + (x * (x * 0.16666666666666666));
	tmp = 0.0;
	if ((x * x) <= 1e+150)
		tmp = 1.0 + ((x * (x * (1.0 - (x * ((x * t_0) * ((x * x) * t_0)))))) / (1.0 - (x * (x * (0.5 + ((x * x) * 0.16666666666666666))))));
	else
		tmp = x * (x * ((x * x) * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 + N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 1e+150], N[(1.0 + N[(N[(x * N[(x * N[(1.0 - N[(x * N[(N[(x * t$95$0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(x * N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 9.99999999999999981e149

   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 27.8%

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

   7. Simplified 27.8%

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

   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. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 20.5%

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

   10. Simplified 20.5%

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

       1. *-commutative N/A

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

       2. flip-+ N/A

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

       3. associate-*l/ N/A

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

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

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

   12. Applied egg-rr 23.5%

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

       1. associate-*r* N/A

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

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

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

   14. Applied egg-rr 23.5%

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

   if 9.99999999999999981e149 < (*.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. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

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

   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

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

       2. pow-plus N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow3 N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

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

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

       11. *-commutative N/A

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+150}:\\ \;\;\;\;1 + \frac{x \cdot \left(x \cdot \left(1 - x \cdot \left(\left(x \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)\right)\right)\right)}{1 - x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.3% accurate, 4.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 5e-10)
   1.0
   (* (* x x) (+ 1.0 (* (* x x) (+ 0.5 (* (* x x) 0.16666666666666666)))))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 5e-10) {
		tmp = 1.0;
	} else {
		tmp = (x * x) * (1.0 + ((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) <= 5d-10) then
        tmp = 1.0d0
    else
        tmp = (x * x) * (1.0d0 + ((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) <= 5e-10) {
		tmp = 1.0;
	} else {
		tmp = (x * x) * (1.0 + ((x * x) * (0.5 + ((x * x) * 0.16666666666666666))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 5e-10:
		tmp = 1.0
	else:
		tmp = (x * x) * (1.0 + ((x * x) * (0.5 + ((x * x) * 0.16666666666666666))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 5e-10)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x * x) * Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(Float64(x * x) * 0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 5e-10)
		tmp = 1.0;
	else
		tmp = (x * x) * (1.0 + ((x * x) * (0.5 + ((x * x) * 0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-10], 1.0, N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.00000000000000031e-10

   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%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 17.8%

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

   if 5.00000000000000031e-10 < (*.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 99.3%

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

   7. Simplified 99.3%

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

   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. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 89.9%

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

   10. Simplified 89.9%

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

   11. Taylor expanded in x around inf

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

   13. Simplified 89.9%

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

Alternative 6: 51.3% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 0.5

   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%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 17.8%

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

   if 0.5 < (*.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. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 90.5%

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

   10. Simplified 90.5%

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

   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 90.5%

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

4. Final simplification 51.6%

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

Alternative 7: 51.3% accurate, 5.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 0.5)
   1.0
   (* (* x x) (* x (* 0.16666666666666666 (* x (* x x)))))))

C

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

Fortran

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

Java

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

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.5:
		tmp = 1.0
	else:
		tmp = (x * x) * (x * (0.16666666666666666 * (x * (x * x))))
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 0.5

   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%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 17.8%

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

   if 0.5 < (*.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. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 90.5%

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

   10. Simplified 90.5%

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

   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. *-commutative N/A

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

       10. associate-*l* N/A

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

       11. metadata-eval N/A

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

       12. pow-sqr N/A

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

       13. associate-*r* N/A

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

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

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

       15. unpow2 N/A

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

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

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

       17. unpow2 N/A

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

       18. associate-*l* N/A

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

       19. *-commutative N/A

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

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

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

       21. associate-*l* N/A

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

       22. unpow2 N/A

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

       23. unpow3 N/A

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

       24. *-commutative N/A

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

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

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

   13. Simplified 90.5%

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

4. Final simplification 51.6%

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

Alternative 8: 51.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ 1 + \left(x \cdot x\right) \cdot \left(1 + \frac{1}{\frac{6}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ 1.0 (* (* x x) (+ 1.0 (/ 1.0 (/ 6.0 (* x (* x (* x x)))))))))

C

double code(double x) {
	return 1.0 + ((x * x) * (1.0 + (1.0 / (6.0 / (x * (x * (x * x)))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + ((x * x) * (1.0d0 + (1.0d0 / (6.0d0 / (x * (x * (x * x)))))))
end function

Java

public static double code(double x) {
	return 1.0 + ((x * x) * (1.0 + (1.0 / (6.0 / (x * (x * (x * x)))))));
}

Python

def code(x):
	return 1.0 + ((x * x) * (1.0 + (1.0 / (6.0 / (x * (x * (x * x)))))))

Julia

function code(x)
	return Float64(1.0 + Float64(Float64(x * x) * Float64(1.0 + Float64(1.0 / Float64(6.0 / Float64(x * Float64(x * Float64(x * x))))))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + ((x * x) * (1.0 + (1.0 / (6.0 / (x * (x * (x * x)))))));
end

Wolfram

code[x_] := N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(1.0 / N[(6.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $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 56.0%

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

7. Simplified 56.0%

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

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. *-lowering-*.f64 N/A

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

   7. unpow2 N/A

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

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

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

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

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

   10. *-commutative N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 51.6%

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

10. Simplified 51.6%

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

    1. *-commutative N/A

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

    2. flip-+ N/A

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

    3. associate-*l/ N/A

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

    4. clear-num N/A

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

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

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

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

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

    7. *-commutative N/A

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

    8. cancel-sign-sub-inv N/A

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

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

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

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

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

    11. metadata-eval N/A

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

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

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

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

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

12. Applied egg-rr 25.4%

    \[\leadsto 1 + \left(x \cdot x\right) \cdot \left(1 + \color{blue}{\frac{1}{\frac{0.5 + -0.16666666666666666 \cdot \left(x \cdot x\right)}{\left(0.25 - \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.027777777777777776\right) \cdot \left(x \cdot x\right)}}}\right) \]

13. Taylor expanded in x around inf

    \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{6}{{x}^{4}}\right)}\right)\right)\right)\right) \]
14. Step-by-step derivation

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

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(6, \color{blue}{\left({x}^{4}\right)}\right)\right)\right)\right)\right) \]

    2. metadata-eval N/A

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

    3. pow-plus N/A

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

    4. *-commutative N/A

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

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

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

    6. cube-mult N/A

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

    7. unpow2 N/A

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

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

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

    9. unpow2 N/A

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

    10. *-lowering-*.f64 51.6%

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

15. Simplified 51.6%

    \[\leadsto 1 + \left(x \cdot x\right) \cdot \left(1 + \frac{1}{\color{blue}{\frac{6}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}}}\right) \]
16. Add Preprocessing

Alternative 9: 51.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ 1 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  1.0
  (* (* x x) (+ 1.0 (* (* x x) (+ 0.5 (* x (* x 0.16666666666666666))))))))

C

double code(double x) {
	return 1.0 + ((x * x) * (1.0 + ((x * x) * (0.5 + (x * (x * 0.16666666666666666))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + ((x * x) * (1.0d0 + ((x * x) * (0.5d0 + (x * (x * 0.16666666666666666d0))))))
end function

Java

public static double code(double x) {
	return 1.0 + ((x * x) * (1.0 + ((x * x) * (0.5 + (x * (x * 0.16666666666666666))))));
}

Python

def code(x):
	return 1.0 + ((x * x) * (1.0 + ((x * x) * (0.5 + (x * (x * 0.16666666666666666))))))

Julia

function code(x)
	return Float64(1.0 + Float64(Float64(x * x) * Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(x * 0.16666666666666666)))))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + ((x * x) * (1.0 + ((x * x) * (0.5 + (x * (x * 0.16666666666666666))))));
end

Wolfram

code[x_] := N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $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 56.0%

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

7. Simplified 56.0%

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

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. *-lowering-*.f64 N/A

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

   7. unpow2 N/A

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

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

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

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

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

   10. *-commutative N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 51.6%

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

10. Simplified 51.6%

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

    1. associate-*l* N/A

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

    2. *-commutative N/A

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

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

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

    4. *-lowering-*.f64 51.6%

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

12. Applied egg-rr 51.6%

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

13. Final simplification 51.6%

    \[\leadsto 1 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\right) \]
14. Add Preprocessing

Alternative 10: 47.3% accurate, 5.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 5e-10) 1.0 (* x (* x (+ 1.0 (* (* x x) 0.5))))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 5e-10) {
		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) <= 5d-10) 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) <= 5e-10) {
		tmp = 1.0;
	} else {
		tmp = x * (x * (1.0 + ((x * x) * 0.5)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 5e-10:
		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) <= 5e-10)
		tmp = 1.0;
	else
		tmp = Float64(x * Float64(x * Float64(1.0 + Float64(Float64(x * x) * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 5e-10)
		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], 5e-10], 1.0, N[(x * N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.00000000000000031e-10

   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%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 17.8%

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

   if 5.00000000000000031e-10 < (*.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 99.3%

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

   7. Simplified 99.3%

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

   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. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 84.4%

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

   10. Simplified 84.4%

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

   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. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. unpow2 N/A

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

       6. cube-mult N/A

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

       7. associate-*r* N/A

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

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

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

       9. cube-mult N/A

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

       10. unpow2 N/A

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

       11. associate-*l* N/A

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

       12. distribute-rgt-in N/A

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

       13. lft-mult-inverse N/A

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

       14. +-commutative N/A

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

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

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

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

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

       17. *-commutative N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 84.4%

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

   13. Simplified 84.4%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 47.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 0.5) 1.0 (* x (* x (* (* x x) 0.5)))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.5) {
		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) <= 0.5d0) 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) <= 0.5) {
		tmp = 1.0;
	} else {
		tmp = x * (x * ((x * x) * 0.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.5:
		tmp = 1.0
	else:
		tmp = x * (x * ((x * x) * 0.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.5)
		tmp = 1.0;
	else
		tmp = Float64(x * Float64(x * Float64(Float64(x * x) * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 0.5)
		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], 0.5], 1.0, N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 0.5

   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%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 17.8%

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

   if 0.5 < (*.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. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 84.9%

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

   10. Simplified 84.9%

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

   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

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

       2. pow-plus N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow3 N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

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

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

       11. *-commutative N/A

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 84.9%

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

   13. Simplified 84.9%

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

Alternative 12: 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%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 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 99.3%

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

   7. Simplified 99.3%

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

   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 58.6%

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

   10. Simplified 58.6%

       \[\leadsto \color{blue}{1 + x \cdot x} \]

   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 58.6%

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

   13. Simplified 58.6%

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

Alternative 13: 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 56.0%

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

7. Simplified 56.0%

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

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 36.9%

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

10. Simplified 36.9%

    \[\leadsto \color{blue}{1 + x \cdot x} \]

11. Final simplification 36.9%

    \[\leadsto x \cdot x + 1 \]
12. Add Preprocessing

Alternative 14: 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 56.0%

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

7. Simplified 56.0%

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

8. Taylor expanded in x around 0

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

10. Simplified 11.0%

    \[\leadsto \color{blue}{1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))