Statistics.Distribution.Poisson.Internal:probability from math-functions-0.1.5.2

Percentage Accurate: 100.0% → 100.0%

Time: 10.0s

Alternatives: 21

Speedup: 1.0×

• 57.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ e^{\left(x + y \cdot \log y\right) - z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (exp (- (+ x (* y (log y))) z)))

C

double code(double x, double y, double z) {
	return exp(((x + (y * log(y))) - z));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = exp(((x + (y * log(y))) - z))
end function

Java

public static double code(double x, double y, double z) {
	return Math.exp(((x + (y * Math.log(y))) - z));
}

Python

def code(x, y, z):
	return math.exp(((x + (y * math.log(y))) - z))

Julia

function code(x, y, z)
	return exp(Float64(Float64(x + Float64(y * log(y))) - z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = exp(((x + (y * log(y))) - z));
end

Wolfram

code[x_, y_, z_] := N[Exp[N[(N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]], $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(x + y \cdot \log y\right) - z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (exp (- (+ x (* y (log y))) z)))

C

double code(double x, double y, double z) {
	return exp(((x + (y * log(y))) - z));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = exp(((x + (y * log(y))) - z))
end function

Java

public static double code(double x, double y, double z) {
	return Math.exp(((x + (y * Math.log(y))) - z));
}

Python

def code(x, y, z):
	return math.exp(((x + (y * math.log(y))) - z))

Julia

function code(x, y, z)
	return exp(Float64(Float64(x + Float64(y * log(y))) - z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = exp(((x + (y * log(y))) - z));
end

Wolfram

code[x_, y_, z_] := N[Exp[N[(N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]], $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(x + y \cdot \log y\right) - z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (exp (- (+ x (* y (log y))) z)))

C

double code(double x, double y, double z) {
	return exp(((x + (y * log(y))) - z));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = exp(((x + (y * log(y))) - z))
end function

Java

public static double code(double x, double y, double z) {
	return Math.exp(((x + (y * Math.log(y))) - z));
}

Python

def code(x, y, z):
	return math.exp(((x + (y * math.log(y))) - z))

Julia

function code(x, y, z)
	return exp(Float64(Float64(x + Float64(y * log(y))) - z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = exp(((x + (y * log(y))) - z));
end

Wolfram

code[x_, y_, z_] := N[Exp[N[(N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{\left(x + y \cdot \log y\right) - z} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 87.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \log y\\ t_1 := e^{z - x}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+72}:\\ \;\;\;\;\frac{{y}^{y}}{t\_1}\\ \mathbf{elif}\;t\_0 \leq 10^{+179}:\\ \;\;\;\;\frac{1}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (log y))) (t_1 (exp (- z x))))
   (if (<= t_0 5e+72)
     (/ (pow y y) t_1)
     (if (<= t_0 1e+179) (/ 1.0 t_1) (pow y y)))))

C

double code(double x, double y, double z) {
	double t_0 = y * log(y);
	double t_1 = exp((z - x));
	double tmp;
	if (t_0 <= 5e+72) {
		tmp = pow(y, y) / t_1;
	} else if (t_0 <= 1e+179) {
		tmp = 1.0 / t_1;
	} else {
		tmp = pow(y, y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y * log(y)
    t_1 = exp((z - x))
    if (t_0 <= 5d+72) then
        tmp = (y ** y) / t_1
    else if (t_0 <= 1d+179) then
        tmp = 1.0d0 / t_1
    else
        tmp = y ** y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * Math.log(y);
	double t_1 = Math.exp((z - x));
	double tmp;
	if (t_0 <= 5e+72) {
		tmp = Math.pow(y, y) / t_1;
	} else if (t_0 <= 1e+179) {
		tmp = 1.0 / t_1;
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * math.log(y)
	t_1 = math.exp((z - x))
	tmp = 0
	if t_0 <= 5e+72:
		tmp = math.pow(y, y) / t_1
	elif t_0 <= 1e+179:
		tmp = 1.0 / t_1
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * log(y))
	t_1 = exp(Float64(z - x))
	tmp = 0.0
	if (t_0 <= 5e+72)
		tmp = Float64((y ^ y) / t_1);
	elseif (t_0 <= 1e+179)
		tmp = Float64(1.0 / t_1);
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * log(y);
	t_1 = exp((z - x));
	tmp = 0.0;
	if (t_0 <= 5e+72)
		tmp = (y ^ y) / t_1;
	elseif (t_0 <= 1e+179)
		tmp = 1.0 / t_1;
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(z - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 5e+72], N[(N[Power[y, y], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 1e+179], N[(1.0 / t$95$1), $MachinePrecision], N[Power[y, y], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (log.f64 y)) < 4.99999999999999992e72

   1. Initial program 100.0%

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

      1. associate--l+ N/A

         \[\leadsto e^{x + \left(y \cdot \log y - z\right)} \]

      2. +-commutative N/A

         \[\leadsto e^{\left(y \cdot \log y - z\right) + x} \]

      3. associate-+l- N/A

         \[\leadsto e^{y \cdot \log y - \left(z - x\right)} \]

      4. exp-diff N/A

         \[\leadsto \frac{e^{y \cdot \log y}}{\color{blue}{e^{z - x}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{y \cdot \log y}\right), \color{blue}{\left(e^{z - x}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log y \cdot y}\right), \left(e^{\color{blue}{z} - x}\right)\right) \]

      7. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({y}^{y}\right), \left(e^{\color{blue}{z - x}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(y, y\right), \left(e^{\color{blue}{z - x}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(y, y\right), \mathsf{exp.f64}\left(\left(z - x\right)\right)\right) \]

      10. --lowering--.f64 94.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(y, y\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(z, x\right)\right)\right) \]

   3. Simplified 94.4%

      \[\leadsto \color{blue}{\frac{{y}^{y}}{e^{z - x}}} \]
   4. Add Preprocessing

   if 4.99999999999999992e72 < (*.f64 y (log.f64 y)) < 9.9999999999999998e178

   1. Initial program 100.0%

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

      1. associate--l+ N/A

         \[\leadsto e^{x + \left(y \cdot \log y - z\right)} \]

      2. +-commutative N/A

         \[\leadsto e^{\left(y \cdot \log y - z\right) + x} \]

      3. associate-+l- N/A

         \[\leadsto e^{y \cdot \log y - \left(z - x\right)} \]

      4. exp-diff N/A

         \[\leadsto \frac{e^{y \cdot \log y}}{\color{blue}{e^{z - x}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{y \cdot \log y}\right), \color{blue}{\left(e^{z - x}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log y \cdot y}\right), \left(e^{\color{blue}{z} - x}\right)\right) \]

      7. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({y}^{y}\right), \left(e^{\color{blue}{z - x}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(y, y\right), \left(e^{\color{blue}{z - x}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(y, y\right), \mathsf{exp.f64}\left(\left(z - x\right)\right)\right) \]

      10. --lowering--.f64 45.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(y, y\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(z, x\right)\right)\right) \]

   3. Simplified 45.5%

      \[\leadsto \color{blue}{\frac{{y}^{y}}{e^{z - x}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(z, x\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 73.4%

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

   if 9.9999999999999998e178 < (*.f64 y (log.f64 y))

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(y \cdot \log \left(\frac{1}{y}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y \cdot \log \left(\frac{1}{y}\right)\right)\right)\right) \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right)\right) \]

      4. remove-double-neg N/A

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

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

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

      6. log-lowering-log.f64 94.2%

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

   5. Simplified 94.2%

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

      1. *-commutative N/A

         \[\leadsto e^{\log y \cdot y} \]

      2. exp-to-pow N/A

         \[\leadsto {y}^{\color{blue}{y}} \]

      3. pow-lowering-pow.f64 94.2%

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

   7. Applied egg-rr 94.2%

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

Alternative 3: 88.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \log y \leq 10^{+179}:\\ \;\;\;\;\frac{1}{e^{z - x}}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* y (log y)) 1e+179) (/ 1.0 (exp (- z x))) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y * log(y)) <= 1e+179) {
		tmp = 1.0 / exp((z - x));
	} else {
		tmp = pow(y, y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y * log(y)) <= 1d+179) then
        tmp = 1.0d0 / exp((z - x))
    else
        tmp = y ** y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y * Math.log(y)) <= 1e+179) {
		tmp = 1.0 / Math.exp((z - x));
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y * math.log(y)) <= 1e+179:
		tmp = 1.0 / math.exp((z - x))
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(y * log(y)) <= 1e+179)
		tmp = Float64(1.0 / exp(Float64(z - x)));
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y * log(y)) <= 1e+179)
		tmp = 1.0 / exp((z - x));
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision], 1e+179], N[(1.0 / N[Exp[N[(z - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (log.f64 y)) < 9.9999999999999998e178

   1. Initial program 100.0%

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

      1. associate--l+ N/A

         \[\leadsto e^{x + \left(y \cdot \log y - z\right)} \]

      2. +-commutative N/A

         \[\leadsto e^{\left(y \cdot \log y - z\right) + x} \]

      3. associate-+l- N/A

         \[\leadsto e^{y \cdot \log y - \left(z - x\right)} \]

      4. exp-diff N/A

         \[\leadsto \frac{e^{y \cdot \log y}}{\color{blue}{e^{z - x}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{y \cdot \log y}\right), \color{blue}{\left(e^{z - x}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log y \cdot y}\right), \left(e^{\color{blue}{z} - x}\right)\right) \]

      7. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({y}^{y}\right), \left(e^{\color{blue}{z - x}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(y, y\right), \left(e^{\color{blue}{z - x}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(y, y\right), \mathsf{exp.f64}\left(\left(z - x\right)\right)\right) \]

      10. --lowering--.f64 83.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(y, y\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(z, x\right)\right)\right) \]

   3. Simplified 83.9%

      \[\leadsto \color{blue}{\frac{{y}^{y}}{e^{z - x}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(z, x\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 87.2%

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

   if 9.9999999999999998e178 < (*.f64 y (log.f64 y))

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(y \cdot \log \left(\frac{1}{y}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y \cdot \log \left(\frac{1}{y}\right)\right)\right)\right) \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right)\right) \]

      4. remove-double-neg N/A

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

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

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

      6. log-lowering-log.f64 94.2%

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

   5. Simplified 94.2%

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

      1. *-commutative N/A

         \[\leadsto e^{\log y \cdot y} \]

      2. exp-to-pow N/A

         \[\leadsto {y}^{\color{blue}{y}} \]

      3. pow-lowering-pow.f64 94.2%

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

   7. Applied egg-rr 94.2%

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

Alternative 4: 72.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{0 - z}\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{+41}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-28}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (exp (- 0.0 z))))
   (if (<= z -1.8e+41) t_0 (if (<= z 1.55e-28) (exp x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = exp((0.0 - z));
	double tmp;
	if (z <= -1.8e+41) {
		tmp = t_0;
	} else if (z <= 1.55e-28) {
		tmp = exp(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((0.0d0 - z))
    if (z <= (-1.8d+41)) then
        tmp = t_0
    else if (z <= 1.55d-28) then
        tmp = exp(x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.exp((0.0 - z));
	double tmp;
	if (z <= -1.8e+41) {
		tmp = t_0;
	} else if (z <= 1.55e-28) {
		tmp = Math.exp(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.exp((0.0 - z))
	tmp = 0
	if z <= -1.8e+41:
		tmp = t_0
	elif z <= 1.55e-28:
		tmp = math.exp(x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = exp(Float64(0.0 - z))
	tmp = 0.0
	if (z <= -1.8e+41)
		tmp = t_0;
	elseif (z <= 1.55e-28)
		tmp = exp(x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = exp((0.0 - z));
	tmp = 0.0;
	if (z <= -1.8e+41)
		tmp = t_0;
	elseif (z <= 1.55e-28)
		tmp = exp(x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Exp[N[(0.0 - z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1.8e+41], t$95$0, If[LessEqual[z, 1.55e-28], N[Exp[x], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.80000000000000013e41 or 1.54999999999999996e-28 < z

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 83.2%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 83.2%

      \[\leadsto e^{\color{blue}{0 - z}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-lowering-neg.f64 83.2%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(z\right)\right) \]

   7. Applied egg-rr 83.2%

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

   if -1.80000000000000013e41 < z < 1.54999999999999996e-28

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 70.4%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+41}:\\ \;\;\;\;e^{0 - z}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-28}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;e^{0 - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-192}:\\ \;\;\;\;e^{0 - z}\\ \mathbf{elif}\;y \leq 200:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.5e-192) (exp (- 0.0 z)) (if (<= y 200.0) (exp x) (pow y y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.5e-192) {
		tmp = exp((0.0 - z));
	} else if (y <= 200.0) {
		tmp = exp(x);
	} else {
		tmp = pow(y, y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 4.5d-192) then
        tmp = exp((0.0d0 - z))
    else if (y <= 200.0d0) then
        tmp = exp(x)
    else
        tmp = y ** y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.5e-192) {
		tmp = Math.exp((0.0 - z));
	} else if (y <= 200.0) {
		tmp = Math.exp(x);
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.5e-192:
		tmp = math.exp((0.0 - z))
	elif y <= 200.0:
		tmp = math.exp(x)
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.5e-192)
		tmp = exp(Float64(0.0 - z));
	elseif (y <= 200.0)
		tmp = exp(x);
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.5e-192)
		tmp = exp((0.0 - z));
	elseif (y <= 200.0)
		tmp = exp(x);
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.5e-192], N[Exp[N[(0.0 - z), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 200.0], N[Exp[x], $MachinePrecision], N[Power[y, y], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.50000000000000024e-192

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 84.7%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 84.7%

      \[\leadsto e^{\color{blue}{0 - z}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-lowering-neg.f64 84.7%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(z\right)\right) \]

   7. Applied egg-rr 84.7%

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

   if 4.50000000000000024e-192 < y < 200

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 79.0%

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

   if 200 < y

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \left(y \cdot \log \left(\frac{1}{y}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y \cdot \log \left(\frac{1}{y}\right)\right)\right)\right) \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right)\right) \]

      4. remove-double-neg N/A

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

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

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

      6. log-lowering-log.f64 77.3%

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

   5. Simplified 77.3%

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

      1. *-commutative N/A

         \[\leadsto e^{\log y \cdot y} \]

      2. exp-to-pow N/A

         \[\leadsto {y}^{\color{blue}{y}} \]

      3. pow-lowering-pow.f64 77.3%

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

   7. Applied egg-rr 77.3%

      \[\leadsto \color{blue}{{y}^{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-192}:\\ \;\;\;\;e^{0 - z}\\ \mathbf{elif}\;y \leq 200:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\\ t_1 := z \cdot \left(-1 + t\_0\right)\\ \mathbf{if}\;z \leq -4 \cdot 10^{+43}:\\ \;\;\;\;\left(t\_1 \cdot \left(z \cdot \left(1 - t\_0\right)\right) + 1\right) \cdot \frac{-1}{-1 - z}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-1 + t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (+ 0.5 (* z -0.16666666666666666))))
        (t_1 (* z (+ -1.0 t_0))))
   (if (<= z -4e+43)
     (* (+ (* t_1 (* z (- 1.0 t_0))) 1.0) (/ -1.0 (- -1.0 z)))
     (if (<= z 1.05e+103) (exp x) (/ -1.0 (+ -1.0 t_1))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (0.5 + (z * -0.16666666666666666));
	double t_1 = z * (-1.0 + t_0);
	double tmp;
	if (z <= -4e+43) {
		tmp = ((t_1 * (z * (1.0 - t_0))) + 1.0) * (-1.0 / (-1.0 - z));
	} else if (z <= 1.05e+103) {
		tmp = exp(x);
	} else {
		tmp = -1.0 / (-1.0 + t_1);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = z * (0.5d0 + (z * (-0.16666666666666666d0)))
    t_1 = z * ((-1.0d0) + t_0)
    if (z <= (-4d+43)) then
        tmp = ((t_1 * (z * (1.0d0 - t_0))) + 1.0d0) * ((-1.0d0) / ((-1.0d0) - z))
    else if (z <= 1.05d+103) then
        tmp = exp(x)
    else
        tmp = (-1.0d0) / ((-1.0d0) + t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (0.5 + (z * -0.16666666666666666));
	double t_1 = z * (-1.0 + t_0);
	double tmp;
	if (z <= -4e+43) {
		tmp = ((t_1 * (z * (1.0 - t_0))) + 1.0) * (-1.0 / (-1.0 - z));
	} else if (z <= 1.05e+103) {
		tmp = Math.exp(x);
	} else {
		tmp = -1.0 / (-1.0 + t_1);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (0.5 + (z * -0.16666666666666666))
	t_1 = z * (-1.0 + t_0)
	tmp = 0
	if z <= -4e+43:
		tmp = ((t_1 * (z * (1.0 - t_0))) + 1.0) * (-1.0 / (-1.0 - z))
	elif z <= 1.05e+103:
		tmp = math.exp(x)
	else:
		tmp = -1.0 / (-1.0 + t_1)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(0.5 + Float64(z * -0.16666666666666666)))
	t_1 = Float64(z * Float64(-1.0 + t_0))
	tmp = 0.0
	if (z <= -4e+43)
		tmp = Float64(Float64(Float64(t_1 * Float64(z * Float64(1.0 - t_0))) + 1.0) * Float64(-1.0 / Float64(-1.0 - z)));
	elseif (z <= 1.05e+103)
		tmp = exp(x);
	else
		tmp = Float64(-1.0 / Float64(-1.0 + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (0.5 + (z * -0.16666666666666666));
	t_1 = z * (-1.0 + t_0);
	tmp = 0.0;
	if (z <= -4e+43)
		tmp = ((t_1 * (z * (1.0 - t_0))) + 1.0) * (-1.0 / (-1.0 - z));
	elseif (z <= 1.05e+103)
		tmp = exp(x);
	else
		tmp = -1.0 / (-1.0 + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(0.5 + N[(z * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4e+43], N[(N[(N[(t$95$1 * N[(z * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-1.0 / N[(-1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.05e+103], N[Exp[x], $MachinePrecision], N[(-1.0 / N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.00000000000000006e43

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 91.7%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 91.7%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 77.4%

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

   8. Simplified 77.4%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

   10. Applied egg-rr 13.7%

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

   11. Taylor expanded in z around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \color{blue}{\left(-1 \cdot z\right)}\right)\right)\right) \]
   12. Step-by-step derivation

       1. neg-mul-1 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(z\right)\right)\right)\right)\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \left(0 - \color{blue}{z}\right)\right)\right)\right) \]

       3. --lowering--.f64 90.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{\_.f64}\left(0, \color{blue}{z}\right)\right)\right)\right) \]

   13. Simplified 90.1%

       \[\leadsto \left(1 - \left(z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)\right) \cdot \left(z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)\right)\right) \cdot \frac{1}{1 - \color{blue}{\left(0 - z\right)}} \]

   if -4.00000000000000006e43 < z < 1.0500000000000001e103

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 66.5%

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

   if 1.0500000000000001e103 < z

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 82.1%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 82.1%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 1.3%

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

   8. Simplified 1.3%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

   10. Applied egg-rr 0.0%

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

   11. Taylor expanded in z around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 82.1%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+43}:\\ \;\;\;\;\left(\left(z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)\right) \cdot \left(z \cdot \left(1 - z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)\right) + 1\right) \cdot \frac{-1}{-1 - z}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-1 + z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\\ t_1 := z \cdot \left(-1 + t\_0\right)\\ t_2 := t\_1 \cdot \left(z \cdot \left(1 - t\_0\right)\right) + 1\\ \mathbf{if}\;z \leq -1.45 \cdot 10^{+40}:\\ \;\;\;\;t\_2 \cdot \frac{-1}{-1 - z}\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-286}:\\ \;\;\;\;x \cdot \left(x \cdot 0.5 + 1\right) + 1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-46}:\\ \;\;\;\;t\_2 \cdot \frac{6 + \frac{18}{z}}{z \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-1 + t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (+ 0.5 (* z -0.16666666666666666))))
        (t_1 (* z (+ -1.0 t_0)))
        (t_2 (+ (* t_1 (* z (- 1.0 t_0))) 1.0)))
   (if (<= z -1.45e+40)
     (* t_2 (/ -1.0 (- -1.0 z)))
     (if (<= z -1.1e-286)
       (+ (* x (+ (* x 0.5) 1.0)) 1.0)
       (if (<= z 1.8e-46)
         (* t_2 (/ (+ 6.0 (/ 18.0 z)) (* z (* z z))))
         (/ -1.0 (+ -1.0 t_1)))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (0.5 + (z * -0.16666666666666666));
	double t_1 = z * (-1.0 + t_0);
	double t_2 = (t_1 * (z * (1.0 - t_0))) + 1.0;
	double tmp;
	if (z <= -1.45e+40) {
		tmp = t_2 * (-1.0 / (-1.0 - z));
	} else if (z <= -1.1e-286) {
		tmp = (x * ((x * 0.5) + 1.0)) + 1.0;
	} else if (z <= 1.8e-46) {
		tmp = t_2 * ((6.0 + (18.0 / z)) / (z * (z * z)));
	} else {
		tmp = -1.0 / (-1.0 + t_1);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = z * (0.5d0 + (z * (-0.16666666666666666d0)))
    t_1 = z * ((-1.0d0) + t_0)
    t_2 = (t_1 * (z * (1.0d0 - t_0))) + 1.0d0
    if (z <= (-1.45d+40)) then
        tmp = t_2 * ((-1.0d0) / ((-1.0d0) - z))
    else if (z <= (-1.1d-286)) then
        tmp = (x * ((x * 0.5d0) + 1.0d0)) + 1.0d0
    else if (z <= 1.8d-46) then
        tmp = t_2 * ((6.0d0 + (18.0d0 / z)) / (z * (z * z)))
    else
        tmp = (-1.0d0) / ((-1.0d0) + t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (0.5 + (z * -0.16666666666666666));
	double t_1 = z * (-1.0 + t_0);
	double t_2 = (t_1 * (z * (1.0 - t_0))) + 1.0;
	double tmp;
	if (z <= -1.45e+40) {
		tmp = t_2 * (-1.0 / (-1.0 - z));
	} else if (z <= -1.1e-286) {
		tmp = (x * ((x * 0.5) + 1.0)) + 1.0;
	} else if (z <= 1.8e-46) {
		tmp = t_2 * ((6.0 + (18.0 / z)) / (z * (z * z)));
	} else {
		tmp = -1.0 / (-1.0 + t_1);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (0.5 + (z * -0.16666666666666666))
	t_1 = z * (-1.0 + t_0)
	t_2 = (t_1 * (z * (1.0 - t_0))) + 1.0
	tmp = 0
	if z <= -1.45e+40:
		tmp = t_2 * (-1.0 / (-1.0 - z))
	elif z <= -1.1e-286:
		tmp = (x * ((x * 0.5) + 1.0)) + 1.0
	elif z <= 1.8e-46:
		tmp = t_2 * ((6.0 + (18.0 / z)) / (z * (z * z)))
	else:
		tmp = -1.0 / (-1.0 + t_1)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(0.5 + Float64(z * -0.16666666666666666)))
	t_1 = Float64(z * Float64(-1.0 + t_0))
	t_2 = Float64(Float64(t_1 * Float64(z * Float64(1.0 - t_0))) + 1.0)
	tmp = 0.0
	if (z <= -1.45e+40)
		tmp = Float64(t_2 * Float64(-1.0 / Float64(-1.0 - z)));
	elseif (z <= -1.1e-286)
		tmp = Float64(Float64(x * Float64(Float64(x * 0.5) + 1.0)) + 1.0);
	elseif (z <= 1.8e-46)
		tmp = Float64(t_2 * Float64(Float64(6.0 + Float64(18.0 / z)) / Float64(z * Float64(z * z))));
	else
		tmp = Float64(-1.0 / Float64(-1.0 + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (0.5 + (z * -0.16666666666666666));
	t_1 = z * (-1.0 + t_0);
	t_2 = (t_1 * (z * (1.0 - t_0))) + 1.0;
	tmp = 0.0;
	if (z <= -1.45e+40)
		tmp = t_2 * (-1.0 / (-1.0 - z));
	elseif (z <= -1.1e-286)
		tmp = (x * ((x * 0.5) + 1.0)) + 1.0;
	elseif (z <= 1.8e-46)
		tmp = t_2 * ((6.0 + (18.0 / z)) / (z * (z * z)));
	else
		tmp = -1.0 / (-1.0 + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(0.5 + N[(z * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[(z * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[z, -1.45e+40], N[(t$95$2 * N[(-1.0 / N[(-1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.1e-286], N[(N[(x * N[(N[(x * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 1.8e-46], N[(t$95$2 * N[(N[(6.0 + N[(18.0 / z), $MachinePrecision]), $MachinePrecision] / N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.45000000000000009e40

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 90.2%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 90.2%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 76.2%

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

   8. Simplified 76.2%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

   10. Applied egg-rr 13.5%

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

   11. Taylor expanded in z around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \color{blue}{\left(-1 \cdot z\right)}\right)\right)\right) \]
   12. Step-by-step derivation

       1. neg-mul-1 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(z\right)\right)\right)\right)\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \left(0 - \color{blue}{z}\right)\right)\right)\right) \]

       3. --lowering--.f64 88.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{\_.f64}\left(0, \color{blue}{z}\right)\right)\right)\right) \]

   13. Simplified 88.6%

       \[\leadsto \left(1 - \left(z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)\right) \cdot \left(z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)\right)\right) \cdot \frac{1}{1 - \color{blue}{\left(0 - z\right)}} \]

   if -1.45000000000000009e40 < z < -1.1e-286

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 70.9%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 51.6%

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

   8. Simplified 51.6%

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

   if -1.1e-286 < z < 1.8e-46

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 19.5%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 19.5%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 19.5%

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

   8. Simplified 19.5%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

   10. Applied egg-rr 19.5%

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

   11. Taylor expanded in z around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \color{blue}{\left(\frac{6 + 18 \cdot \frac{1}{z}}{{z}^{3}}\right)}\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\left(6 + 18 \cdot \frac{1}{z}\right), \color{blue}{\left({z}^{3}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(6, \left(18 \cdot \frac{1}{z}\right)\right), \left({\color{blue}{z}}^{3}\right)\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(6, \left(\frac{18 \cdot 1}{z}\right)\right), \left({z}^{3}\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(6, \left(\frac{18}{z}\right)\right), \left({z}^{3}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(6, \mathsf{/.f64}\left(18, z\right)\right), \left({z}^{3}\right)\right)\right) \]

       6. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(6, \mathsf{/.f64}\left(18, z\right)\right), \left(z \cdot \color{blue}{\left(z \cdot z\right)}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(6, \mathsf{/.f64}\left(18, z\right)\right), \left(z \cdot {z}^{\color{blue}{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(6, \mathsf{/.f64}\left(18, z\right)\right), \mathsf{*.f64}\left(z, \color{blue}{\left({z}^{2}\right)}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(6, \mathsf{/.f64}\left(18, z\right)\right), \mathsf{*.f64}\left(z, \left(z \cdot \color{blue}{z}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 43.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(6, \mathsf{/.f64}\left(18, z\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right)\right) \]

   13. Simplified 43.9%

       \[\leadsto \left(1 - \left(z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)\right) \cdot \left(z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)\right)\right) \cdot \color{blue}{\frac{6 + \frac{18}{z}}{z \cdot \left(z \cdot z\right)}} \]

   if 1.8e-46 < z

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 71.5%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 71.5%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 7.9%

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

   8. Simplified 7.9%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

   10. Applied egg-rr 7.2%

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

   11. Taylor expanded in z around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 51.7%

       \[\leadsto \color{blue}{1} \cdot \frac{1}{1 - z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+40}:\\ \;\;\;\;\left(\left(z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)\right) \cdot \left(z \cdot \left(1 - z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)\right) + 1\right) \cdot \frac{-1}{-1 - z}\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-286}:\\ \;\;\;\;x \cdot \left(x \cdot 0.5 + 1\right) + 1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-46}:\\ \;\;\;\;\left(\left(z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)\right) \cdot \left(z \cdot \left(1 - z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)\right) + 1\right) \cdot \frac{6 + \frac{18}{z}}{z \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-1 + z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + z \cdot -0.16666666666666666\\ t_1 := z \cdot \left(z \cdot z\right)\\ t_2 := z \cdot t\_0\\ t_3 := \frac{-1}{-1 + z \cdot \left(-1 + t\_2\right)}\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{+99}:\\ \;\;\;\;-0.16666666666666666 \cdot t\_1\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{+41}:\\ \;\;\;\;\frac{z \cdot \left(1 - t\_0 \cdot \left(t\_0 \cdot \left(z \cdot z\right)\right)\right)}{-1 - t\_2} + 1\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-186}:\\ \;\;\;\;x \cdot \left(x \cdot 0.5 + 1\right) + 1\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-53}:\\ \;\;\;\;t\_3 \cdot \left(-0.027777777777777776 \cdot \left(t\_1 \cdot t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* z -0.16666666666666666)))
        (t_1 (* z (* z z)))
        (t_2 (* z t_0))
        (t_3 (/ -1.0 (+ -1.0 (* z (+ -1.0 t_2))))))
   (if (<= z -2.4e+99)
     (* -0.16666666666666666 t_1)
     (if (<= z -1.45e+41)
       (+ (/ (* z (- 1.0 (* t_0 (* t_0 (* z z))))) (- -1.0 t_2)) 1.0)
       (if (<= z 2.7e-186)
         (+ (* x (+ (* x 0.5) 1.0)) 1.0)
         (if (<= z 9e-53)
           (* t_3 (* -0.027777777777777776 (* t_1 t_1)))
           t_3))))))

C

double code(double x, double y, double z) {
	double t_0 = 0.5 + (z * -0.16666666666666666);
	double t_1 = z * (z * z);
	double t_2 = z * t_0;
	double t_3 = -1.0 / (-1.0 + (z * (-1.0 + t_2)));
	double tmp;
	if (z <= -2.4e+99) {
		tmp = -0.16666666666666666 * t_1;
	} else if (z <= -1.45e+41) {
		tmp = ((z * (1.0 - (t_0 * (t_0 * (z * z))))) / (-1.0 - t_2)) + 1.0;
	} else if (z <= 2.7e-186) {
		tmp = (x * ((x * 0.5) + 1.0)) + 1.0;
	} else if (z <= 9e-53) {
		tmp = t_3 * (-0.027777777777777776 * (t_1 * t_1));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 0.5d0 + (z * (-0.16666666666666666d0))
    t_1 = z * (z * z)
    t_2 = z * t_0
    t_3 = (-1.0d0) / ((-1.0d0) + (z * ((-1.0d0) + t_2)))
    if (z <= (-2.4d+99)) then
        tmp = (-0.16666666666666666d0) * t_1
    else if (z <= (-1.45d+41)) then
        tmp = ((z * (1.0d0 - (t_0 * (t_0 * (z * z))))) / ((-1.0d0) - t_2)) + 1.0d0
    else if (z <= 2.7d-186) then
        tmp = (x * ((x * 0.5d0) + 1.0d0)) + 1.0d0
    else if (z <= 9d-53) then
        tmp = t_3 * ((-0.027777777777777776d0) * (t_1 * t_1))
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 0.5 + (z * -0.16666666666666666);
	double t_1 = z * (z * z);
	double t_2 = z * t_0;
	double t_3 = -1.0 / (-1.0 + (z * (-1.0 + t_2)));
	double tmp;
	if (z <= -2.4e+99) {
		tmp = -0.16666666666666666 * t_1;
	} else if (z <= -1.45e+41) {
		tmp = ((z * (1.0 - (t_0 * (t_0 * (z * z))))) / (-1.0 - t_2)) + 1.0;
	} else if (z <= 2.7e-186) {
		tmp = (x * ((x * 0.5) + 1.0)) + 1.0;
	} else if (z <= 9e-53) {
		tmp = t_3 * (-0.027777777777777776 * (t_1 * t_1));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 0.5 + (z * -0.16666666666666666)
	t_1 = z * (z * z)
	t_2 = z * t_0
	t_3 = -1.0 / (-1.0 + (z * (-1.0 + t_2)))
	tmp = 0
	if z <= -2.4e+99:
		tmp = -0.16666666666666666 * t_1
	elif z <= -1.45e+41:
		tmp = ((z * (1.0 - (t_0 * (t_0 * (z * z))))) / (-1.0 - t_2)) + 1.0
	elif z <= 2.7e-186:
		tmp = (x * ((x * 0.5) + 1.0)) + 1.0
	elif z <= 9e-53:
		tmp = t_3 * (-0.027777777777777776 * (t_1 * t_1))
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(0.5 + Float64(z * -0.16666666666666666))
	t_1 = Float64(z * Float64(z * z))
	t_2 = Float64(z * t_0)
	t_3 = Float64(-1.0 / Float64(-1.0 + Float64(z * Float64(-1.0 + t_2))))
	tmp = 0.0
	if (z <= -2.4e+99)
		tmp = Float64(-0.16666666666666666 * t_1);
	elseif (z <= -1.45e+41)
		tmp = Float64(Float64(Float64(z * Float64(1.0 - Float64(t_0 * Float64(t_0 * Float64(z * z))))) / Float64(-1.0 - t_2)) + 1.0);
	elseif (z <= 2.7e-186)
		tmp = Float64(Float64(x * Float64(Float64(x * 0.5) + 1.0)) + 1.0);
	elseif (z <= 9e-53)
		tmp = Float64(t_3 * Float64(-0.027777777777777776 * Float64(t_1 * t_1)));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 0.5 + (z * -0.16666666666666666);
	t_1 = z * (z * z);
	t_2 = z * t_0;
	t_3 = -1.0 / (-1.0 + (z * (-1.0 + t_2)));
	tmp = 0.0;
	if (z <= -2.4e+99)
		tmp = -0.16666666666666666 * t_1;
	elseif (z <= -1.45e+41)
		tmp = ((z * (1.0 - (t_0 * (t_0 * (z * z))))) / (-1.0 - t_2)) + 1.0;
	elseif (z <= 2.7e-186)
		tmp = (x * ((x * 0.5) + 1.0)) + 1.0;
	elseif (z <= 9e-53)
		tmp = t_3 * (-0.027777777777777776 * (t_1 * t_1));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(0.5 + N[(z * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(-1.0 / N[(-1.0 + N[(z * N[(-1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.4e+99], N[(-0.16666666666666666 * t$95$1), $MachinePrecision], If[LessEqual[z, -1.45e+41], N[(N[(N[(z * N[(1.0 - N[(t$95$0 * N[(t$95$0 * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 2.7e-186], N[(N[(x * N[(N[(x * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 9e-53], N[(t$95$3 * N[(-0.027777777777777776 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -2.4000000000000001e99

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 92.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 92.0%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 92.0%

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

   8. Simplified 92.0%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]
   10. Step-by-step derivation

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 92.0%

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

   11. Simplified 92.0%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)} \]

   if -2.4000000000000001e99 < z < -1.44999999999999994e41

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 82.1%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 82.1%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 5.8%

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

   8. Simplified 5.8%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   10. Applied egg-rr 73.5%

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

   if -1.44999999999999994e41 < z < 2.6999999999999999e-186

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 69.1%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 46.3%

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

   8. Simplified 46.3%

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

   if 2.6999999999999999e-186 < z < 8.9999999999999997e-53

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 12.5%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 12.5%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 12.5%

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

   8. Simplified 12.5%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

   10. Applied egg-rr 12.5%

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

   11. Taylor expanded in z around inf

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{-1}{36} \cdot {z}^{6}\right)}, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \left({z}^{6}\right)\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \left({z}^{\left(2 \cdot 3\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \left({z}^{3} \cdot {z}^{3}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\left({z}^{3}\right), \left({z}^{3}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       5. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\left(z \cdot \left(z \cdot z\right)\right), \left({z}^{3}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\left(z \cdot {z}^{2}\right), \left({z}^{3}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \left({z}^{2}\right)\right), \left({z}^{3}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \left(z \cdot z\right)\right), \left({z}^{3}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \left({z}^{3}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       10. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \left(z \cdot \left(z \cdot z\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \left(z \cdot {z}^{2}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{*.f64}\left(z, \left({z}^{2}\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{*.f64}\left(z, \left(z \cdot z\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 46.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

   13. Simplified 46.2%

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

   if 8.9999999999999997e-53 < z

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 70.5%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 70.5%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 7.9%

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

   8. Simplified 7.9%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

   10. Applied egg-rr 7.1%

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

   11. Taylor expanded in z around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 50.9%

       \[\leadsto \color{blue}{1} \cdot \frac{1}{1 - z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+99}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{+41}:\\ \;\;\;\;\frac{z \cdot \left(1 - \left(0.5 + z \cdot -0.16666666666666666\right) \cdot \left(\left(0.5 + z \cdot -0.16666666666666666\right) \cdot \left(z \cdot z\right)\right)\right)}{-1 - z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)} + 1\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-186}:\\ \;\;\;\;x \cdot \left(x \cdot 0.5 + 1\right) + 1\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-53}:\\ \;\;\;\;\frac{-1}{-1 + z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)} \cdot \left(-0.027777777777777776 \cdot \left(\left(z \cdot \left(z \cdot z\right)\right) \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-1 + z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.9% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot z\right)\\ t_1 := \frac{-1}{-1 + z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)}\\ t_2 := t\_1 \cdot \left(-0.027777777777777776 \cdot \left(t\_0 \cdot t\_0\right)\right)\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{+99}:\\ \;\;\;\;-0.16666666666666666 \cdot t\_0\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{+41}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-187}:\\ \;\;\;\;x \cdot \left(x \cdot 0.5 + 1\right) + 1\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-54}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* z z)))
        (t_1
         (/
          -1.0
          (+ -1.0 (* z (+ -1.0 (* z (+ 0.5 (* z -0.16666666666666666))))))))
        (t_2 (* t_1 (* -0.027777777777777776 (* t_0 t_0)))))
   (if (<= z -2.4e+99)
     (* -0.16666666666666666 t_0)
     (if (<= z -1.25e+41)
       t_2
       (if (<= z 1.95e-187)
         (+ (* x (+ (* x 0.5) 1.0)) 1.0)
         (if (<= z 2.8e-54) t_2 t_1))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (z * z);
	double t_1 = -1.0 / (-1.0 + (z * (-1.0 + (z * (0.5 + (z * -0.16666666666666666))))));
	double t_2 = t_1 * (-0.027777777777777776 * (t_0 * t_0));
	double tmp;
	if (z <= -2.4e+99) {
		tmp = -0.16666666666666666 * t_0;
	} else if (z <= -1.25e+41) {
		tmp = t_2;
	} else if (z <= 1.95e-187) {
		tmp = (x * ((x * 0.5) + 1.0)) + 1.0;
	} else if (z <= 2.8e-54) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = z * (z * z)
    t_1 = (-1.0d0) / ((-1.0d0) + (z * ((-1.0d0) + (z * (0.5d0 + (z * (-0.16666666666666666d0)))))))
    t_2 = t_1 * ((-0.027777777777777776d0) * (t_0 * t_0))
    if (z <= (-2.4d+99)) then
        tmp = (-0.16666666666666666d0) * t_0
    else if (z <= (-1.25d+41)) then
        tmp = t_2
    else if (z <= 1.95d-187) then
        tmp = (x * ((x * 0.5d0) + 1.0d0)) + 1.0d0
    else if (z <= 2.8d-54) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (z * z);
	double t_1 = -1.0 / (-1.0 + (z * (-1.0 + (z * (0.5 + (z * -0.16666666666666666))))));
	double t_2 = t_1 * (-0.027777777777777776 * (t_0 * t_0));
	double tmp;
	if (z <= -2.4e+99) {
		tmp = -0.16666666666666666 * t_0;
	} else if (z <= -1.25e+41) {
		tmp = t_2;
	} else if (z <= 1.95e-187) {
		tmp = (x * ((x * 0.5) + 1.0)) + 1.0;
	} else if (z <= 2.8e-54) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (z * z)
	t_1 = -1.0 / (-1.0 + (z * (-1.0 + (z * (0.5 + (z * -0.16666666666666666))))))
	t_2 = t_1 * (-0.027777777777777776 * (t_0 * t_0))
	tmp = 0
	if z <= -2.4e+99:
		tmp = -0.16666666666666666 * t_0
	elif z <= -1.25e+41:
		tmp = t_2
	elif z <= 1.95e-187:
		tmp = (x * ((x * 0.5) + 1.0)) + 1.0
	elif z <= 2.8e-54:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(z * z))
	t_1 = Float64(-1.0 / Float64(-1.0 + Float64(z * Float64(-1.0 + Float64(z * Float64(0.5 + Float64(z * -0.16666666666666666)))))))
	t_2 = Float64(t_1 * Float64(-0.027777777777777776 * Float64(t_0 * t_0)))
	tmp = 0.0
	if (z <= -2.4e+99)
		tmp = Float64(-0.16666666666666666 * t_0);
	elseif (z <= -1.25e+41)
		tmp = t_2;
	elseif (z <= 1.95e-187)
		tmp = Float64(Float64(x * Float64(Float64(x * 0.5) + 1.0)) + 1.0);
	elseif (z <= 2.8e-54)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (z * z);
	t_1 = -1.0 / (-1.0 + (z * (-1.0 + (z * (0.5 + (z * -0.16666666666666666))))));
	t_2 = t_1 * (-0.027777777777777776 * (t_0 * t_0));
	tmp = 0.0;
	if (z <= -2.4e+99)
		tmp = -0.16666666666666666 * t_0;
	elseif (z <= -1.25e+41)
		tmp = t_2;
	elseif (z <= 1.95e-187)
		tmp = (x * ((x * 0.5) + 1.0)) + 1.0;
	elseif (z <= 2.8e-54)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 / N[(-1.0 + N[(z * N[(-1.0 + N[(z * N[(0.5 + N[(z * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(-0.027777777777777776 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.4e+99], N[(-0.16666666666666666 * t$95$0), $MachinePrecision], If[LessEqual[z, -1.25e+41], t$95$2, If[LessEqual[z, 1.95e-187], N[(N[(x * N[(N[(x * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 2.8e-54], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.4000000000000001e99

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 92.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 92.0%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 92.0%

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

   8. Simplified 92.0%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]
   10. Step-by-step derivation

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 92.0%

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

   11. Simplified 92.0%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)} \]

   if -2.4000000000000001e99 < z < -1.25000000000000006e41 or 1.9499999999999999e-187 < z < 2.8000000000000002e-54

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 30.7%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 30.7%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 10.7%

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

   8. Simplified 10.7%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

   10. Applied egg-rr 28.5%

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

   11. Taylor expanded in z around inf

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{-1}{36} \cdot {z}^{6}\right)}, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \left({z}^{6}\right)\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \left({z}^{\left(2 \cdot 3\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \left({z}^{3} \cdot {z}^{3}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\left({z}^{3}\right), \left({z}^{3}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       5. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\left(z \cdot \left(z \cdot z\right)\right), \left({z}^{3}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\left(z \cdot {z}^{2}\right), \left({z}^{3}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \left({z}^{2}\right)\right), \left({z}^{3}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \left(z \cdot z\right)\right), \left({z}^{3}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \left({z}^{3}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       10. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \left(z \cdot \left(z \cdot z\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \left(z \cdot {z}^{2}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{*.f64}\left(z, \left({z}^{2}\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{*.f64}\left(z, \left(z \cdot z\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 53.3%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

   13. Simplified 53.3%

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

   if -1.25000000000000006e41 < z < 1.9499999999999999e-187

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 69.1%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 46.3%

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

   8. Simplified 46.3%

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

   if 2.8000000000000002e-54 < z

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 70.5%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 70.5%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 7.9%

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

   8. Simplified 7.9%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

   10. Applied egg-rr 7.1%

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

   11. Taylor expanded in z around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 50.9%

       \[\leadsto \color{blue}{1} \cdot \frac{1}{1 - z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+99}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{+41}:\\ \;\;\;\;\frac{-1}{-1 + z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)} \cdot \left(-0.027777777777777776 \cdot \left(\left(z \cdot \left(z \cdot z\right)\right) \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)\right)\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-187}:\\ \;\;\;\;x \cdot \left(x \cdot 0.5 + 1\right) + 1\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-54}:\\ \;\;\;\;\frac{-1}{-1 + z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)} \cdot \left(-0.027777777777777776 \cdot \left(\left(z \cdot \left(z \cdot z\right)\right) \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-1 + z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.1% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot z\right)\\ t_1 := z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\\ t_2 := z \cdot \left(-1 + t\_1\right)\\ t_3 := \frac{-1}{-1 + t\_2}\\ \mathbf{if}\;z \leq -1.45 \cdot 10^{+41}:\\ \;\;\;\;\left(t\_2 \cdot \left(z \cdot \left(1 - t\_1\right)\right) + 1\right) \cdot \frac{-1}{-1 - z}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-186}:\\ \;\;\;\;x \cdot \left(x \cdot 0.5 + 1\right) + 1\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-53}:\\ \;\;\;\;t\_3 \cdot \left(-0.027777777777777776 \cdot \left(t\_0 \cdot t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* z z)))
        (t_1 (* z (+ 0.5 (* z -0.16666666666666666))))
        (t_2 (* z (+ -1.0 t_1)))
        (t_3 (/ -1.0 (+ -1.0 t_2))))
   (if (<= z -1.45e+41)
     (* (+ (* t_2 (* z (- 1.0 t_1))) 1.0) (/ -1.0 (- -1.0 z)))
     (if (<= z 1.3e-186)
       (+ (* x (+ (* x 0.5) 1.0)) 1.0)
       (if (<= z 1.45e-53)
         (* t_3 (* -0.027777777777777776 (* t_0 t_0)))
         t_3)))))

C

double code(double x, double y, double z) {
	double t_0 = z * (z * z);
	double t_1 = z * (0.5 + (z * -0.16666666666666666));
	double t_2 = z * (-1.0 + t_1);
	double t_3 = -1.0 / (-1.0 + t_2);
	double tmp;
	if (z <= -1.45e+41) {
		tmp = ((t_2 * (z * (1.0 - t_1))) + 1.0) * (-1.0 / (-1.0 - z));
	} else if (z <= 1.3e-186) {
		tmp = (x * ((x * 0.5) + 1.0)) + 1.0;
	} else if (z <= 1.45e-53) {
		tmp = t_3 * (-0.027777777777777776 * (t_0 * t_0));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = z * (z * z)
    t_1 = z * (0.5d0 + (z * (-0.16666666666666666d0)))
    t_2 = z * ((-1.0d0) + t_1)
    t_3 = (-1.0d0) / ((-1.0d0) + t_2)
    if (z <= (-1.45d+41)) then
        tmp = ((t_2 * (z * (1.0d0 - t_1))) + 1.0d0) * ((-1.0d0) / ((-1.0d0) - z))
    else if (z <= 1.3d-186) then
        tmp = (x * ((x * 0.5d0) + 1.0d0)) + 1.0d0
    else if (z <= 1.45d-53) then
        tmp = t_3 * ((-0.027777777777777776d0) * (t_0 * t_0))
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (z * z);
	double t_1 = z * (0.5 + (z * -0.16666666666666666));
	double t_2 = z * (-1.0 + t_1);
	double t_3 = -1.0 / (-1.0 + t_2);
	double tmp;
	if (z <= -1.45e+41) {
		tmp = ((t_2 * (z * (1.0 - t_1))) + 1.0) * (-1.0 / (-1.0 - z));
	} else if (z <= 1.3e-186) {
		tmp = (x * ((x * 0.5) + 1.0)) + 1.0;
	} else if (z <= 1.45e-53) {
		tmp = t_3 * (-0.027777777777777776 * (t_0 * t_0));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (z * z)
	t_1 = z * (0.5 + (z * -0.16666666666666666))
	t_2 = z * (-1.0 + t_1)
	t_3 = -1.0 / (-1.0 + t_2)
	tmp = 0
	if z <= -1.45e+41:
		tmp = ((t_2 * (z * (1.0 - t_1))) + 1.0) * (-1.0 / (-1.0 - z))
	elif z <= 1.3e-186:
		tmp = (x * ((x * 0.5) + 1.0)) + 1.0
	elif z <= 1.45e-53:
		tmp = t_3 * (-0.027777777777777776 * (t_0 * t_0))
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(z * z))
	t_1 = Float64(z * Float64(0.5 + Float64(z * -0.16666666666666666)))
	t_2 = Float64(z * Float64(-1.0 + t_1))
	t_3 = Float64(-1.0 / Float64(-1.0 + t_2))
	tmp = 0.0
	if (z <= -1.45e+41)
		tmp = Float64(Float64(Float64(t_2 * Float64(z * Float64(1.0 - t_1))) + 1.0) * Float64(-1.0 / Float64(-1.0 - z)));
	elseif (z <= 1.3e-186)
		tmp = Float64(Float64(x * Float64(Float64(x * 0.5) + 1.0)) + 1.0);
	elseif (z <= 1.45e-53)
		tmp = Float64(t_3 * Float64(-0.027777777777777776 * Float64(t_0 * t_0)));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (z * z);
	t_1 = z * (0.5 + (z * -0.16666666666666666));
	t_2 = z * (-1.0 + t_1);
	t_3 = -1.0 / (-1.0 + t_2);
	tmp = 0.0;
	if (z <= -1.45e+41)
		tmp = ((t_2 * (z * (1.0 - t_1))) + 1.0) * (-1.0 / (-1.0 - z));
	elseif (z <= 1.3e-186)
		tmp = (x * ((x * 0.5) + 1.0)) + 1.0;
	elseif (z <= 1.45e-53)
		tmp = t_3 * (-0.027777777777777776 * (t_0 * t_0));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(0.5 + N[(z * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-1.0 / N[(-1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.45e+41], N[(N[(N[(t$95$2 * N[(z * N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-1.0 / N[(-1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.3e-186], N[(N[(x * N[(N[(x * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 1.45e-53], N[(t$95$3 * N[(-0.027777777777777776 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.44999999999999994e41

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 90.2%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 90.2%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 76.2%

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

   8. Simplified 76.2%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

   10. Applied egg-rr 13.5%

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

   11. Taylor expanded in z around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \color{blue}{\left(-1 \cdot z\right)}\right)\right)\right) \]
   12. Step-by-step derivation

       1. neg-mul-1 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(z\right)\right)\right)\right)\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \left(0 - \color{blue}{z}\right)\right)\right)\right) \]

       3. --lowering--.f64 88.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{\_.f64}\left(0, \color{blue}{z}\right)\right)\right)\right) \]

   13. Simplified 88.6%

       \[\leadsto \left(1 - \left(z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)\right) \cdot \left(z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)\right)\right) \cdot \frac{1}{1 - \color{blue}{\left(0 - z\right)}} \]

   if -1.44999999999999994e41 < z < 1.29999999999999997e-186

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 69.1%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 46.3%

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

   8. Simplified 46.3%

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

   if 1.29999999999999997e-186 < z < 1.4499999999999999e-53

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 12.5%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 12.5%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 12.5%

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

   8. Simplified 12.5%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

   10. Applied egg-rr 12.5%

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

   11. Taylor expanded in z around inf

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{-1}{36} \cdot {z}^{6}\right)}, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \left({z}^{6}\right)\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \left({z}^{\left(2 \cdot 3\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \left({z}^{3} \cdot {z}^{3}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\left({z}^{3}\right), \left({z}^{3}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       5. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\left(z \cdot \left(z \cdot z\right)\right), \left({z}^{3}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\left(z \cdot {z}^{2}\right), \left({z}^{3}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \left({z}^{2}\right)\right), \left({z}^{3}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \left(z \cdot z\right)\right), \left({z}^{3}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \left({z}^{3}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       10. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \left(z \cdot \left(z \cdot z\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \left(z \cdot {z}^{2}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{*.f64}\left(z, \left({z}^{2}\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{*.f64}\left(z, \left(z \cdot z\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 46.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

   13. Simplified 46.2%

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

   if 1.4499999999999999e-53 < z

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 70.5%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 70.5%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 7.9%

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

   8. Simplified 7.9%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

   10. Applied egg-rr 7.1%

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

   11. Taylor expanded in z around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 50.9%

       \[\leadsto \color{blue}{1} \cdot \frac{1}{1 - z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+41}:\\ \;\;\;\;\left(\left(z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)\right) \cdot \left(z \cdot \left(1 - z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)\right) + 1\right) \cdot \frac{-1}{-1 - z}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-186}:\\ \;\;\;\;x \cdot \left(x \cdot 0.5 + 1\right) + 1\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-53}:\\ \;\;\;\;\frac{-1}{-1 + z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)} \cdot \left(-0.027777777777777776 \cdot \left(\left(z \cdot \left(z \cdot z\right)\right) \cdot \left(z \cdot \left(z \cdot z\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-1 + z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.4% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -380:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+98}:\\ \;\;\;\;z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 + x \cdot 0.16666666666666666\right) + 1\right) + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -380.0)
   (* -0.16666666666666666 (* z (* z z)))
   (if (<= x 1.15e+98)
     (+ (* z (+ -1.0 (* z (+ 0.5 (* z -0.16666666666666666))))) 1.0)
     (+ (* x (+ (* x (+ 0.5 (* x 0.16666666666666666))) 1.0)) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -380.0) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (x <= 1.15e+98) {
		tmp = (z * (-1.0 + (z * (0.5 + (z * -0.16666666666666666))))) + 1.0;
	} else {
		tmp = (x * ((x * (0.5 + (x * 0.16666666666666666))) + 1.0)) + 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-380.0d0)) then
        tmp = (-0.16666666666666666d0) * (z * (z * z))
    else if (x <= 1.15d+98) then
        tmp = (z * ((-1.0d0) + (z * (0.5d0 + (z * (-0.16666666666666666d0)))))) + 1.0d0
    else
        tmp = (x * ((x * (0.5d0 + (x * 0.16666666666666666d0))) + 1.0d0)) + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -380.0) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (x <= 1.15e+98) {
		tmp = (z * (-1.0 + (z * (0.5 + (z * -0.16666666666666666))))) + 1.0;
	} else {
		tmp = (x * ((x * (0.5 + (x * 0.16666666666666666))) + 1.0)) + 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -380.0:
		tmp = -0.16666666666666666 * (z * (z * z))
	elif x <= 1.15e+98:
		tmp = (z * (-1.0 + (z * (0.5 + (z * -0.16666666666666666))))) + 1.0
	else:
		tmp = (x * ((x * (0.5 + (x * 0.16666666666666666))) + 1.0)) + 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -380.0)
		tmp = Float64(-0.16666666666666666 * Float64(z * Float64(z * z)));
	elseif (x <= 1.15e+98)
		tmp = Float64(Float64(z * Float64(-1.0 + Float64(z * Float64(0.5 + Float64(z * -0.16666666666666666))))) + 1.0);
	else
		tmp = Float64(Float64(x * Float64(Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666))) + 1.0)) + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -380.0)
		tmp = -0.16666666666666666 * (z * (z * z));
	elseif (x <= 1.15e+98)
		tmp = (z * (-1.0 + (z * (0.5 + (z * -0.16666666666666666))))) + 1.0;
	else
		tmp = (x * ((x * (0.5 + (x * 0.16666666666666666))) + 1.0)) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -380.0], N[(-0.16666666666666666 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.15e+98], N[(N[(z * N[(-1.0 + N[(z * N[(0.5 + N[(z * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x * N[(N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -380

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 36.1%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 36.1%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 17.4%

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

   8. Simplified 17.4%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]
   10. Step-by-step derivation

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 49.4%

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

   11. Simplified 49.4%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)} \]

   if -380 < x < 1.15000000000000007e98

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 68.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 68.0%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 46.1%

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

   8. Simplified 46.1%

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

   if 1.15000000000000007e98 < x

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 96.7%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 96.7%

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

   8. Simplified 96.7%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -380:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+98}:\\ \;\;\;\;z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 + x \cdot 0.16666666666666666\right) + 1\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 48.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -470:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+97}:\\ \;\;\;\;z \cdot \left(z \cdot \left(z \cdot -0.16666666666666666\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 + x \cdot 0.16666666666666666\right) + 1\right) + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -470.0)
   (* -0.16666666666666666 (* z (* z z)))
   (if (<= x 7e+97)
     (+ (* z (* z (* z -0.16666666666666666))) 1.0)
     (+ (* x (+ (* x (+ 0.5 (* x 0.16666666666666666))) 1.0)) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -470.0) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (x <= 7e+97) {
		tmp = (z * (z * (z * -0.16666666666666666))) + 1.0;
	} else {
		tmp = (x * ((x * (0.5 + (x * 0.16666666666666666))) + 1.0)) + 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-470.0d0)) then
        tmp = (-0.16666666666666666d0) * (z * (z * z))
    else if (x <= 7d+97) then
        tmp = (z * (z * (z * (-0.16666666666666666d0)))) + 1.0d0
    else
        tmp = (x * ((x * (0.5d0 + (x * 0.16666666666666666d0))) + 1.0d0)) + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -470.0) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (x <= 7e+97) {
		tmp = (z * (z * (z * -0.16666666666666666))) + 1.0;
	} else {
		tmp = (x * ((x * (0.5 + (x * 0.16666666666666666))) + 1.0)) + 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -470.0:
		tmp = -0.16666666666666666 * (z * (z * z))
	elif x <= 7e+97:
		tmp = (z * (z * (z * -0.16666666666666666))) + 1.0
	else:
		tmp = (x * ((x * (0.5 + (x * 0.16666666666666666))) + 1.0)) + 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -470.0)
		tmp = Float64(-0.16666666666666666 * Float64(z * Float64(z * z)));
	elseif (x <= 7e+97)
		tmp = Float64(Float64(z * Float64(z * Float64(z * -0.16666666666666666))) + 1.0);
	else
		tmp = Float64(Float64(x * Float64(Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666))) + 1.0)) + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -470.0)
		tmp = -0.16666666666666666 * (z * (z * z));
	elseif (x <= 7e+97)
		tmp = (z * (z * (z * -0.16666666666666666))) + 1.0;
	else
		tmp = (x * ((x * (0.5 + (x * 0.16666666666666666))) + 1.0)) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -470.0], N[(-0.16666666666666666 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7e+97], N[(N[(z * N[(z * N[(z * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x * N[(N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -470

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 36.1%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 36.1%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 17.4%

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

   8. Simplified 17.4%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]
   10. Step-by-step derivation

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 49.4%

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

   11. Simplified 49.4%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)} \]

   if -470 < x < 7.0000000000000001e97

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 68.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 68.0%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 46.1%

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

   8. Simplified 46.1%

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

   9. Taylor expanded in z around inf

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{-1}{6} \cdot {z}^{2}\right)}\right)\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 45.7%

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

   11. Simplified 45.7%

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

   if 7.0000000000000001e97 < x

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 96.7%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 96.7%

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

   8. Simplified 96.7%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -470:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+97}:\\ \;\;\;\;z \cdot \left(z \cdot \left(z \cdot -0.16666666666666666\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 + x \cdot 0.16666666666666666\right) + 1\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 36.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-88}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 1500:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+131}:\\ \;\;\;\;0.5 \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.5 + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.7e-88)
   (* -0.16666666666666666 (* z (* z z)))
   (if (<= x 1500.0)
     (+ x 1.0)
     (if (<= x 5.2e+131) (* 0.5 (* z z)) (* x (+ (* x 0.5) 1.0))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.7e-88) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (x <= 1500.0) {
		tmp = x + 1.0;
	} else if (x <= 5.2e+131) {
		tmp = 0.5 * (z * z);
	} else {
		tmp = x * ((x * 0.5) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.7d-88)) then
        tmp = (-0.16666666666666666d0) * (z * (z * z))
    else if (x <= 1500.0d0) then
        tmp = x + 1.0d0
    else if (x <= 5.2d+131) then
        tmp = 0.5d0 * (z * z)
    else
        tmp = x * ((x * 0.5d0) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.7e-88) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (x <= 1500.0) {
		tmp = x + 1.0;
	} else if (x <= 5.2e+131) {
		tmp = 0.5 * (z * z);
	} else {
		tmp = x * ((x * 0.5) + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.7e-88:
		tmp = -0.16666666666666666 * (z * (z * z))
	elif x <= 1500.0:
		tmp = x + 1.0
	elif x <= 5.2e+131:
		tmp = 0.5 * (z * z)
	else:
		tmp = x * ((x * 0.5) + 1.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.7e-88)
		tmp = Float64(-0.16666666666666666 * Float64(z * Float64(z * z)));
	elseif (x <= 1500.0)
		tmp = Float64(x + 1.0);
	elseif (x <= 5.2e+131)
		tmp = Float64(0.5 * Float64(z * z));
	else
		tmp = Float64(x * Float64(Float64(x * 0.5) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.7e-88)
		tmp = -0.16666666666666666 * (z * (z * z));
	elseif (x <= 1500.0)
		tmp = x + 1.0;
	elseif (x <= 5.2e+131)
		tmp = 0.5 * (z * z);
	else
		tmp = x * ((x * 0.5) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.7e-88], N[(-0.16666666666666666 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1500.0], N[(x + 1.0), $MachinePrecision], If[LessEqual[x, 5.2e+131], N[(0.5 * N[(z * z), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(x * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.69999999999999987e-88

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 42.7%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 42.7%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 25.2%

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

   8. Simplified 25.2%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]
   10. Step-by-step derivation

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 47.7%

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

   11. Simplified 47.7%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)} \]

   if -1.69999999999999987e-88 < x < 1500

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 33.1%

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

   6. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 32.0%

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

   8. Simplified 32.0%

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

   if 1500 < x < 5.2e131

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 60.4%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 60.4%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

      7. *-lowering-*.f64 35.8%

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

   8. Simplified 35.8%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]
   10. Step-by-step derivation

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 35.3%

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

   11. Simplified 35.3%

       \[\leadsto \color{blue}{0.5 \cdot \left(z \cdot z\right)} \]

   if 5.2e131 < x

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 96.1%

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

   8. Simplified 96.1%

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

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt-in N/A

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

       5. lft-mult-inverse N/A

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

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

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

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

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

       8. *-lowering-*.f64 96.1%

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

   11. Simplified 96.1%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-88}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 1500:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+131}:\\ \;\;\;\;0.5 \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.5 + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 36.5% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-91}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 1500:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+131}:\\ \;\;\;\;0.5 \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.6e-91)
   (* -0.16666666666666666 (* z (* z z)))
   (if (<= x 1500.0)
     (+ x 1.0)
     (if (<= x 5.2e+131) (* 0.5 (* z z)) (* 0.5 (* x x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.6e-91) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (x <= 1500.0) {
		tmp = x + 1.0;
	} else if (x <= 5.2e+131) {
		tmp = 0.5 * (z * z);
	} else {
		tmp = 0.5 * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-3.6d-91)) then
        tmp = (-0.16666666666666666d0) * (z * (z * z))
    else if (x <= 1500.0d0) then
        tmp = x + 1.0d0
    else if (x <= 5.2d+131) then
        tmp = 0.5d0 * (z * z)
    else
        tmp = 0.5d0 * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.6e-91) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (x <= 1500.0) {
		tmp = x + 1.0;
	} else if (x <= 5.2e+131) {
		tmp = 0.5 * (z * z);
	} else {
		tmp = 0.5 * (x * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.6e-91:
		tmp = -0.16666666666666666 * (z * (z * z))
	elif x <= 1500.0:
		tmp = x + 1.0
	elif x <= 5.2e+131:
		tmp = 0.5 * (z * z)
	else:
		tmp = 0.5 * (x * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.6e-91)
		tmp = Float64(-0.16666666666666666 * Float64(z * Float64(z * z)));
	elseif (x <= 1500.0)
		tmp = Float64(x + 1.0);
	elseif (x <= 5.2e+131)
		tmp = Float64(0.5 * Float64(z * z));
	else
		tmp = Float64(0.5 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.6e-91)
		tmp = -0.16666666666666666 * (z * (z * z));
	elseif (x <= 1500.0)
		tmp = x + 1.0;
	elseif (x <= 5.2e+131)
		tmp = 0.5 * (z * z);
	else
		tmp = 0.5 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.6e-91], N[(-0.16666666666666666 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1500.0], N[(x + 1.0), $MachinePrecision], If[LessEqual[x, 5.2e+131], N[(0.5 * N[(z * z), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.6e-91

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 42.7%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 42.7%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 25.2%

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

   8. Simplified 25.2%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]
   10. Step-by-step derivation

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 47.7%

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

   11. Simplified 47.7%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)} \]

   if -3.6e-91 < x < 1500

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 33.1%

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

   6. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 32.0%

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

   8. Simplified 32.0%

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

   if 1500 < x < 5.2e131

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 60.4%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 60.4%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

      7. *-lowering-*.f64 35.8%

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

   8. Simplified 35.8%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]
   10. Step-by-step derivation

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 35.3%

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

   11. Simplified 35.3%

       \[\leadsto \color{blue}{0.5 \cdot \left(z \cdot z\right)} \]

   if 5.2e131 < x

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 96.1%

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

   8. Simplified 96.1%

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

   9. Taylor expanded in x around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 96.1%

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

   11. Simplified 96.1%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-91}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 1500:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+131}:\\ \;\;\;\;0.5 \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 34.9% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(z \cdot z\right)\\ \mathbf{if}\;x \leq -4.3 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 750000000000:\\ \;\;\;\;1 - z\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+131}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 0.5 (* z z))))
   (if (<= x -4.3e-14)
     t_0
     (if (<= x 750000000000.0)
       (- 1.0 z)
       (if (<= x 5e+131) t_0 (* 0.5 (* x x)))))))

C

double code(double x, double y, double z) {
	double t_0 = 0.5 * (z * z);
	double tmp;
	if (x <= -4.3e-14) {
		tmp = t_0;
	} else if (x <= 750000000000.0) {
		tmp = 1.0 - z;
	} else if (x <= 5e+131) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (z * z)
    if (x <= (-4.3d-14)) then
        tmp = t_0
    else if (x <= 750000000000.0d0) then
        tmp = 1.0d0 - z
    else if (x <= 5d+131) then
        tmp = t_0
    else
        tmp = 0.5d0 * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 0.5 * (z * z);
	double tmp;
	if (x <= -4.3e-14) {
		tmp = t_0;
	} else if (x <= 750000000000.0) {
		tmp = 1.0 - z;
	} else if (x <= 5e+131) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (x * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 0.5 * (z * z)
	tmp = 0
	if x <= -4.3e-14:
		tmp = t_0
	elif x <= 750000000000.0:
		tmp = 1.0 - z
	elif x <= 5e+131:
		tmp = t_0
	else:
		tmp = 0.5 * (x * x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(0.5 * Float64(z * z))
	tmp = 0.0
	if (x <= -4.3e-14)
		tmp = t_0;
	elseif (x <= 750000000000.0)
		tmp = Float64(1.0 - z);
	elseif (x <= 5e+131)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 0.5 * (z * z);
	tmp = 0.0;
	if (x <= -4.3e-14)
		tmp = t_0;
	elseif (x <= 750000000000.0)
		tmp = 1.0 - z;
	elseif (x <= 5e+131)
		tmp = t_0;
	else
		tmp = 0.5 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(0.5 * N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.3e-14], t$95$0, If[LessEqual[x, 750000000000.0], N[(1.0 - z), $MachinePrecision], If[LessEqual[x, 5e+131], t$95$0, N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.29999999999999998e-14 or 7.5e11 < x < 4.99999999999999995e131

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 41.9%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 41.9%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

      7. *-lowering-*.f64 20.2%

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

   8. Simplified 20.2%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]
   10. Step-by-step derivation

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 38.1%

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

   11. Simplified 38.1%

       \[\leadsto \color{blue}{0.5 \cdot \left(z \cdot z\right)} \]

   if -4.29999999999999998e-14 < x < 7.5e11

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 68.3%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 68.3%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

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

      1. neg-mul-1 N/A

         \[\leadsto 1 + \left(\mathsf{neg}\left(z\right)\right) \]

      2. unsub-neg N/A

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

      3. --lowering--.f64 30.4%

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{z}\right) \]

   8. Simplified 30.4%

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

   if 4.99999999999999995e131 < x

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 96.1%

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

   8. Simplified 96.1%

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

   9. Taylor expanded in x around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 96.1%

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

   11. Simplified 96.1%

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

Alternative 16: 44.9% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -465:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+131}:\\ \;\;\;\;z \cdot \left(z \cdot \left(z \cdot -0.16666666666666666\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.5 + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -465.0)
   (* -0.16666666666666666 (* z (* z z)))
   (if (<= x 4.5e+131)
     (+ (* z (* z (* z -0.16666666666666666))) 1.0)
     (* x (+ (* x 0.5) 1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -465.0) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (x <= 4.5e+131) {
		tmp = (z * (z * (z * -0.16666666666666666))) + 1.0;
	} else {
		tmp = x * ((x * 0.5) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-465.0d0)) then
        tmp = (-0.16666666666666666d0) * (z * (z * z))
    else if (x <= 4.5d+131) then
        tmp = (z * (z * (z * (-0.16666666666666666d0)))) + 1.0d0
    else
        tmp = x * ((x * 0.5d0) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -465.0) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (x <= 4.5e+131) {
		tmp = (z * (z * (z * -0.16666666666666666))) + 1.0;
	} else {
		tmp = x * ((x * 0.5) + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -465.0:
		tmp = -0.16666666666666666 * (z * (z * z))
	elif x <= 4.5e+131:
		tmp = (z * (z * (z * -0.16666666666666666))) + 1.0
	else:
		tmp = x * ((x * 0.5) + 1.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -465.0)
		tmp = Float64(-0.16666666666666666 * Float64(z * Float64(z * z)));
	elseif (x <= 4.5e+131)
		tmp = Float64(Float64(z * Float64(z * Float64(z * -0.16666666666666666))) + 1.0);
	else
		tmp = Float64(x * Float64(Float64(x * 0.5) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -465.0)
		tmp = -0.16666666666666666 * (z * (z * z));
	elseif (x <= 4.5e+131)
		tmp = (z * (z * (z * -0.16666666666666666))) + 1.0;
	else
		tmp = x * ((x * 0.5) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -465.0], N[(-0.16666666666666666 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.5e+131], N[(N[(z * N[(z * N[(z * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(N[(x * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -465

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 36.1%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 36.1%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 17.4%

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

   8. Simplified 17.4%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]
   10. Step-by-step derivation

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 49.4%

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

   11. Simplified 49.4%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)} \]

   if -465 < x < 4.5000000000000002e131

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 67.4%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 67.4%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(z \cdot \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + -1\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \left(-1 + \color{blue}{z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \color{blue}{\left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right)\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{1}{2} + \frac{-1}{6} \cdot z\right)}\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{6} \cdot z\right)}\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \left(z \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 45.7%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 45.7%

      \[\leadsto \color{blue}{1 + z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{-1}{6} \cdot {z}^{2}\right)}\right)\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \left(\frac{-1}{6} \cdot \left(z \cdot \color{blue}{z}\right)\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \left(\left(\frac{-1}{6} \cdot z\right) \cdot \color{blue}{z}\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \left(z \cdot \color{blue}{\left(\frac{-1}{6} \cdot z\right)}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{-1}{6} \cdot z\right)}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \left(z \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 45.3%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   11. Simplified 45.3%

       \[\leadsto 1 + z \cdot \color{blue}{\left(z \cdot \left(z \cdot -0.16666666666666666\right)\right)} \]

   if 4.5000000000000002e131 < x

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{x}\right) \]
   4. Step-by-step derivation

   5. Simplified 100.0%

      \[\leadsto e^{\color{blue}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + \frac{1}{2} \cdot x\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{1}{2} \cdot x\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 96.1%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   8. Simplified 96.1%

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + x \cdot 0.5\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{x}\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{x}\right) \]

       2. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{x}\right)\right)} \]

       3. +-commutative N/A

          \[\leadsto x \cdot \left(x \cdot \left(\frac{1}{x} + \color{blue}{\frac{1}{2}}\right)\right) \]

       4. distribute-rgt-in N/A

          \[\leadsto x \cdot \left(\frac{1}{x} \cdot x + \color{blue}{\frac{1}{2} \cdot x}\right) \]

       5. lft-mult-inverse N/A

          \[\leadsto x \cdot \left(1 + \color{blue}{\frac{1}{2}} \cdot x\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{1}{2} \cdot x\right)}\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right) \]

       8. *-lowering-*.f64 96.1%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{x}\right)\right)\right) \]

   11. Simplified 96.1%

       \[\leadsto \color{blue}{x \cdot \left(1 + 0.5 \cdot x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -465:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+131}:\\ \;\;\;\;z \cdot \left(z \cdot \left(z \cdot -0.16666666666666666\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.5 + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 42.5% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-56}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+131}:\\ \;\;\;\;z \cdot \left(z \cdot 0.5\right) + 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.5 + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.45e-56)
   (* -0.16666666666666666 (* z (* z z)))
   (if (<= x 2.8e+131) (+ (* z (* z 0.5)) 1.0) (* x (+ (* x 0.5) 1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.45e-56) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (x <= 2.8e+131) {
		tmp = (z * (z * 0.5)) + 1.0;
	} else {
		tmp = x * ((x * 0.5) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.45d-56)) then
        tmp = (-0.16666666666666666d0) * (z * (z * z))
    else if (x <= 2.8d+131) then
        tmp = (z * (z * 0.5d0)) + 1.0d0
    else
        tmp = x * ((x * 0.5d0) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.45e-56) {
		tmp = -0.16666666666666666 * (z * (z * z));
	} else if (x <= 2.8e+131) {
		tmp = (z * (z * 0.5)) + 1.0;
	} else {
		tmp = x * ((x * 0.5) + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.45e-56:
		tmp = -0.16666666666666666 * (z * (z * z))
	elif x <= 2.8e+131:
		tmp = (z * (z * 0.5)) + 1.0
	else:
		tmp = x * ((x * 0.5) + 1.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.45e-56)
		tmp = Float64(-0.16666666666666666 * Float64(z * Float64(z * z)));
	elseif (x <= 2.8e+131)
		tmp = Float64(Float64(z * Float64(z * 0.5)) + 1.0);
	else
		tmp = Float64(x * Float64(Float64(x * 0.5) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.45e-56)
		tmp = -0.16666666666666666 * (z * (z * z));
	elseif (x <= 2.8e+131)
		tmp = (z * (z * 0.5)) + 1.0;
	else
		tmp = x * ((x * 0.5) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.45e-56], N[(-0.16666666666666666 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8e+131], N[(N[(z * N[(z * 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(N[(x * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999996e-56

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - z\right)\right) \]

      3. --lowering--.f64 39.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 39.0%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + z \cdot \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(z \cdot \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) - 1\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) + -1\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \left(-1 + \color{blue}{z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \color{blue}{\left(z \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot z\right)\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{1}{2} + \frac{-1}{6} \cdot z\right)}\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{6} \cdot z\right)}\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \left(z \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 19.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(z, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 19.6%

      \[\leadsto \color{blue}{1 + z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({z}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \left(z \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \left(z \cdot {z}^{\color{blue}{2}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(z, \color{blue}{\left({z}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(z, \left(z \cdot \color{blue}{z}\right)\right)\right) \]

       6. *-lowering-*.f64 47.8%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   11. Simplified 47.8%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)} \]

   if -1.44999999999999996e-56 < x < 2.8000000000000001e131

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - z\right)\right) \]

      3. --lowering--.f64 67.1%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 67.1%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + z \cdot \left(\frac{1}{2} \cdot z - 1\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(z \cdot \left(\frac{1}{2} \cdot z - 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{1}{2} \cdot z - 1\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \left(\frac{1}{2} \cdot z + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \left(\frac{1}{2} \cdot z + -1\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \left(-1 + \color{blue}{\frac{1}{2} \cdot z}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{1}{2} \cdot z\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 43.5%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{z}\right)\right)\right)\right) \]

   8. Simplified 43.5%

      \[\leadsto \color{blue}{1 + z \cdot \left(-1 + 0.5 \cdot z\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{1}{2} \cdot z\right)}\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 43.1%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{z}\right)\right)\right) \]

   11. Simplified 43.1%

       \[\leadsto 1 + z \cdot \color{blue}{\left(0.5 \cdot z\right)} \]

   if 2.8000000000000001e131 < x

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{x}\right) \]
   4. Step-by-step derivation

   5. Simplified 100.0%

      \[\leadsto e^{\color{blue}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + \frac{1}{2} \cdot x\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{1}{2} \cdot x\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 96.1%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   8. Simplified 96.1%

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + x \cdot 0.5\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{x}\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{x}\right) \]

       2. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{x}\right)\right)} \]

       3. +-commutative N/A

          \[\leadsto x \cdot \left(x \cdot \left(\frac{1}{x} + \color{blue}{\frac{1}{2}}\right)\right) \]

       4. distribute-rgt-in N/A

          \[\leadsto x \cdot \left(\frac{1}{x} \cdot x + \color{blue}{\frac{1}{2} \cdot x}\right) \]

       5. lft-mult-inverse N/A

          \[\leadsto x \cdot \left(1 + \color{blue}{\frac{1}{2}} \cdot x\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{1}{2} \cdot x\right)}\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right) \]

       8. *-lowering-*.f64 96.1%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{x}\right)\right)\right) \]

   11. Simplified 96.1%

       \[\leadsto \color{blue}{x \cdot \left(1 + 0.5 \cdot x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-56}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(z \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+131}:\\ \;\;\;\;z \cdot \left(z \cdot 0.5\right) + 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.5 + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 28.3% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{+156}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+76}:\\ \;\;\;\;1 - z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 0.5 (* x x))))
   (if (<= x -1.5e+156) t_0 (if (<= x 3.2e+76) (- 1.0 z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = 0.5 * (x * x);
	double tmp;
	if (x <= -1.5e+156) {
		tmp = t_0;
	} else if (x <= 3.2e+76) {
		tmp = 1.0 - z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (x * x)
    if (x <= (-1.5d+156)) then
        tmp = t_0
    else if (x <= 3.2d+76) then
        tmp = 1.0d0 - z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 0.5 * (x * x);
	double tmp;
	if (x <= -1.5e+156) {
		tmp = t_0;
	} else if (x <= 3.2e+76) {
		tmp = 1.0 - z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 0.5 * (x * x)
	tmp = 0
	if x <= -1.5e+156:
		tmp = t_0
	elif x <= 3.2e+76:
		tmp = 1.0 - z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(0.5 * Float64(x * x))
	tmp = 0.0
	if (x <= -1.5e+156)
		tmp = t_0;
	elseif (x <= 3.2e+76)
		tmp = Float64(1.0 - z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 0.5 * (x * x);
	tmp = 0.0;
	if (x <= -1.5e+156)
		tmp = t_0;
	elseif (x <= 3.2e+76)
		tmp = 1.0 - z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.5e+156], t$95$0, If[LessEqual[x, 3.2e+76], N[(1.0 - z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5e156 or 3.19999999999999976e76 < x

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{x}\right) \]
   4. Step-by-step derivation

   5. Simplified 86.5%

      \[\leadsto e^{\color{blue}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + \frac{1}{2} \cdot x\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{1}{2} \cdot x\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 42.7%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   8. Simplified 42.7%

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + x \cdot 0.5\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{x}\right)\right) \]

       3. *-lowering-*.f64 42.7%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   11. Simplified 42.7%

       \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot x\right)} \]

   if -1.5e156 < x < 3.19999999999999976e76

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - z\right)\right) \]

      3. --lowering--.f64 61.7%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

   5. Simplified 61.7%

      \[\leadsto e^{\color{blue}{0 - z}} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + -1 \cdot z} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto 1 + \left(\mathsf{neg}\left(z\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto 1 - \color{blue}{z} \]

      3. --lowering--.f64 23.5%

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{z}\right) \]

   8. Simplified 23.5%

      \[\leadsto \color{blue}{1 - z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 14.7% accurate, 69.0× speedup

Math

\[\begin{array}{l} \\ 1 - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- 1.0 z))

C

double code(double x, double y, double z) {
	return 1.0 - z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 - z
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 - z;
}

Python

def code(x, y, z):
	return 1.0 - z

Julia

function code(x, y, z)
	return Float64(1.0 - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 - z;
end

Wolfram

code[x_, y_, z_] := N[(1.0 - z), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{\left(x + y \cdot \log y\right) - z} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot z\right)}\right) \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(z\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(0 - z\right)\right) \]

   3. --lowering--.f64 53.9%

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, z\right)\right) \]

5. Simplified 53.9%

   \[\leadsto e^{\color{blue}{0 - z}} \]

6. Taylor expanded in z around 0

   \[\leadsto \color{blue}{1 + -1 \cdot z} \]
7. Step-by-step derivation

   1. neg-mul-1 N/A

      \[\leadsto 1 + \left(\mathsf{neg}\left(z\right)\right) \]

   2. unsub-neg N/A

      \[\leadsto 1 - \color{blue}{z} \]

   3. --lowering--.f64 17.2%

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{z}\right) \]

8. Simplified 17.2%

   \[\leadsto \color{blue}{1 - z} \]
9. Add Preprocessing

Alternative 20: 14.9% accurate, 69.0× speedup

Math

\[\begin{array}{l} \\ x + 1 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x 1.0))

C

double code(double x, double y, double z) {
	return x + 1.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + 1.0d0
end function

Java

public static double code(double x, double y, double z) {
	return x + 1.0;
}

Python

def code(x, y, z):
	return x + 1.0

Julia

function code(x, y, z)
	return Float64(x + 1.0)
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + 1.0;
end

Wolfram

code[x_, y_, z_] := N[(x + 1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{\left(x + y \cdot \log y\right) - z} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{exp.f64}\left(\color{blue}{x}\right) \]
4. Step-by-step derivation

5. Simplified 56.1%

   \[\leadsto e^{\color{blue}{x}} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + x} \]
7. Step-by-step derivation

   1. +-lowering-+.f64 16.8%

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{x}\right) \]

8. Simplified 16.8%

   \[\leadsto \color{blue}{1 + x} \]

9. Final simplification 16.8%

   \[\leadsto x + 1 \]
10. Add Preprocessing

Alternative 21: 14.6% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 1.0)

C

double code(double x, double y, double z) {
	return 1.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 1.0;
}

Python

def code(x, y, z):
	return 1.0

Julia

function code(x, y, z)
	return 1.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0;
end

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 100.0%

   \[e^{\left(x + y \cdot \log y\right) - z} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{exp.f64}\left(\color{blue}{x}\right) \]
4. Step-by-step derivation

5. Simplified 56.1%

   \[\leadsto e^{\color{blue}{x}} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 16.7%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(x - z\right) + \log y \cdot y} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (exp (+ (- x z) (* (log y) y))))

C

double code(double x, double y, double z) {
	return exp(((x - z) + (log(y) * y)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = exp(((x - z) + (log(y) * y)))
end function

Java

public static double code(double x, double y, double z) {
	return Math.exp(((x - z) + (Math.log(y) * y)));
}

Python

def code(x, y, z):
	return math.exp(((x - z) + (math.log(y) * y)))

Julia

function code(x, y, z)
	return exp(Float64(Float64(x - z) + Float64(log(y) * y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = exp(((x - z) + (log(y) * y)));
end

Wolfram

code[x_, y_, z_] := N[Exp[N[(N[(x - z), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Reproduce

herbie shell --seed 2024138 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson.Internal:probability from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (exp (+ (- x z) (* (log y) y))))

  (exp (- (+ x (* y (log y))) z)))