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

Percentage Accurate: 100.0% → 100.0%

Time: 9.3s

Alternatives: 18

Speedup: 1.0×

• 58.8% 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: 90.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \log y \leq 1000:\\ \;\;\;\;\frac{{y}^{y}}{e^{z - x}}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* y (log y)) 1000.0) (/ (pow y y) (exp (- z x))) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y * log(y)) <= 1000.0) {
		tmp = pow(y, y) / 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)) <= 1000.0d0) then
        tmp = (y ** y) / 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)) <= 1000.0) {
		tmp = Math.pow(y, y) / 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)) <= 1000.0:
		tmp = math.pow(y, y) / 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)) <= 1000.0)
		tmp = Float64((y ^ y) / 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)) <= 1000.0)
		tmp = (y ^ y) / 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], 1000.0], N[(N[Power[y, y], $MachinePrecision] / 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)) < 1e3

   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 100.0%

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

   3. Simplified 100.0%

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

   if 1e3 < (*.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 84.0%

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

   5. Simplified 84.0%

      \[\leadsto e^{\color{blue}{y \cdot \log y}} \]
   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 84.0%

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

   7. Applied egg-rr 84.0%

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

Alternative 3: 73.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+40}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-27}:\\ \;\;\;\;{y}^{y}\\ \mathbf{elif}\;x \leq 6.5:\\ \;\;\;\;e^{0 - z}\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.7e+40)
   (exp x)
   (if (<= x -1.55e-27) (pow y y) (if (<= x 6.5) (exp (- 0.0 z)) (exp x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.7e+40) {
		tmp = exp(x);
	} else if (x <= -1.55e-27) {
		tmp = pow(y, y);
	} else if (x <= 6.5) {
		tmp = exp((0.0 - z));
	} else {
		tmp = exp(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 <= (-1.7d+40)) then
        tmp = exp(x)
    else if (x <= (-1.55d-27)) then
        tmp = y ** y
    else if (x <= 6.5d0) then
        tmp = exp((0.0d0 - z))
    else
        tmp = exp(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.7e+40) {
		tmp = Math.exp(x);
	} else if (x <= -1.55e-27) {
		tmp = Math.pow(y, y);
	} else if (x <= 6.5) {
		tmp = Math.exp((0.0 - z));
	} else {
		tmp = Math.exp(x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.7e+40:
		tmp = math.exp(x)
	elif x <= -1.55e-27:
		tmp = math.pow(y, y)
	elif x <= 6.5:
		tmp = math.exp((0.0 - z))
	else:
		tmp = math.exp(x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.7e+40)
		tmp = exp(x);
	elseif (x <= -1.55e-27)
		tmp = y ^ y;
	elseif (x <= 6.5)
		tmp = exp(Float64(0.0 - z));
	else
		tmp = exp(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.7e+40)
		tmp = exp(x);
	elseif (x <= -1.55e-27)
		tmp = y ^ y;
	elseif (x <= 6.5)
		tmp = exp((0.0 - z));
	else
		tmp = exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.7e+40], N[Exp[x], $MachinePrecision], If[LessEqual[x, -1.55e-27], N[Power[y, y], $MachinePrecision], If[LessEqual[x, 6.5], N[Exp[N[(0.0 - z), $MachinePrecision]], $MachinePrecision], N[Exp[x], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.69999999999999994e40 or 6.5 < 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 85.6%

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

   if -1.69999999999999994e40 < x < -1.5499999999999999e-27

   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 77.1%

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

   5. Simplified 77.1%

      \[\leadsto e^{\color{blue}{y \cdot \log y}} \]
   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 77.1%

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

   7. Applied egg-rr 77.1%

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

   if -1.5499999999999999e-27 < x < 6.5

   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 76.9%

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

   5. Simplified 76.9%

      \[\leadsto e^{\color{blue}{0 - z}} \]
   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 76.9%

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

   7. Applied egg-rr 76.9%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+40}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-27}:\\ \;\;\;\;{y}^{y}\\ \mathbf{elif}\;x \leq 6.5:\\ \;\;\;\;e^{0 - z}\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{0 - z}\\ \mathbf{if}\;z \leq -1.08 \cdot 10^{-24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+23}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (exp (- 0.0 z))))
   (if (<= z -1.08e-24) t_0 (if (<= z 4.6e+23) (exp x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = exp((0.0 - z));
	double tmp;
	if (z <= -1.08e-24) {
		tmp = t_0;
	} else if (z <= 4.6e+23) {
		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 <= (-1.08d-24)) then
        tmp = t_0
    else if (z <= 4.6d+23) 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 <= -1.08e-24) {
		tmp = t_0;
	} else if (z <= 4.6e+23) {
		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 <= -1.08e-24:
		tmp = t_0
	elif z <= 4.6e+23:
		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 <= -1.08e-24)
		tmp = t_0;
	elseif (z <= 4.6e+23)
		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 <= -1.08e-24)
		tmp = t_0;
	elseif (z <= 4.6e+23)
		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, -1.08e-24], t$95$0, If[LessEqual[z, 4.6e+23], N[Exp[x], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.08000000000000006e-24 or 4.6000000000000001e23 < 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 86.5%

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

   5. Simplified 86.5%

      \[\leadsto e^{\color{blue}{0 - z}} \]
   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 86.5%

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

   7. Applied egg-rr 86.5%

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

   if -1.08000000000000006e-24 < z < 4.6000000000000001e23

   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 68.4%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.08 \cdot 10^{-24}:\\ \;\;\;\;e^{0 - z}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+23}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;e^{0 - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + z \cdot -0.16666666666666666\\ t_1 := z \cdot t\_0\\ \mathbf{if}\;z \leq -2 \cdot 10^{+77}:\\ \;\;\;\;1 + z \cdot \frac{1}{\frac{1}{-1 + 0.125 \cdot \left(z \cdot \left(z \cdot z\right)\right)}}\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{+46}:\\ \;\;\;\;1 + \frac{z \cdot \left(-1 + t\_1 \cdot \left(z \cdot \left(t\_0 \cdot t\_1\right)\right)\right)}{1 + t\_1 \cdot \left(1 + t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* z -0.16666666666666666))) (t_1 (* z t_0)))
   (if (<= z -2e+77)
     (+ 1.0 (* z (/ 1.0 (/ 1.0 (+ -1.0 (* 0.125 (* z (* z z))))))))
     (if (<= z -2.3e+46)
       (+
        1.0
        (/
         (* z (+ -1.0 (* t_1 (* z (* t_0 t_1)))))
         (+ 1.0 (* t_1 (+ 1.0 t_1)))))
       (exp x)))))

C

double code(double x, double y, double z) {
	double t_0 = 0.5 + (z * -0.16666666666666666);
	double t_1 = z * t_0;
	double tmp;
	if (z <= -2e+77) {
		tmp = 1.0 + (z * (1.0 / (1.0 / (-1.0 + (0.125 * (z * (z * z)))))));
	} else if (z <= -2.3e+46) {
		tmp = 1.0 + ((z * (-1.0 + (t_1 * (z * (t_0 * t_1))))) / (1.0 + (t_1 * (1.0 + t_1))));
	} else {
		tmp = exp(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) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 + (z * (-0.16666666666666666d0))
    t_1 = z * t_0
    if (z <= (-2d+77)) then
        tmp = 1.0d0 + (z * (1.0d0 / (1.0d0 / ((-1.0d0) + (0.125d0 * (z * (z * z)))))))
    else if (z <= (-2.3d+46)) then
        tmp = 1.0d0 + ((z * ((-1.0d0) + (t_1 * (z * (t_0 * t_1))))) / (1.0d0 + (t_1 * (1.0d0 + t_1))))
    else
        tmp = exp(x)
    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 * t_0;
	double tmp;
	if (z <= -2e+77) {
		tmp = 1.0 + (z * (1.0 / (1.0 / (-1.0 + (0.125 * (z * (z * z)))))));
	} else if (z <= -2.3e+46) {
		tmp = 1.0 + ((z * (-1.0 + (t_1 * (z * (t_0 * t_1))))) / (1.0 + (t_1 * (1.0 + t_1))));
	} else {
		tmp = Math.exp(x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 0.5 + (z * -0.16666666666666666)
	t_1 = z * t_0
	tmp = 0
	if z <= -2e+77:
		tmp = 1.0 + (z * (1.0 / (1.0 / (-1.0 + (0.125 * (z * (z * z)))))))
	elif z <= -2.3e+46:
		tmp = 1.0 + ((z * (-1.0 + (t_1 * (z * (t_0 * t_1))))) / (1.0 + (t_1 * (1.0 + t_1))))
	else:
		tmp = math.exp(x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(0.5 + Float64(z * -0.16666666666666666))
	t_1 = Float64(z * t_0)
	tmp = 0.0
	if (z <= -2e+77)
		tmp = Float64(1.0 + Float64(z * Float64(1.0 / Float64(1.0 / Float64(-1.0 + Float64(0.125 * Float64(z * Float64(z * z))))))));
	elseif (z <= -2.3e+46)
		tmp = Float64(1.0 + Float64(Float64(z * Float64(-1.0 + Float64(t_1 * Float64(z * Float64(t_0 * t_1))))) / Float64(1.0 + Float64(t_1 * Float64(1.0 + t_1)))));
	else
		tmp = exp(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 0.5 + (z * -0.16666666666666666);
	t_1 = z * t_0;
	tmp = 0.0;
	if (z <= -2e+77)
		tmp = 1.0 + (z * (1.0 / (1.0 / (-1.0 + (0.125 * (z * (z * z)))))));
	elseif (z <= -2.3e+46)
		tmp = 1.0 + ((z * (-1.0 + (t_1 * (z * (t_0 * t_1))))) / (1.0 + (t_1 * (1.0 + t_1))));
	else
		tmp = exp(x);
	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 * t$95$0), $MachinePrecision]}, If[LessEqual[z, -2e+77], N[(1.0 + N[(z * N[(1.0 / N[(1.0 / N[(-1.0 + N[(0.125 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -2.3e+46], N[(1.0 + N[(N[(z * N[(-1.0 + N[(t$95$1 * N[(z * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$1 * N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[x], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.99999999999999997e77

   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 93.3%

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

   5. Simplified 93.3%

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

   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 65.3%

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

   8. Simplified 65.3%

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

      1. +-commutative N/A

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

      2. flip3-+ N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. clear-num N/A

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

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

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

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

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

   10. Applied egg-rr 22.5%

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

   11. Taylor expanded in z around 0

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right)\right)\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 93.3%

       \[\leadsto 1 + z \cdot \frac{1}{\frac{\color{blue}{1}}{-1 + 0.125 \cdot \left(z \cdot \left(z \cdot z\right)\right)}} \]

   if -1.99999999999999997e77 < z < -2.3000000000000001e46

   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 100.0%

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

   5. Simplified 100.0%

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

   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 5.1%

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

   8. Simplified 5.1%

      \[\leadsto \color{blue}{1 + z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)} \]
   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. flip3-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   10. Applied egg-rr 100.0%

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

   if -2.3000000000000001e46 < z

   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 55.6%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+77}:\\ \;\;\;\;1 + z \cdot \frac{1}{\frac{1}{-1 + 0.125 \cdot \left(z \cdot \left(z \cdot z\right)\right)}}\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{+46}:\\ \;\;\;\;1 + \frac{z \cdot \left(-1 + \left(z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right) \cdot \left(z \cdot \left(\left(0.5 + z \cdot -0.16666666666666666\right) \cdot \left(z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)\right)\right)\right)}{1 + \left(z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right) \cdot \left(1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.9% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -800:\\ \;\;\;\;z \cdot \left(\left(z \cdot z\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;x \leq 7.1 \cdot 10^{+99}:\\ \;\;\;\;1 + z \cdot \frac{1}{\frac{1}{-1 + 0.125 \cdot \left(z \cdot \left(z \cdot z\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -800.0)
   (* z (* (* z z) -0.16666666666666666))
   (if (<= x 7.1e+99)
     (+ 1.0 (* z (/ 1.0 (/ 1.0 (+ -1.0 (* 0.125 (* z (* z z))))))))
     (+ 1.0 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666)))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -800.0) {
		tmp = z * ((z * z) * -0.16666666666666666);
	} else if (x <= 7.1e+99) {
		tmp = 1.0 + (z * (1.0 / (1.0 / (-1.0 + (0.125 * (z * (z * z)))))));
	} else {
		tmp = 1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))));
	}
	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 <= (-800.0d0)) then
        tmp = z * ((z * z) * (-0.16666666666666666d0))
    else if (x <= 7.1d+99) then
        tmp = 1.0d0 + (z * (1.0d0 / (1.0d0 / ((-1.0d0) + (0.125d0 * (z * (z * z)))))))
    else
        tmp = 1.0d0 + (x * (1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -800.0) {
		tmp = z * ((z * z) * -0.16666666666666666);
	} else if (x <= 7.1e+99) {
		tmp = 1.0 + (z * (1.0 / (1.0 / (-1.0 + (0.125 * (z * (z * z)))))));
	} else {
		tmp = 1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -800.0:
		tmp = z * ((z * z) * -0.16666666666666666)
	elif x <= 7.1e+99:
		tmp = 1.0 + (z * (1.0 / (1.0 / (-1.0 + (0.125 * (z * (z * z)))))))
	else:
		tmp = 1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -800.0)
		tmp = Float64(z * Float64(Float64(z * z) * -0.16666666666666666));
	elseif (x <= 7.1e+99)
		tmp = Float64(1.0 + Float64(z * Float64(1.0 / Float64(1.0 / Float64(-1.0 + Float64(0.125 * Float64(z * Float64(z * z))))))));
	else
		tmp = Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -800.0)
		tmp = z * ((z * z) * -0.16666666666666666);
	elseif (x <= 7.1e+99)
		tmp = 1.0 + (z * (1.0 / (1.0 / (-1.0 + (0.125 * (z * (z * z)))))));
	else
		tmp = 1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -800.0], N[(z * N[(N[(z * z), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.1e+99], N[(1.0 + N[(z * N[(1.0 / N[(1.0 / N[(-1.0 + N[(0.125 * N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -800

   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 34.1%

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

   5. Simplified 34.1%

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

   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 15.7%

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

   8. Simplified 15.7%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

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

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

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

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

      9. *-lowering-*.f64 15.7%

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

   10. Applied egg-rr 15.7%

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

   11. Taylor expanded in z around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

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

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 45.5%

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

   13. Simplified 45.5%

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

   if -800 < x < 7.09999999999999994e99

   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 73.2%

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

   5. Simplified 73.2%

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

   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 35.7%

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

   8. Simplified 35.7%

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

      1. +-commutative N/A

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

      2. flip3-+ N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. clear-num N/A

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

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

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

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

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

   10. Applied egg-rr 25.1%

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

   11. Taylor expanded in z around 0

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right)\right)\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 46.3%

       \[\leadsto 1 + z \cdot \frac{1}{\frac{\color{blue}{1}}{-1 + 0.125 \cdot \left(z \cdot \left(z \cdot z\right)\right)}} \]

   if 7.09999999999999994e99 < 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 88.0%

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

   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 83.7%

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

   8. Simplified 83.7%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -800:\\ \;\;\;\;z \cdot \left(\left(z \cdot z\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;x \leq 7.1 \cdot 10^{+99}:\\ \;\;\;\;1 + z \cdot \frac{1}{\frac{1}{-1 + 0.125 \cdot \left(z \cdot \left(z \cdot z\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.9% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -680:\\ \;\;\;\;z \cdot \left(\left(z \cdot z\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+99}:\\ \;\;\;\;1 + z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -680.0)
   (* z (* (* z z) -0.16666666666666666))
   (if (<= x 6e+99)
     (+ 1.0 (* z (+ -1.0 (* z (+ 0.5 (* z -0.16666666666666666))))))
     (+ 1.0 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666)))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -680.0) {
		tmp = z * ((z * z) * -0.16666666666666666);
	} else if (x <= 6e+99) {
		tmp = 1.0 + (z * (-1.0 + (z * (0.5 + (z * -0.16666666666666666)))));
	} else {
		tmp = 1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))));
	}
	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 <= (-680.0d0)) then
        tmp = z * ((z * z) * (-0.16666666666666666d0))
    else if (x <= 6d+99) then
        tmp = 1.0d0 + (z * ((-1.0d0) + (z * (0.5d0 + (z * (-0.16666666666666666d0))))))
    else
        tmp = 1.0d0 + (x * (1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -680.0) {
		tmp = z * ((z * z) * -0.16666666666666666);
	} else if (x <= 6e+99) {
		tmp = 1.0 + (z * (-1.0 + (z * (0.5 + (z * -0.16666666666666666)))));
	} else {
		tmp = 1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -680.0:
		tmp = z * ((z * z) * -0.16666666666666666)
	elif x <= 6e+99:
		tmp = 1.0 + (z * (-1.0 + (z * (0.5 + (z * -0.16666666666666666)))))
	else:
		tmp = 1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -680.0)
		tmp = Float64(z * Float64(Float64(z * z) * -0.16666666666666666));
	elseif (x <= 6e+99)
		tmp = Float64(1.0 + Float64(z * Float64(-1.0 + Float64(z * Float64(0.5 + Float64(z * -0.16666666666666666))))));
	else
		tmp = Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -680.0)
		tmp = z * ((z * z) * -0.16666666666666666);
	elseif (x <= 6e+99)
		tmp = 1.0 + (z * (-1.0 + (z * (0.5 + (z * -0.16666666666666666)))));
	else
		tmp = 1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -680.0], N[(z * N[(N[(z * z), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6e+99], N[(1.0 + N[(z * N[(-1.0 + N[(z * N[(0.5 + N[(z * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -680

   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 34.1%

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

   5. Simplified 34.1%

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

   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 15.7%

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

   8. Simplified 15.7%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

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

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

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

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

      9. *-lowering-*.f64 15.7%

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

   10. Applied egg-rr 15.7%

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

   11. Taylor expanded in z around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

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

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 45.5%

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

   13. Simplified 45.5%

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

   if -680 < x < 6.00000000000000029e99

   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 73.2%

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

   5. Simplified 73.2%

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

   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 42.1%

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

   8. Simplified 42.1%

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

   if 6.00000000000000029e99 < 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 88.0%

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

   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 83.7%

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

   8. Simplified 83.7%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -680:\\ \;\;\;\;z \cdot \left(\left(z \cdot z\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+99}:\\ \;\;\;\;1 + z \cdot \left(-1 + z \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.8% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -480:\\ \;\;\;\;z \cdot \left(\left(z \cdot z\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;x \leq 7.1 \cdot 10^{+99}:\\ \;\;\;\;1 + z \cdot \left(-1 + \frac{z}{\frac{-6}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -480.0)
   (* z (* (* z z) -0.16666666666666666))
   (if (<= x 7.1e+99)
     (+ 1.0 (* z (+ -1.0 (/ z (/ -6.0 z)))))
     (+ 1.0 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666)))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -480.0) {
		tmp = z * ((z * z) * -0.16666666666666666);
	} else if (x <= 7.1e+99) {
		tmp = 1.0 + (z * (-1.0 + (z / (-6.0 / z))));
	} else {
		tmp = 1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))));
	}
	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 <= (-480.0d0)) then
        tmp = z * ((z * z) * (-0.16666666666666666d0))
    else if (x <= 7.1d+99) then
        tmp = 1.0d0 + (z * ((-1.0d0) + (z / ((-6.0d0) / z))))
    else
        tmp = 1.0d0 + (x * (1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -480.0) {
		tmp = z * ((z * z) * -0.16666666666666666);
	} else if (x <= 7.1e+99) {
		tmp = 1.0 + (z * (-1.0 + (z / (-6.0 / z))));
	} else {
		tmp = 1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -480.0:
		tmp = z * ((z * z) * -0.16666666666666666)
	elif x <= 7.1e+99:
		tmp = 1.0 + (z * (-1.0 + (z / (-6.0 / z))))
	else:
		tmp = 1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -480.0)
		tmp = Float64(z * Float64(Float64(z * z) * -0.16666666666666666));
	elseif (x <= 7.1e+99)
		tmp = Float64(1.0 + Float64(z * Float64(-1.0 + Float64(z / Float64(-6.0 / z)))));
	else
		tmp = Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -480.0)
		tmp = z * ((z * z) * -0.16666666666666666);
	elseif (x <= 7.1e+99)
		tmp = 1.0 + (z * (-1.0 + (z / (-6.0 / z))));
	else
		tmp = 1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -480.0], N[(z * N[(N[(z * z), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.1e+99], N[(1.0 + N[(z * N[(-1.0 + N[(z / N[(-6.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -480

   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 34.1%

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

   5. Simplified 34.1%

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

   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 15.7%

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

   8. Simplified 15.7%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

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

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

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

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

      9. *-lowering-*.f64 15.7%

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

   10. Applied egg-rr 15.7%

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

   11. Taylor expanded in z around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

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

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 45.5%

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

   13. Simplified 45.5%

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

   if -480 < x < 7.09999999999999994e99

   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 73.2%

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

   5. Simplified 73.2%

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

   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 42.1%

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

   8. Simplified 42.1%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

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

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

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

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

      9. *-lowering-*.f64 42.1%

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

   10. Applied egg-rr 42.1%

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

   11. Taylor expanded in z around inf

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

       1. /-lowering-/.f64 41.9%

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

   13. Simplified 41.9%

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

   if 7.09999999999999994e99 < 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 88.0%

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

   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 83.7%

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

   8. Simplified 83.7%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -480:\\ \;\;\;\;z \cdot \left(\left(z \cdot z\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;x \leq 7.1 \cdot 10^{+99}:\\ \;\;\;\;1 + z \cdot \left(-1 + \frac{z}{\frac{-6}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.1% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -420:\\ \;\;\;\;z \cdot \left(\left(z \cdot z\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;1 + z \cdot \left(-1 + \frac{z}{\frac{-6}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -420.0)
   (* z (* (* z z) -0.16666666666666666))
   (if (<= x 1.32e+154)
     (+ 1.0 (* z (+ -1.0 (/ z (/ -6.0 z)))))
     (* 0.5 (* x x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -420.0) {
		tmp = z * ((z * z) * -0.16666666666666666);
	} else if (x <= 1.32e+154) {
		tmp = 1.0 + (z * (-1.0 + (z / (-6.0 / 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 <= (-420.0d0)) then
        tmp = z * ((z * z) * (-0.16666666666666666d0))
    else if (x <= 1.32d+154) then
        tmp = 1.0d0 + (z * ((-1.0d0) + (z / ((-6.0d0) / 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 <= -420.0) {
		tmp = z * ((z * z) * -0.16666666666666666);
	} else if (x <= 1.32e+154) {
		tmp = 1.0 + (z * (-1.0 + (z / (-6.0 / z))));
	} else {
		tmp = 0.5 * (x * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -420.0:
		tmp = z * ((z * z) * -0.16666666666666666)
	elif x <= 1.32e+154:
		tmp = 1.0 + (z * (-1.0 + (z / (-6.0 / z))))
	else:
		tmp = 0.5 * (x * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -420.0)
		tmp = Float64(z * Float64(Float64(z * z) * -0.16666666666666666));
	elseif (x <= 1.32e+154)
		tmp = Float64(1.0 + Float64(z * Float64(-1.0 + Float64(z / Float64(-6.0 / 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 <= -420.0)
		tmp = z * ((z * z) * -0.16666666666666666);
	elseif (x <= 1.32e+154)
		tmp = 1.0 + (z * (-1.0 + (z / (-6.0 / z))));
	else
		tmp = 0.5 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -420.0], N[(z * N[(N[(z * z), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.32e+154], N[(1.0 + N[(z * N[(-1.0 + N[(z / N[(-6.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -420

   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 34.1%

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

   5. Simplified 34.1%

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

   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 15.7%

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

   8. Simplified 15.7%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

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

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

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

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

      9. *-lowering-*.f64 15.7%

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

   10. Applied egg-rr 15.7%

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

   11. Taylor expanded in z around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

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

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 45.5%

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

   13. Simplified 45.5%

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

   if -420 < x < 1.31999999999999998e154

   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 69.9%

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

   5. Simplified 69.9%

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

   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 39.7%

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

   8. Simplified 39.7%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

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

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

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

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

      9. *-lowering-*.f64 39.7%

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

   10. Applied egg-rr 39.7%

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

   11. Taylor expanded in z around inf

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

       1. /-lowering-/.f64 39.5%

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

   13. Simplified 39.5%

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

   if 1.31999999999999998e154 < 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 91.4%

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

   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 91.4%

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

   8. Simplified 91.4%

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

   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 91.4%

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

   11. Simplified 91.4%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -420:\\ \;\;\;\;z \cdot \left(\left(z \cdot z\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;1 + z \cdot \left(-1 + \frac{z}{\frac{-6}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 44.1% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -10500:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-7}:\\ \;\;\;\;1 + x \cdot \left(1 + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -10500.0)
   (* (* z z) (+ 0.5 (* z -0.16666666666666666)))
   (if (<= z 2.2e-7) (+ 1.0 (* x (+ 1.0 (* x 0.5)))) (* 0.5 (* x x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -10500.0) {
		tmp = (z * z) * (0.5 + (z * -0.16666666666666666));
	} else if (z <= 2.2e-7) {
		tmp = 1.0 + (x * (1.0 + (x * 0.5)));
	} 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 (z <= (-10500.0d0)) then
        tmp = (z * z) * (0.5d0 + (z * (-0.16666666666666666d0)))
    else if (z <= 2.2d-7) then
        tmp = 1.0d0 + (x * (1.0d0 + (x * 0.5d0)))
    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 (z <= -10500.0) {
		tmp = (z * z) * (0.5 + (z * -0.16666666666666666));
	} else if (z <= 2.2e-7) {
		tmp = 1.0 + (x * (1.0 + (x * 0.5)));
	} else {
		tmp = 0.5 * (x * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -10500.0:
		tmp = (z * z) * (0.5 + (z * -0.16666666666666666))
	elif z <= 2.2e-7:
		tmp = 1.0 + (x * (1.0 + (x * 0.5)))
	else:
		tmp = 0.5 * (x * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -10500.0)
		tmp = Float64(Float64(z * z) * Float64(0.5 + Float64(z * -0.16666666666666666)));
	elseif (z <= 2.2e-7)
		tmp = Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * 0.5))));
	else
		tmp = Float64(0.5 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -10500.0)
		tmp = (z * z) * (0.5 + (z * -0.16666666666666666));
	elseif (z <= 2.2e-7)
		tmp = 1.0 + (x * (1.0 + (x * 0.5)));
	else
		tmp = 0.5 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -10500.0], N[(N[(z * z), $MachinePrecision] * N[(0.5 + N[(z * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.2e-7], N[(1.0 + N[(x * N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -10500

   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 93.4%

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

   5. Simplified 93.4%

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

   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 69.5%

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

   8. Simplified 69.5%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{z} - \frac{1}{6}\right)} \]
   10. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto {z}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{z} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right) \]

       2. metadata-eval N/A

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

       3. distribute-rgt-in N/A

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

       4. unpow3 N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. metadata-eval N/A

          \[\leadsto \left(\left(\frac{1}{2} \cdot \frac{1}{z}\right) \cdot {z}^{2}\right) \cdot z + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot {\color{blue}{z}}^{3} \]

       8. distribute-lft-neg-in N/A

          \[\leadsto \left(\left(\frac{1}{2} \cdot \frac{1}{z}\right) \cdot {z}^{2}\right) \cdot z + \left(\mathsf{neg}\left(\frac{1}{6} \cdot {z}^{3}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} \cdot \frac{1}{z}\right) \cdot {z}^{2}\right) \cdot z + \left(\mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) \]

       10. fma-define N/A

           \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} \cdot \frac{1}{z}\right) \cdot {z}^{2}, \color{blue}{z}, \mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \left(\frac{1}{z} \cdot {z}^{2}\right), z, \mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) \]

       12. associate-*l/ N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \frac{1 \cdot {z}^{2}}{z}, z, \mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) \]

       13. *-lft-identity N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \frac{{z}^{2}}{z}, z, \mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \frac{z \cdot z}{z}, z, \mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) \]

       15. associate-/l* N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \left(z \cdot \frac{z}{z}\right), z, \mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) \]

       16. *-rgt-identity N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \left(z \cdot \frac{z \cdot 1}{z}\right), z, \mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) \]

       17. associate-*r/ N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \left(z \cdot \left(z \cdot \frac{1}{z}\right)\right), z, \mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) \]

       18. rgt-mult-inverse N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \left(z \cdot 1\right), z, \mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) \]

       19. *-rgt-identity N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot z, z, \mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) \]

       20. fma-define N/A

           \[\leadsto \left(\frac{1}{2} \cdot z\right) \cdot z + \color{blue}{\left(\mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right)} \]

   11. Simplified 69.5%

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

   if -10500 < z < 2.2000000000000001e-7

   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 69.4%

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

   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 33.3%

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

   8. Simplified 33.3%

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

   if 2.2000000000000001e-7 < z

   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 35.2%

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

   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 16.9%

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

   8. Simplified 16.9%

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

   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 36.3%

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

   11. Simplified 36.3%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -10500:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-7}:\\ \;\;\;\;1 + x \cdot \left(1 + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 44.0% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -24:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;z \leq 370:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -24.0)
   (* (* z z) (+ 0.5 (* z -0.16666666666666666)))
   (if (<= z 370.0) (+ 1.0 (* x (* x 0.5))) (* 0.5 (* x x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -24.0) {
		tmp = (z * z) * (0.5 + (z * -0.16666666666666666));
	} else if (z <= 370.0) {
		tmp = 1.0 + (x * (x * 0.5));
	} 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 (z <= (-24.0d0)) then
        tmp = (z * z) * (0.5d0 + (z * (-0.16666666666666666d0)))
    else if (z <= 370.0d0) then
        tmp = 1.0d0 + (x * (x * 0.5d0))
    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 (z <= -24.0) {
		tmp = (z * z) * (0.5 + (z * -0.16666666666666666));
	} else if (z <= 370.0) {
		tmp = 1.0 + (x * (x * 0.5));
	} else {
		tmp = 0.5 * (x * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -24.0:
		tmp = (z * z) * (0.5 + (z * -0.16666666666666666))
	elif z <= 370.0:
		tmp = 1.0 + (x * (x * 0.5))
	else:
		tmp = 0.5 * (x * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -24.0)
		tmp = Float64(Float64(z * z) * Float64(0.5 + Float64(z * -0.16666666666666666)));
	elseif (z <= 370.0)
		tmp = Float64(1.0 + Float64(x * Float64(x * 0.5)));
	else
		tmp = Float64(0.5 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -24.0)
		tmp = (z * z) * (0.5 + (z * -0.16666666666666666));
	elseif (z <= 370.0)
		tmp = 1.0 + (x * (x * 0.5));
	else
		tmp = 0.5 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -24.0], N[(N[(z * z), $MachinePrecision] * N[(0.5 + N[(z * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 370.0], N[(1.0 + N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -24

   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 93.4%

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

   5. Simplified 93.4%

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

   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 69.5%

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

   8. Simplified 69.5%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{z} - \frac{1}{6}\right)} \]
   10. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto {z}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{z} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right) \]

       2. metadata-eval N/A

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

       3. distribute-rgt-in N/A

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

       4. unpow3 N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. metadata-eval N/A

          \[\leadsto \left(\left(\frac{1}{2} \cdot \frac{1}{z}\right) \cdot {z}^{2}\right) \cdot z + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot {\color{blue}{z}}^{3} \]

       8. distribute-lft-neg-in N/A

          \[\leadsto \left(\left(\frac{1}{2} \cdot \frac{1}{z}\right) \cdot {z}^{2}\right) \cdot z + \left(\mathsf{neg}\left(\frac{1}{6} \cdot {z}^{3}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} \cdot \frac{1}{z}\right) \cdot {z}^{2}\right) \cdot z + \left(\mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) \]

       10. fma-define N/A

           \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} \cdot \frac{1}{z}\right) \cdot {z}^{2}, \color{blue}{z}, \mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \left(\frac{1}{z} \cdot {z}^{2}\right), z, \mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) \]

       12. associate-*l/ N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \frac{1 \cdot {z}^{2}}{z}, z, \mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) \]

       13. *-lft-identity N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \frac{{z}^{2}}{z}, z, \mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \frac{z \cdot z}{z}, z, \mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) \]

       15. associate-/l* N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \left(z \cdot \frac{z}{z}\right), z, \mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) \]

       16. *-rgt-identity N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \left(z \cdot \frac{z \cdot 1}{z}\right), z, \mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) \]

       17. associate-*r/ N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \left(z \cdot \left(z \cdot \frac{1}{z}\right)\right), z, \mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) \]

       18. rgt-mult-inverse N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \left(z \cdot 1\right), z, \mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) \]

       19. *-rgt-identity N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot z, z, \mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right) \]

       20. fma-define N/A

           \[\leadsto \left(\frac{1}{2} \cdot z\right) \cdot z + \color{blue}{\left(\mathsf{neg}\left({z}^{3} \cdot \frac{1}{6}\right)\right)} \]

   11. Simplified 69.5%

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

   if -24 < z < 370

   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 69.3%

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

   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 33.3%

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

   8. Simplified 33.3%

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

   9. Taylor expanded in x around inf

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

       1. *-lowering-*.f64 33.1%

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

   11. Simplified 33.1%

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

   if 370 < z

   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 33.7%

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

   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 16.1%

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

   8. Simplified 16.1%

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

   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 36.3%

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

   11. Simplified 36.3%

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

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -24:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(0.5 + z \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;z \leq 370:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.1% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4500000:\\ \;\;\;\;z \cdot \left(\left(z \cdot z\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;z \leq 390:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4500000.0)
   (* z (* (* z z) -0.16666666666666666))
   (if (<= z 390.0) (+ 1.0 (* x (* x 0.5))) (* 0.5 (* x x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4500000.0) {
		tmp = z * ((z * z) * -0.16666666666666666);
	} else if (z <= 390.0) {
		tmp = 1.0 + (x * (x * 0.5));
	} 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 (z <= (-4500000.0d0)) then
        tmp = z * ((z * z) * (-0.16666666666666666d0))
    else if (z <= 390.0d0) then
        tmp = 1.0d0 + (x * (x * 0.5d0))
    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 (z <= -4500000.0) {
		tmp = z * ((z * z) * -0.16666666666666666);
	} else if (z <= 390.0) {
		tmp = 1.0 + (x * (x * 0.5));
	} else {
		tmp = 0.5 * (x * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4500000.0:
		tmp = z * ((z * z) * -0.16666666666666666)
	elif z <= 390.0:
		tmp = 1.0 + (x * (x * 0.5))
	else:
		tmp = 0.5 * (x * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4500000.0)
		tmp = Float64(z * Float64(Float64(z * z) * -0.16666666666666666));
	elseif (z <= 390.0)
		tmp = Float64(1.0 + Float64(x * Float64(x * 0.5)));
	else
		tmp = Float64(0.5 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4500000.0)
		tmp = z * ((z * z) * -0.16666666666666666);
	elseif (z <= 390.0)
		tmp = 1.0 + (x * (x * 0.5));
	else
		tmp = 0.5 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4500000.0], N[(z * N[(N[(z * z), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 390.0], N[(1.0 + N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.5e6

   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 93.4%

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

   5. Simplified 93.4%

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

   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 69.5%

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

   8. Simplified 69.5%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

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

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

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

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

      9. *-lowering-*.f64 69.5%

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

   10. Applied egg-rr 69.5%

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

   11. Taylor expanded in z around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

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

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 69.5%

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

   13. Simplified 69.5%

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

   if -4.5e6 < z < 390

   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 69.3%

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

   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 33.3%

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

   8. Simplified 33.3%

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

   9. Taylor expanded in x around inf

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

       1. *-lowering-*.f64 33.1%

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

   11. Simplified 33.1%

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

   if 390 < z

   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 33.7%

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

   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 16.1%

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

   8. Simplified 16.1%

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

   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 36.3%

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

   11. Simplified 36.3%

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

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4500000:\\ \;\;\;\;z \cdot \left(\left(z \cdot z\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;z \leq 390:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 36.7% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1260000:\\ \;\;\;\;z \cdot \left(\left(z \cdot z\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-90}:\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1260000.0)
   (* z (* (* z z) -0.16666666666666666))
   (if (<= z 4.5e-90) (+ x 1.0) (* 0.5 (* x x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1260000.0) {
		tmp = z * ((z * z) * -0.16666666666666666);
	} else if (z <= 4.5e-90) {
		tmp = x + 1.0;
	} 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) :: tmp
    if (z <= (-1260000.0d0)) then
        tmp = z * ((z * z) * (-0.16666666666666666d0))
    else if (z <= 4.5d-90) then
        tmp = x + 1.0d0
    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 (z <= -1260000.0) {
		tmp = z * ((z * z) * -0.16666666666666666);
	} else if (z <= 4.5e-90) {
		tmp = x + 1.0;
	} else {
		tmp = 0.5 * (x * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1260000.0:
		tmp = z * ((z * z) * -0.16666666666666666)
	elif z <= 4.5e-90:
		tmp = x + 1.0
	else:
		tmp = 0.5 * (x * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1260000.0)
		tmp = Float64(z * Float64(Float64(z * z) * -0.16666666666666666));
	elseif (z <= 4.5e-90)
		tmp = Float64(x + 1.0);
	else
		tmp = Float64(0.5 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1260000.0)
		tmp = z * ((z * z) * -0.16666666666666666);
	elseif (z <= 4.5e-90)
		tmp = x + 1.0;
	else
		tmp = 0.5 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1260000.0], N[(z * N[(N[(z * z), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.5e-90], N[(x + 1.0), $MachinePrecision], N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.26e6

   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 93.4%

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

   5. Simplified 93.4%

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

   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 69.5%

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

   8. Simplified 69.5%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

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

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

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

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

      9. *-lowering-*.f64 69.5%

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

   10. Applied egg-rr 69.5%

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

   11. Taylor expanded in z around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

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

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 69.5%

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

   13. Simplified 69.5%

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

   if -1.26e6 < z < 4.50000000000000009e-90

   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 70.6%

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

   6. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 27.0%

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

   8. Simplified 27.0%

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

   if 4.50000000000000009e-90 < z

   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 39.8%

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

   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 15.7%

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

   8. Simplified 15.7%

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

   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 31.6%

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

   11. Simplified 31.6%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1260000:\\ \;\;\;\;z \cdot \left(\left(z \cdot z\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-90}:\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 33.6% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -250:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-90}:\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -250.0)
   (* (* z z) 0.5)
   (if (<= z 4.4e-90) (+ x 1.0) (* 0.5 (* x x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -250.0) {
		tmp = (z * z) * 0.5;
	} else if (z <= 4.4e-90) {
		tmp = x + 1.0;
	} 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) :: tmp
    if (z <= (-250.0d0)) then
        tmp = (z * z) * 0.5d0
    else if (z <= 4.4d-90) then
        tmp = x + 1.0d0
    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 (z <= -250.0) {
		tmp = (z * z) * 0.5;
	} else if (z <= 4.4e-90) {
		tmp = x + 1.0;
	} else {
		tmp = 0.5 * (x * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -250.0:
		tmp = (z * z) * 0.5
	elif z <= 4.4e-90:
		tmp = x + 1.0
	else:
		tmp = 0.5 * (x * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -250.0)
		tmp = Float64(Float64(z * z) * 0.5);
	elseif (z <= 4.4e-90)
		tmp = Float64(x + 1.0);
	else
		tmp = Float64(0.5 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -250.0)
		tmp = (z * z) * 0.5;
	elseif (z <= 4.4e-90)
		tmp = x + 1.0;
	else
		tmp = 0.5 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -250.0], N[(N[(z * z), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[z, 4.4e-90], N[(x + 1.0), $MachinePrecision], N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -250

   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 93.4%

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

   5. Simplified 93.4%

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

   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 52.2%

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

   8. Simplified 52.2%

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

   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 53.2%

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

   11. Simplified 53.2%

       \[\leadsto \color{blue}{0.5 \cdot \left(z \cdot z\right)} \]

   if -250 < z < 4.39999999999999972e-90

   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 70.6%

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

   6. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 27.0%

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

   8. Simplified 27.0%

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

   if 4.39999999999999972e-90 < z

   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 39.8%

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

   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 15.7%

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

   8. Simplified 15.7%

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

   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 31.6%

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

   11. Simplified 31.6%

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

4. Final simplification 36.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -250:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-90}:\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 24.9% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;z \leq -1.5 \cdot 10^{-92}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-90}:\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 0.5 (* x x))))
   (if (<= z -1.5e-92) t_0 (if (<= z 4.5e-90) (+ x 1.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = 0.5 * (x * x);
	double tmp;
	if (z <= -1.5e-92) {
		tmp = t_0;
	} else if (z <= 4.5e-90) {
		tmp = x + 1.0;
	} 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 (z <= (-1.5d-92)) then
        tmp = t_0
    else if (z <= 4.5d-90) then
        tmp = x + 1.0d0
    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 (z <= -1.5e-92) {
		tmp = t_0;
	} else if (z <= 4.5e-90) {
		tmp = x + 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 0.5 * (x * x)
	tmp = 0
	if z <= -1.5e-92:
		tmp = t_0
	elif z <= 4.5e-90:
		tmp = x + 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(0.5 * Float64(x * x))
	tmp = 0.0
	if (z <= -1.5e-92)
		tmp = t_0;
	elseif (z <= 4.5e-90)
		tmp = Float64(x + 1.0);
	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 (z <= -1.5e-92)
		tmp = t_0;
	elseif (z <= 4.5e-90)
		tmp = x + 1.0;
	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[z, -1.5e-92], t$95$0, If[LessEqual[z, 4.5e-90], N[(x + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.50000000000000007e-92 or 4.50000000000000009e-90 < z

   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 36.7%

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

   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 15.9%

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

   8. Simplified 15.9%

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

   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 21.7%

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

   11. Simplified 21.7%

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

   if -1.50000000000000007e-92 < z < 4.50000000000000009e-90

   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 71.6%

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

   6. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 28.1%

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

   8. Simplified 28.1%

      \[\leadsto \color{blue}{1 + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-92}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-90}:\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 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 56.7%

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

5. Simplified 56.7%

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

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 12.4%

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{z}\right) \]

8. Simplified 12.4%

   \[\leadsto \color{blue}{1 - z} \]
9. Add Preprocessing

Alternative 17: 14.7% 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 47.9%

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

6. Taylor expanded in x around 0

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

   1. +-lowering-+.f64 12.2%

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

8. Simplified 12.2%

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

9. Final simplification 12.2%

   \[\leadsto x + 1 \]
10. Add Preprocessing

Alternative 18: 14.4% 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 47.9%

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

6. Taylor expanded in x around 0

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

8. Simplified 12.0%

   \[\leadsto \color{blue}{1} \]
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 2024161 
(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)))